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Machines of life: catalogue, stochastic process modeling,

probabilistic reverse engineering and the PIs- from Aristotle to Alberts

Debashish Chowdhury1, 2, ∗

1Department of Physics, Indian Institute of Technology, Kanpur 208016, India2Mathematical Biosciences Institute, The Ohio State University, Columbus, OH 43210, USA.

(Dated: July 2, 2018)

Molecular machines consist of either a single protein or a macromolecular complex composed ofprotein and RNA molecules. Just like their macroscopic counterparts, each of these nano-machineshas an engine that “transduces” input energy into an output form which is then utilized by its cou-pling to a transmission system for appropriate operations. The theory of heat engines, pioneered byCarnot, rests on the second law of equilibrium thermodynamics. However, the engines of molecularmachines, operate under isothermal conditions far from thermodynamic equilibrium. Moreover, oneof the possible mechanisms of energy transduction, popularized by Feynman and called Brownianratchet, does not even have any macroscopic counterpart. But, molecular machine is not synony-mous with Brownian ratchet; a large number of molecular machines actually execute a noisy powerstroke, rather than operating as Brownian ratchet. The man-machine analogy, a topic of intensephilosophical debate in which many leading philosophers like Aristotle and Descartes participated,was extended to similar analogies at the cellular and subcellular levels after the invention of opti-cal microscope. The idea of molecular machine, pioneered by Marcelo Malpighi, has been pursuedvigorously in the last fifty years. It has become a well established topic of current interdisciplinaryresearch as evident from the publication of a very influential paper by Alberts towards the end ofthe twentieth century. Here we give a non-technical overview of the strategies for (a) stochasticmodeling of mechano-chemical kinetic processes, and (b) model selection based on statistical in-ference drawn from analysis of experimental data. It is written for non-experts and from a broadperspective, showing overlapping concepts from several different branches of physics and from otherareas of science and technology.

PACS numbers: 87.16.Ac 89.20.-a

∗Electronic address: debch@iitk.ac.in

Table of contents:

1. Introduction

II. A catalogue: typical examples of molecular machines and fuels

II.A. Fuels for molecular machines

II.B. Cytoskeletal motors

II.B.1. Porters

II.B.2. Machines for chipping filamentous tracks

II.B.3. Contractility: motor-filament crossbridge and

collective dynamics of sliders and rowers

II.B.4. Push and pull of cytoskeletal filaments: nano-pistons and nano-springs

II.C. Machines for synthesis, manipulation and degradation of macromolecules of life

II.C.1. Membrane-associated machines for macromolecule

translocation: exporters, importers and packers

II.C.2. Machines for degrading macromolecules of life

II.C.3. Machines for template-dictated polymerization

II.C.4. Unwrappers and unzippers of packaged DNA: chromatin remodellers and helicases

II.D. Rotary motors

II.E. Processivity, duty ratio, stall force and dwell time

II.F. Fundamental general questions

III. Motoring in a viscous fluid: from Newton to Langevin

III.A. Newton’s equation: deterministic dynamics

III.B. Langevin equation: stochastic dynamics

III.B.1. Motoring in a ‘‘sticky’’ medium: Purcell’s idea

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III.B.2. Motoring under random impacts from surroundings: Brownian force

IV. Energy transduction by molecular machines: from Carnot to Feynman

IV.A. Molecular machines are run by isothermal engines

IV.B. Inadequacy of equilibrium thermodynamics

IV.C. Inadequacy of endo-reversible thermodynamics

IV.D. Domain of non-equilibrium statistical mechanics

IV.E. Defining efficiency: from Carnot to Stokes

IV.F. Noisy Power stroke and Brownian ratchet

IV.G. Physical realizations of Brownian ratchet in molecular motors

V. Mathematical description of mechano-chemical kinetics:

continuous landscape versus discrete networks

V.A. Motor kinetics as wandering in a time-dependent mechano-chemical free-energy landscape

V.B. Motor kinetics as wandering in the time-dependent real-space potential landscape

V.C. Motor kinetics as a jump process in a fully discrete mechano-chemical network

V.D. Balance conditions for mechano-chemical kinetics: cycles and flux

VI. Solving forward problem by stochastic process modeling: from model to data

VI.A. Average speed and load-velocity relation

VI.B. Jamming on a crowded track: flux-density relation

VI.C. Information processing machines: fidelity versus power and efficiency

VI.D. Beyond average: dwell time distribution

VI.D.1. DTD for a motor that never steps backward

VI.D.2. Conditional DTD for motors with both forward and backward stepping

VII. A summary of experimental techniques: ensemble versus single-machine

VIII. Solving inverse problem by probabilistic reverse engineering: from data to model

VIII.A. Frequentist versus Bayesian approach

VIII.A.1. Maximum-likelihood estimate

VIII.A.2. Bayesian estimate

VIII.A.3. Hidden Markov models

VIII.B. Extracting FP-based models from data?

IX. Overlapping research areas

IX.A. Symmetry braeking: directed motility and cell polarity

IX.B. Self-organization and pattern formation: assembling

machines and cellular morphogenesis

IX.C. Dissipationless computation: polymerases as ‘‘tape-copying Turing machines’’

IX.D. Enzymatic processes: conformational fluctuations, static and dynamic disorder

IX.E. Applications in biomimetics and nano-technology

X. Concept of biological machines: from Aristotle to Alberts

XI. Summary and outlook

I. INTRODUCTION

Cell is the structural and functional units of life. How“active” is the interior of a living cell? Imagine an un-der water “metro city” which is, however, only about10µm long in each direction! In this city, there are “high-ways” and “railroad” tracks on which motorized “ve-hicles” transport cargo to various destinations. It hasan elaborate mechanism of preserving the integrity ofthe chemically encoded blueprint of the construction andmaintenance of the city. The “factories” not only sup-ply their products for the construction and repair works,but also manufacture the components of the machines.This eco-friendly city re-charges spent “chemical fuel” inuniquely designed “power plants”. This city also usesa few “alternative energy” sources in some operations.Finally, it has special “waste-disposal plants” which de-

grade waste into products that are recycled as raw mate-rials for fresh synthesis. All the automated processes inthis high-tech micro-city are run by nano-machines [1].This is not the plot of a science fiction, but a dramatizedpicture of the dynamic interior of a cell.

A molecular machine is either a single protein or amacromolecular complex consisting of proteins and RNAmolecules [2]. Just like their macroscopic counterparts,these nano-machines take an input (most often, chemicalenergy) and it transduces input energy into an output. Ifthe output is mechanical work the machine is usually re-ferred to as a molecular motor [3–5]. Similarly, a cell canalso be regarded as a micron-size “energy transformingdevice”. The concept of “machine” is not restricted onlyto subcellular or cellular levels of biological organization.In ancient times, philosophers proposed the concept of“living machine” to describe a whole animal, including a

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man; we’ll discuss the evolution of this concept towardsthe end of this article.

Not only animals, but even plants also move in re-sponse to external stimuli [6, 7] and movements takeplace in plants at all levels- from whole plant and plantcell to the subcellular level. I cannot resist the tempta-tion of quoting Thomas Huxley’s poetic description of cy-toplasmic streaming [8], in particular, and of intracellularmotor traffic, in general, in plants [9]: “..the wonderful

noonday silence of a tropical forest is, after all, due only

to the dulness of our hearing; and could our ears catch

the murmur of these tiny maelstroms, as they whirl in the

innumerable myriads of living cells which constitute each

tree, we should be stunned, as with the roar of a great

city”.

In this article, however, we focus attention almost ex-clusively on molecular machines that drive subcellularprocesses within living cells. The processes driven bymolecular machines include not only intracellular mo-tor transport, but also manipulation, polymerization anddegradation of the bio-molecules [10, 11]. Explaining thephysical principles that govern these machine-driven pro-cesses will bring us closer to the ultimate goal of biologi-cal physics: understanding “what is life” in terms of thelaws of physics (and chemistry).

Biomolecular machines operate in a domain where theappropriate units of length, time, force and energy are,nano-meter, milli-second, pico-Newton and kBT , respec-tively (kB being the Boltzmann constant and T is the ab-solute temperature). Aren’t the operational mechanismof molecular machines similar to their macroscopic coun-terparts except, perhaps, the difference of scale? NO. Inspite of the striking similarities, it is the differences be-tween molecular machines and their macroscopic coun-terparts that makes the studies of these systems so inter-esting from the perspective of physicists.

This article is a brief guide for a beginner embark-ing on an exploration of the exciting frontier of researchon molecular machines. It begins with a catalog ofthe known molecular machines and a list of fundamen-tal questions on their operational mechanism that theexplorer should try to address. Like any travel guide,this article also cautions the explorer about the counter-intuitive phenomena that await him/her in this new ter-ritory of his/her exploration where he/she may need newsophisticated technical tools that were not needed forhandling macroscopic machines. For the explorer, thisarticle also charts a tentative road map along with a sum-mary of the mathematical strategies that he/she mightuse to make progress in this frontier territory. Excursionsto some of the listed border areas that overlap with areasof research in other branches of science and engineeringcould be enjoyable and intellectually rewarding for theexplorer. For any explorer, it is good to know the experi-ence of the pioneers. The final section, on the evolutionof the concept of biological machines, is likely to be en-joyed as a dessert by physicists, and as a fresh food forthought by historians and philosophers of science.

II. A CATALOGUE: TYPICAL EXAMPLES OFMOLECULAR MACHINES AND FUELS

Based on the mode of operation, the biomolecular ma-chines can be divided broadly into two groups. Cyclic

machines operate in repetitive cycles in a manner thatis very similar to that of the cyclic engines which runour cars. In contrast, some other molecular machines areone-shot machines that exhaust an internal source of freeenergy in a single round. Force exerted by a compressedspring, upon its release, is a typical example of a one-shotmachine.

A. Fuels for molecular machines

The most common way of supplying energy to a nat-ural nanomachine is to utilize the chemical energy (or,more appropriately, free energy) released by a chemi-cal reaction. Most of the machines use the so-calledhigh-energy compounds- particularly, nucleoside triphos-phates (NTPs)- as an energy source to generate the me-chanical energy required for their directed movement.The most common chemical reaction is the hydrolysis ofATP (ADP): ATP → ADP+Pi. Some other high-energycompounds can also supply input energy; one typical ex-ample being the hydrolysis of Guanosine Triphosphate(GTP) to Guanosine Diphosphate (GDP). Under normalphysiological conditions, hydrolysis of ATP is extremelyslow. Most of the molecular motors fuelled by ATP func-tion also as ATPase enzyme (i.e., catalyzes hydrolysis ofATP) thereby speeding up the energy-supplying reactionby several orders of magnitude.If a cyclic machine runs on a specific chemical fuel then

the spent fuel must be removed as waste products andfresh fuel must be supplied to the machine. Fortunately,normal cells have machineries for recycling waste prod-ucts to manufacture fresh fuel, e.g., synthesizing ATPfrom ADP. This raises an important question: since ATPis a higher-energy compound than ADP, how are theATP-synthesizing machines driven to perform this ener-getically “uphill” task? Fortunately, chemical fuel is notthe only means by which input energy can be supplied tointracellular molecular machines.A cell gets its energy from external sources. It has

special machines to convert the input energy into some“energy currency”. For example, chemical energy sup-plied by the food we consume is converted into an electro-chemical potential ∆µ that not only can be used to syn-thesize ATP, but can also directly run some other ma-chines. In plants similar proton-motive forces are gen-erated by machines which are driven by the input sun-light. Only hydrogen ion (H+), i.e., a proton, and sodiumion (Na+) are used in the kingdom of life to create theelectro-chemical potentials, i.e., it used either a proton-motive force (PMF) or a sodium-motive force (SMF) asan energy currency.The advantages of using light, instead of chemical re-

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Motor superfamily Filamentous track Minimum step sizeMyosin F-actin 36 nmKinesin Microtubule 8 nmDynein Microtubule 8 nm

TABLE I: Superfamilies of motor proteins and the corre-sponding tracks.

action, as the input energy for a molecular motor are asfollows: (i) light can be switched on and off easily andrapidly, (ii) usually, no waste product, which would re-quire disposal or recycling, is generated.Thus, study of molecular machines deals with two com-

plementary aspects of bioenergetics: (a) conversion ofenergy input from the external sources into the energycurrency of the cell, and (b) conversion of the energycurrency to drive various active other active processes.

B. Cytoskeletal motors

The cytoskeleton of a cell is the analogue of the hu-man skeleton [3]. However, it not only provides mechan-ical strength to the cell, but its filamentous proteins alsoform the networks of “highways” (or, “tracks”) on whichcytoskeletal motor proteins [3, 4] can move. Filamen-tous actin (F-actin) and microtubules (MT) which serveas tracks are “polar” in the sense that the structure andkinetics of the two ends of each filament are dissimilar.The superfamilies of cytoskeletal motors and the cor-

responding filamentous tracks are listed in table I. Everysuperfamily can be further divided into families. Mem-bers of every family move always in a particular direc-tion on its track; for example, kinesin-1 and cytoplasmicdynein move towards + and - end of MT, respectively.Similarly, myosin-V and myosin-VI move towards the +and - ends of F-actin, respectively.For their operation, each motor must have a track-

binding site and another site that binds and hydrolyzesATP (so-called ATPase site). Both these sites are lo-cated, for example, in the head domain of myosins andkinesins both of which walk on their heads! The motor-binding sites on the tracks are equispaced; the actual stepsize of a motor can be, in principle, an integral multipleof the minimum step size which is the separation betweentwo neighboring motor-binding sites on the correspond-ing track.

1. Porters

Some linear motors are cargo transporters [12]; how-ever, the size of the cargoes are usually much larger than

the motor itself! It is desirable that such a motor “walks”for a significant distance on its track carrying the cargo;for obvious reasons, such motors are referred to as porters[13]. Kinesin and dyneins attached simultaneously to thesame “hard” cargo can get engaged in a tug-of-war lead-ing to a bidirectional movement of the cargo [14, 15].In regions of overlap between MT and F-actin filaments,a large cargo may be hauled simultaneously by kinesinsand myosins which, however, walk on their respectivetracks. Alternatively, along its journey route, a cargomay be transferred from the MT-based transport net-work, which dominate at the cell center, to the F-actinbased network that covers most of the cell periphery [16].A “soft” cargo pulled by many kinesins can get elongatedinto a tube [17].

2. Machines for chipping filamentous tracks

A MT depolymerase is a kinesin motor that chips awayits own track from one end [18]. Members of the kinesin-13 family can reach either end of the MT diffusively(without ATP hydrolysis) and, then, start chipping thetrack from the end where it reaches. In contrast, mem-bers of the kinesin-8 family walk towards the plus end ofthe MT track hydrolyzing ATP and after reaching thatend starts chipping it from there [19, 20]. Chipping byboth families of depolymerase kinesins are energized byATP hydrolysis.

3. Contractility: motor-filament crossbridge and collectivedynamics of sliders and rowers

Some motors are capable of sliding two different fila-ments with respect to each other by stepping simulate-nously on these two filaments [21]. Some sliders work ingroups and each detaches from the filament after everysingle stroke; these are often referred to as rowers becauseof the analogy with rowing with oars [13]. Contractility,rather than motility, at the subcellular and cellular levelare driven by the sliders and rowers [22, 23]. Some exam-ples of this category are listed in the table II; the detailscan be found in the cited review articles.

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Motor Sliding filaments Function (example)Myosin “Thin filaments” (F-actin) of muscle fibers Muscle contraction [24]Myosin “Stress fibers” (F-actin) of non-muscle cells Cell contraction [25]Myosin Cytokinetic “contractile ring” (F-actin) in eukaryotes Cell division [26]Kinesin Interpolar microtubules in mitotic spindle Mitosis [27, 28]Dynein Microtubules of axoneme Beating of eukaryotic flagella [29, 30]Dynein Microtubules of megakaryocytes Blood platelet formation [31]

TABLE II: Few example of cytoskeletal rowers and sliders aswell as their biological functions.

Polymer mode of force generation Function (example)MT polymerizing pistion-like organizing cell interior [35]

F-actin polymerization cell motility [36]FtsZ polymerization bacterial cytokinesis [37]MSP polymerization motility of nematode sperm cells [38]

Type-IV pili polymerization bacterial motility [39]MT de-polymerization Eukaryotic chromosome segregation [33]

spasmin spring-like vorticellid spasmoneme [34]Coiled actin spring-like egg fertilization by sperm cell of the horse-shoe crab Limulus polyphemus [40]

TABLE III: Force generation by polymeriz-ing/depolymerizing, coiling/uncoiling filaments: pistons,hooks and springs.

4. Push and pull of cytoskeletal filaments: nano-pistonsand nano-springs

Elongation of filamentous biopolymers that pressesagainst a light object (e.g., a membrane) can result ina “push” [32]. Similarly, a depolymerizing tubular fila-ment can “pull” a light ring-like object by inserting itshook-like outwardly curled depolymerizing tip into thering [33]. A flexible filament, upon compression by in-put energy, can store energy that can perform mechan-ical work when the filament springs back to its originalrelaxed shape [34]. Some typical examples are given intableIII.

C. Machines for synthesis, manipulation anddegradation of macromolecules of life

1. Membrane-associated machines for macromoleculetranslocation: exporters, importers and packers

In many situations, the motor remains immobile andpulls a macromolecule; the latter are often called translo-

case. Some translocases export (or, import) either a pro-tein [41] or a nucleic acid strand [42, 43] across the plasmamembrane of the cell or, in case of eukaryotes, across in-ternal membranes. A list is provided in table IV.The genome of many viruses are packaged into a pre-

fabricated empty container, called viral capsid, by a pow-erful motor attached to the entrance of the capsid. Asthe capsid gets filled, The pressure inside the capsid in-creases which opposes further filling [44, 45]. The effec-tive force, which opposes packaging, gets contributions

Membrane PolymerNuclear envelope RNA/Protein [43, 47]

Membrane of endoplasmic reticulum Protein [48]Membranes of mitochondria/chloroplasts Protein [49, 50]

Membrane of peroxisome Protein [51]

TABLE IV: Membrane-bound translocases.

from three sources: (a) bending of stiff DNA moleculeinside the capsid; (b) strong electrostatic repulsion be-tween the negatively charged strands of the DNA; (c)loss of entropy caused by the packaging [46]. The pack-aging motor has to be powerful enough to overcome sucha high pressure.

2. Machines for degrading macromolecules of life

Restriction-modification (RM) enzyme defend bacte-rial hosts against bacteriophage infection by cleavingthe phage genome while the DNA of the host bacteriaare not cleaved. Exosome and proteasome are machinesthat shred RNA and proteins into their basic subunits,namely, nucleotides and amino acids, respectively. Sim-ilarly, there are machines for degrading polysachharides,e.g., cellulosome (a cellulose degrading machine), starchdegrading enzymes, chitin degrading enzyme (chitinase),etc. These machines are listed in table V.

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Polymer Examples of MachinesDNA (polynucleotide) RM enzyme [52]RNA (polynucleotide) Exosome [53]Protein (polypeptide) Proteasome [54]

Cellulose (polysachharide) Cellulosome [56]Starch (polysachharide) Starch degrad. enzyme [55]Chitin (polysachharide) Chitinase [57]

TABLE V: Machines for degradation of macromolecules oflife.

Machine Template Product FunctionDdDP DNA DNA DNA replication [58]DdRP DNA RNA Transcription [59–61]RdDP RNA DNA Reverse transcription [62]RdRP RNA RNA RNA replication [63]

Ribosome mRNA Protein Translation [64–66]

TABLE VI: Types of polymerizing machines, the templatesthey use and the corresponding product of polymerization.

3. Machines for template-dictated polymerization

Two classes of biopolymers, namely, polynucleotidesand polypeptides perform wide range of important func-tions in a living cell. DNA and RNA are examples ofpolynucleotides while proteins are polypeptides. Bothpolynucleotides and polypeptides are made from a lim-ited number of different species of monomeric buildingblocks, namely, nucleotides and amino acids,respectively.The sequence of the monomeric subunits to be used forsynthesis of each of these are dictated by that of thecorresponding template. These polymers are elongated,step-by-step, during their birth by successive addition ofmonomers, one at a time. The template itself also servesas the track for the polymerizer machine that takes chem-ical energy as input to polymerize the biopolymer as wellas for its own forward movement. Therefore, these ma-chines are also referred to as motors.Depending on the nature of the template and prod-

uct nucleic acid strands, polymerases can be classifiedas DNA-dependent DNA polymerase (DdDP), DNA-dependent RNA polymerase (DdRP), etc. as listed inthe table VI.

4. Unwrappers and unzippers of packaged DNA: chromatinremodellers and Helicases

In an eukaryotic cell DNA is packaged in a hierarchi-cal structure called chromatin. In order to use a sin-gle strand of the DNA as a template for transcriptionor replication, it has to be unpackaged either locally orglobally. ATP-dependent chromatin remodelers [67] aremotors that perform this unpackaging. However, onlyone of the strands of the unpackaged duplex DNA servesas a template; the duplex DNA is unzipped by a DNAhelicase [68, 69]. Similarly, a RNA helicase unwinds

a RNA secondary structure. During DNA replication,a helicase moves ahead of the polymerase, like a minesweeper, unzipping the duplex DNA and dislodging otherDNA-bound proteins. However, the transcriptional andtranslational machineries do not need assistance of anyhelicase because these are capable of unzipping DNA andunwinding RNA, respectively, on their own. A helicasecan be monomeric, or dimeric or hexameric.

D. Rotary motors

Rotary molecular motors are, at least superficially,very similar to the motor of a hair dryer. Two rotarymotors have been studied most extensively. (i) A rotarymotor embedded in the membrane of bacteria drive thebacterial flagella [70, 71] which, the bacteria use for theirswimming in aqueous media. (ii) A rotary motor, calledATP synthase [72, 73], is embedded on the membraneof mitochondria, the powerhouses of a cell. A synthase

drives a chemical reaction, typically the synthesis of someproduct; the ATP synthase produces ATP, the “energycurrency” of the cell, from ADP.

E. Processivity, duty ratio, stall force and dwelltime

One can define processivity of a motor in three differentways:(i) Average number of chemical cycles in between attach-ment and the next detachment from the track;(ii) mean time of a single run, i.e., in between an attach-ment and the next detachment of the motor from thetrack;(iii) mean distance spanned by the motor on the track ina single run.Since the definitions (ii) and (iii) are directly accessibleto experiments, these are more useful than the definition(i). Another related, but distinct, concept is that of dutyratio which is defined as the average fraction of the timethat each head spends remaining attached to its trackduring one cycle.To translocate processively, a motor may utilize one of

the three following strategies:Strategy I: the motor may have more than one track-binding domain (oligomeric structure can give rise tosuch a possibility quite naturally). Most of the cytoskele-tal motors like conventional two-headed kinesin use sucha strategy [74]. One of the track-binding sites remainsbound to the track while the other searches for its nextbinding site.Strategy II: A motor may possess non-motor extra do-mains or some accessory protein(s) bound to it whichcan interact with the track even when none of the motordomains of the motor is attached to the track. Dyneinseems to exploit dynactin [75] to enhance its own proces-sivity.

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Strategy III: it can use a “clamp-like” device to remainattached to the track; opening of the clamp will be re-quired before the motor detaches from the track. Forexample, DdDP utilizes this strategy [76].

An external force directed opposite to the natural di-rection of walk of a motor is called a load force. Averagevelocity of a motor decreases with increasing load force.The magnitude of the load force at which the averagevelocity of the motor vanishes, is called the stall force.The force-velocity relation is one of the most fundamen-tal characteristic properties of a motor. Its status in bio-physics is comparable, for example, to that of the I-Vcharacteristics of a device in semiconductor physics.

Two motors with identical average velocities may ex-hibits widely different types of fluctuations. Therefore,as we’ll show in detail, a deeper understanding of the op-erational mechanism of a motor can be gained from thedistributions of their “dwells” at the successive spatialpositions on the track.

F. Fundamental general questions

(i) Single-molecule mechanism [77]: How do the in-teractions among the component structural units of anindividual motor, motor-track interactions, motor-ligand(fuel) interactions and the mechano-chemical kinetics ofthe system determine, for example, (a) the directional-ity, (b) processivity, (c) dwell-time distribution, (d) force-velocity relation, and (e) efficiency of transduction? Doesa given dimeric motor walk hand-over-hand or crawl likean inchworm, and why? Do the ATPase domains of ahexameric motor “fire” (a) in series, or (b) in parallel,or, (c) in random sequence?

(ii) Multi-motor coordination in a “workshop”: Themotors do not work in isolation in-vivo. What are themechanisms and consequences of the spatio-temporal co-ordination of the motors? For example, a single mitoticspindle consists of many polymerizing and depolymeriz-ing MTs, several types of cytoskeletal motors, includingdepolymerases [28]. A replisome, the workshop for DNAreplication, has to coordinate the operations of clampsand clamp loaders with those of primases, polymerases,etc. [58]. Similarly, a ribosome is a mobile platform onwhich the operations of several devices have to be coordi-nated properly during protein synthesis [64]. Well knownco-directional as well as head-on collisions of polymerasesand those of ribosomes can create traffic jam under somecircumstances whereas a collision can restart stalled traf-fic in other circumstances [78].

III. MOTORING IN A VISCOUS FLUID: FROMNEWTON TO LANGEVIN

Force is one of the most fundamental quantities inphysics. As we’ll argue in this section, some of the forces

which dominate the dynamics of molecular machines havenegligible effect on macroscopic machines.

A. Newton’s equation: deterministic dynamics

For simplicity, let us first consider a hypothetical sce-nario where neither the tracks nor the fuel molecules arepresent in the aqueous medium in which there is a free(i.e., not bound to any other molecule) motor protein.Suppose the medium consists of only N “particle-like”molecules and the center of mass of the motor proteinis also represented by a “particle”. Then, the exact tra-jectories of all these N + 1 “particles” can be obtainedby solving the corresponding coupled Newton’s equationsthat describe the dynamics of the N + 1 particles. Inreality, this approach is impractical because, even withthe largest and fastest computers available at present,we cannot get the trajectories over time intervals of theorder of 1s which is relevant for majority of the motorproteins as long as N is large (N ∼ 1023).

B. Langevin equation: stochastic dynamics

A more pragmatic approach would be to solve only theequations of motion for the motor protein by treating theN particles of the medium as merely the constituents ofa reservoir. In other words, one monitors the motion in a6-dimensional subspace of the full 6(N + 1)-dimensionalphase space of the system. The price one has to pay forthis simplification is that instead of the original Newton’sequation for the motor protein, one has to now solve aLangevin equation that describes the “Brownian” motionof the motor protein. Since we are soon going to extendthe discussion in the presence of filamentous tracks whichare essentially one-dimensional, we write the Langevinequation for the “Brownian” motion of the motor proteinin one-dimension m(d2x/dt2) = Fd + ξ where m is themass of the protein, Fd = −γv is the viscous drag and andξ is the random Brownian force. Note that the Langevinequation is neither deterministic nor symmetric undertime-reversal (t→ −t).

1. Motoring in a “sticky” medium: Purcell’s idea

Using the Stokes formula γ = 6πηr for a globular pro-tein of radius r ∼ 10nm moving with an average ve-locity v ∼ 1m/sec (corresponding to 1nm/ns) in theaqueous medium of viscosity η ∼ 10−3Pas we get anestimate Fd ∼ 200pN . Surprisingly, this viscous dragforce is about 200 times larger than the elastic force itexperiences when stretched by 1nm ! Consequently, theReynold’s number, which is the ratio of the inertial andviscous forces, is at most of the order O(10−2). TheReynold’s number will be of the order of 10−2 if you evertry (not recommended) to swim in honey!

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Now let us supply the tracks and fuel molecules to themotor protein in the same medium. The Langevin equa-tion will now include additional terms Fext which rep-resents the net force arising from the interaction of themotor protein with its track and the ligand; the ligandcould be the fuel molecule or the molecules produced byits “burning” (e.g., ATP or ADP). Besides, since the dy-namics of motor proteins is expected to be dominated byhydrodynamics at low Reynold’s number [79], one gener-ally drops the inertial term. In this “overdamped” regime

γv = Fext + ξ (1)

2. Motoring under random impacts from surroundings:Brownian force

The energy released by the hydrolysis of a single ATPmolecule is about 10−21J . Interestingly, the mean ther-mal energy kBT associated with a molecule at a temper-ature of the order of T ∼ 100K, is also kBT ∼ 10−21J .Moreover, equating this thermal energy with the workdone by the thermal force Ft in causing a displacementof 1nm we get Ft ∼ 1pN . This is comparable to the elas-tic force experienced by a typical motor protein whenstretched by 1nm. Thus, a motor protein that gets bom-barded from all sides by random thermal forces is similarto a tiny creature is a very strong hurricane! Therefore,in contrast to the deterministic dynamics of the macro-scopic machines, the dynamics of molecular motors isstochastic (i.e., probabilistic).Already in the first half of the twentieth century

D’Arcy Thompson, father of modern bio-mechanics, real-ized the importance of viscous drag and Brownian forces

at the cellular (and subcellular) level. He pointed outthat [80] that in this microscopic world “gravitation is

forgotten, and the viscosity of the liquid,..., the molecu-

lar shocks of the Brownian movement, .... the electric

charges of the ionized medium, make up the physical en-

vironment ... predominant factors are no longer those of

our scale”.

IV. ENERGY TRANSDUCTION BYMOLECULAR MACHINES: FROM CARNOT TO

FEYNMAN

Molecular motors generate force by transducing en-ergy. Interestingly, one of the two mechanisms of en-ergy transduction that I describe here, does not haveany analogous macroscopic counterpart. I explain in con-siderable detail why thermodynamic formalisms, whichhave been successfully utilized to calculate the commonperformance characteristics of macroscopic machines, areinadequate for natural nano-machines.Molecular motors are made of soft matter whereas

macroscopic motors are normally made of hard matterto withstand wear and tear. Nature seems to exploit the

high deformability of molecular motors for its biologicalfunction. But, this difference is minor compared to theothers and will not be elaborated further here.

A. Molecular machines are run by isothermalengines

This is in sharp contrast to the macroscopic thermalengines which require at least two thermal reservoirs atdifferent temperatures and which convert part of the in-put heat energy into mechanical work.Why can’t a molecular machine work as a heat en-

gine? In order to examine the viability of a intracellularheat engine, let us create a temperature gradient on alength scale ℓ ∼ 10nm. An elementary analysis of heatdiffusion equation is adequate to show that a tempera-ture gradient on a length scale ℓ relaxes within a timeinterval τtemp ∼ chℓ

2/κ where ch is the specific heat perunit volume and κ is the thermal conductivity. Usingthe typical characteristic values of ch and κ for water,one finds [81] τtemp ≃ 10−6 − 10−8s. Thus, temperaturegradients cannot be maintained for the entire duration ofeven one single cycle of a cyclic molecular machine whichis, typically, few orders of magnitude longer than τtemp.In other words, for all practical purposes, natural nano-machines operate isothermally. Because of the conditionT = constant, the molecular machines operate as free

energy transducers.

B. Inadequacy of equilibrium thermodynamics

A majority of the molecular motors are chemo-mechanical machines for which input and output arechemical energy and mechanical work, respectively. Re-call that for Carnot’s thermo-mechanical machine, thethermodynamic efficiency is ηeq−th = 1− (TL/TH) whereTH and TL are the temperatures of the two reservoirsat high and low temperatures, respectively. Similarly,for an isothermal chemo-mechanical engine, the thermalreservoirs would be replaced by chemical reservoirs atfixed chemical potentials µH and µL (µH > µL). Con-sequently, the heat flow of the thermal engine would bereplaced by particle flow in the chemical engine. Justas the difference of heat input and output in the heatengine is converted to mechanical work, the differenceµH − µL would be converted into output work in thechemical engine. Obviously, the corresponding thermo-dynamic efficiency would be [82] ηeq−ch = 1 − (µL/µH).Note that, because of the quasi-static nature of each stepin the equilibrium thermodynamic theory of these cyclicengines, each cycle takes infinite time. Naturally, thepower output Pout = 0 for both the thermal and chemi-cal Carnot engines.Can we apply the theory of such isothermal Carnot en-

gines to a chemo-mechanical molecular machine by iden-tifying, for example, µH and µL (µH > µL) as the chem-

9

ical potentials of ATP and ADP, respectively? The an-swer is: NO. The cycle time of the real molecular ma-chines is finite and their power output is non-zero. More-over, motors are examples of open systems [83] whichcontinue to run in repetitive cycles as long as energy ispumped in. Such a system cannot be in thermodynamicsequilibrium.

C. Inadequacy of endo-reversible thermodynamics

For macroscopic engines with finite cycle time, thecharacteristics of performance are often reliably calcu-lated within the theoretical framework of endo-reversiblethermodynamics [84]. In this case, the ratio of the rates

of output and input energies defines the efficiency oftransduction. The rate of output work is called poweroutput of the engine.

The formalism of endo-reversible thermodyamics forheat engines is based on the assumption that the work-ing substance of the engine is coupled to the two thermal

reservoirs by heat conductors of finite thermal conduc-tivity (non-zero thermal resistance). Entire dissipationis assumed to take place in the irreversible process ofheat conduction through these thermal resistors whereasall the processes in the working material of the engineare assumed to be fully reversible (i.e., no entropy is gen-erated internally). In this scenario, the correspondingthermal conductances in the Carnot engine are infinite.

Efficiency at maximum power output η(Pmax), ratherthan maximum efficiency itself, is the most appropriatequantitative measure of the performance of such engines.The upper-bound of η(Pmax) is given by the Curzon-Ahlborn expression [85] ηCA(Pmax) = 1 − (TL/TH)1/2.Similarly, within the framework of the endoreversiblethermodynamics, a phenomenological theory for chem-

ical engines can also be formulated if the cycle time isfinite [86]. Can’t we use this approach for molecular ma-chines which are also chemical engines with finite cycletimes?

Recall that endo-reversible thermodynamics is basedon the assumption that dissipation takes place only inthe process of transfer of matter between reservoirs andthe working substance whereas processes that the work-ing substance goes through in each cycle are perfectlyreversible (and, therefore, non-dissipative). This assump-tion is valid if the relaxations in the working substanceof the engine are very rapid compared to the processesinvolving the coupling of the working substance with thereservoirs. The validity of this assumption requires clearseparation of time scales of relaxation in the workingsubstance and in the working substance-reservoir cou-pling. But, such separation of time scales does not holdin molecular machines which consist of macromolecules.

D. Domain of non-equilibrium statistical mechanics

In general, molecular motors operate far beyond thelinear response regime and, therefore, the formalismof non-equilibrium thermodynamics [87] for coupledmechano-chemical processes is not applicable. More-over, thermodynamic formalisms developed for macro-scopically large systems do not account for the sponta-neous fluctuations. On the other hand, because of thesmall size of a molecular motor and because of the lowconcentrations of the molecules involved in its operation,fluctuations of positions, conformations as well as thecycle times are intrinsic features of their kinetics. There-fore, one has to use the more sophisticated toolbox ofstochastic processes and non-equilibrium statistical me-chanics for theoretical treatment of molecular machines.Interestingly, as we argue below, noise need not be a nui-sance for a motor; instead, a motor can move forward bygainfully exploiting this noise!

E. Defining efficiency: from Carnot to Stokes

The performance of macroscopic motors are charac-terized by a combination of its efficiency, power output,maximum force or torque that it can generate. Just likethe performance of their macroscopic counterparts withfinite cycle time, that of molecular motors [88] have alsobeen characterized in terms of efficiency at maximumpower, rather than maximum efficiency. However, theefficiency of molecular motors can be defined in severaldifferent ways [89].The efficiency of a motor, with finite cycle time, is

generally defined by

η = Pout/Pin (2)

where Pin and Pout are the input and output powers,respectively. The usual definition of thermodynamic effi-

ciency ηT is based on the assumption that, like its macro-scopic counterpart, a molecular motor has an outputpower [90]

Pout = −FextV. (3)

where Fext is the externally applied opposing (load) force.Although this definition is unambiguous, it is unsatisfac-tory for practical use in characterizing the performanceof molecular motors. As explained earlier, a molecularmotor has to work against the omnipresent viscous dragin the intracelluar medium even when no other force op-poses its movement (i.e., even if Fext = 0).A generalized efficiency ηG is also represented by the

same expression (2) where, instead of (3), the outputpower is assumed to be [91]

Pout = FextV + γV 2. (4)

This definition treats the load force and viscous drag onequal footing. In contrast, the “Stokes efficiency” ηS for a

10

molecular motor driven by a chemical reaction is definedas [92]

ηS =γV 2

(∆G)〈r〉 + FextV(5)

where 〈r〉 is the average rate of the chemical reaction and∆G is the chemical free energy consumed in each reactioncycle. This efficiency is named after Stokes because theviscous drag is calculated from Stokes law.As we show in the next subsection, the directional

movement of some motors arises from the rectification ofrandom thermal noise. For such motors, an alternativemeasure of performance is the rectification efficiency [93].Thus, different definitions of the efficiency of a molecularmotor may arise from different interpretations of theirtasks or may characterize distinct aspects of their oper-ation. Nevertheless, for any definition of efficiency, it isessential to ensure that its allowed values lie between 0and 1.

F. Noisy power stroke and Brownian ratchet

If the input energy directly causes a conformationalchange of the protein machinery which manifests itselfas a mechanical stroke of the motor, the operation of themotor is said to be driven by a “power stroke” mecha-nism. This is also the mechanism used by all man mademacroscopic machines. However, in case of molecularmotors, the power stroke is always “noisy” because ofthe Brownian forces acting on it.Let us contrast this with the following alternative sce-

nario: suppose, the machine exhibits both “forward” and“backward” movements because of spontaneous thermalfluctuations. If now energy input is utilized to prevent“backward” movements, but allow the “forward” move-ments, the system will exhibit directed, albeit noisy,movement in the “forward” direction. Note that the for-ward movements in this case are caused directly by thespontaneous thermal fluctuations, the input energy rec-tifies the “backward” movements. This alternative sce-nario is called the Brownian ratchet mechanism [94, 95].The concept of Brownian ratchet [96] was popularizedby Richard Feynman with his ratchet-and-pawl device[97]. However, in the context of molecular motors, theBrownin ratchet mechanism was proposed by severalgroups in the 1990s [94, 98].Thus, in principle, there are two idealized scenarios for

a transition from a conformation A to a conformation B-one is by a pure power stroke and the other by a purelyBrownian ratchet mechanism [99]. However, for a realmolecular motor, it is difficult to unambiguously distin-guish between a power stroke and a Brownian ratchet[100]; the actual mechanism may be a combination ofthe two idealized extremes.•Are Brownian motors Maxwell’s demon?

Brownian ratchet mechanism transduces random ther-mal energy of the reservoir into mechanical work. Does

it imply that these motors are perpetual motors of thesecond kind that violate the second law of thermody-namics? In that case, does it have any similarity withthe Maxwell’s demon [101]? A short answer is: No, aBrownian ratchet is not a Maxwell’s demon because it isan open system far from the state of equilibrium whereasthe second law of thermodynamics is strictly valid onlyfor systems in stable thermodynamic equilibrium.

G. Physical realizations of Brownian ratchet inmolecular motors

Is there any real biomolecular motor which can be re-garded as a true physical realization of Brownian ratch-ets? The answer is an emphatic “yes” and we list a fewtypical examples here.

KIF1A, a single-headed kinesin, is an example of cy-toskeletal motor whose operational mechanism can be in-terpreted as a Brownian ratchet [102–104]. The originalacto-myosin crossbridge model suggested by Huxley [105]can be interpreted as Brownian ratchet [106] althoughHuxley himself did not use this terminology. Brownianratchet can also account for the force generation by apolymerizing cytoskeletal filament [107].

The crossing of a membrane during the export/importof a protein of length L (amino acid monomers) takesplace at a faster rate by the Brownian ratchet mechanismcompared to that by pure translational diffusion [108].It has been claimed that the translocation of a mRNAmolecule across the nuclear membrane of an eukaryoticcell takes place also by a similar Brownian ratchet mech-anism through the nuclear pore complex [109]. The elon-gation of a mRNA during transcription by a RNA poly-merase can also be a physical realization of the Brownianratchet mechanism [110, 111]. Translocation, one of thecrucial steps of the mechano-chemical cycle of a ribosomeis a physical realization of Brownian ratchet [65].

V. MATHEMATICAL DESCRIPTION OFMECHANO-CHEMICAL KINETICS:

CONTINUOUS LANDSCAPES VS. DISCRETENETWORKS

We combine the fundamental principles of (stochas-tic) nano-mechanics and (stochastic) chemical kinetics toformulate the general theoretical framework for a quan-titative description of the mechano-chemistry or chemo-

mechanics of molecular motors. We mention a few alter-native formalisms.

11

A. Motor kinetics as wandering in atime-independent mechano-chemical free-energy

landscape

This approach is useful for an intuitive physical ex-planation of the coupled mechano-chemical kinetics ofmolecular motors. At least one of the independent coor-dinate axes of this landscape represents the position ofthe motor in real space (i.e., its “mechanical coordinate”)while at least another represents its chemical state (.e.,“chemical coordinate”). Therefore, the minimum dimen-sion of this “land” is 2 and the free energy can be rep-resented by the hight at each point on the “land”. Boththe position and chemical state variables are assumedto be continuous. The profiles of the free energy alongboth the position and chemical coordinates, obtained bytaking appropriate cross sections of the free energy land-scape, are periodic with the same periodicity in both thedirections. But, the profile is tiled forward along thechemical direction so that the bottom of the minima arelocated deeper in the forward direction along the “chem-ical” coordinate; this tilt accounts for the lowering of freeenergy caused, for example, by ATP hydrolysis.

B. Motor kinetics as wandering in thetime-dependent real-space potential landscape

Consider those special situations where the chemicalstates of the motor are long lived and change in fast dis-

crete jumps so that the position of the motor can con-tinue to change without alteration in its chemical state,except during chemical transitions when position remainsfrozen. Thus, no mixed mechano-chemical transition isallowed in this scenario. In such situations, we can as-sume that the potential landscape in real space remainsunchanged for a while, during which the wanderings ofthe motor in this landscape (i.e., the positional dynamicsof the motor) is governed by the “frozen” spatial profileof the potential. The profile of the potential switches se-quentially from one form to another and the sequence ofm profiles is repeated in each cycle although the switch-ing times are random. Different profiles correspond todifferent motor-track interactions which is dependent onthe nature of the ligand (if any) bound to the motor.

In this formulation we assume that the allowed po-sitions of the motor on its track form a continuum x.The discrete index µ = 1, 2, ...,M denote the M dis-tinct spatial profiles of the potential Vµ(x) that are pos-tulated apriori. At any instant of time t, the stateof the motor is given by (x, µ). In the overdampedregime, the time-evolution of the position x(t) of the mo-tor obeys the Langevin equation (1) with [112] Fext =−dVµ(x)/dx+Fload. The time-dependence of the profile

of the potential Vµ(x) is governed by the master equation

∂Pµ(x, t)

∂t=

∑

µ′

Pµ′(x, t)Wµ′→µ(x)−∑

µ′

Pµ(x, t)Wµ→µ′ (x)

(6)

where Pµ(x, t) is the probability that the motor protein“sees” the landscape Vµ(x) at time t and Wµ→µ′ (x) is thetransition probability per unit time for a transition fromthe chemical state µ to the chemical state µ′ at positionx.Instead of the Langevin equation, an equivalent

Fokker-Planck equation can be used to describe the wan-dering of the motor protein in the potential energy land-scape Vµ(x). The advantage of this alternative formu-lation is that the Fokker-Planck equation can be com-bined naturally with the master equation that describesthe time-evolution of the potential energy landscape. Inthis hybrid formulation, Pµ(x, t) represents the probabil-ity that, at time t, the motor is located at x, while “see-ing” the potential energy landscape Vµ(x). The equationgoverning the time dependence of Pµ(x, t)

∂Pµ(x, t)

∂t=

1

η

∂

∂x

[

V ′µ(x)− FPµ(x, t)

]

+

(

kBT

η

)

∂2Pµ(x, t)

∂x2

+∑

µ′

Pµ′(x, t)Wµ′→µ(x)

−∑

µ′

Pµ(x, t)Wµ→µ′ (x) (7)

where η is a phenomenological coefficient that capturesthe effect of viscous drag. Note that there is no termin this equation which would correspond to a mixedmechano-chemical transition.

C. Motor kinetics as a jump process in a fullydiscrete mechano-chemical network

In this formulation, both the positions and “internal”(or “chemical”) states of the motors are assumed to bediscrete. Let Pµ(i, t) be the probability of finding themotor at the discrete position labelled by i and in the“chemical” state µ at time t. Then, the master equationfor Pµ(i, t) is given by

∂Pµ(i, t)

∂t= [

∑

j 6=i

Pµ(j, t)kµ(j → i)−∑

j 6=i

Pµ(i, t)kµ(i→ j)]

+ [∑

µ′

Pµ′ (i, t)Wµ′→µ(i)−∑

µ′

Pµ(i, t)Wµ→µ′ (i)]

+ [∑

j 6=i

∑

µ′

Pµ′(j, t)ωµ′→µ(j → i)

−∑

j 6=i

∑

µ′

Pµ(i, t)ωµ→µ′(i→ j)] (8)

12

where the terms enclosed by the three different brackets[.] correspond to the purely mechanical, purely chemicaland mechano-chemical transitions, respectively.As a concrete example, which will be used also for on

several other occasions later in this review, consider a 2-state motor that, at any site j, can exist in one of theonly two possible chemical states labelled by the symbols1j and 2j. We assume the mechano-chemical cycle of thismotor to be

1jω1

ω−1

2jω2→ 1j+1 (9)

where the rates of the allowed transitions are shown ex-plicitly above or below the corresponding arrow. Notethat the transition 1j 2j is purely chemical whereasthe transition 2j → 1j+1 is a mixed mechano-chemicaltransition. The corresponding master equations are givenby

dP1(i)

dt= ω2P2(i− 1) + ω−1P2(i)− ω1P1(i)

dP2(i)

dt= ω1P1(i)− ω−1P2(i)− ω2P2(i)

(10)

Suppose a load force Fext opposes the meacho-chemicaltransition 2j → 1j+1. The effect of the load forcecan be incorporated by assuming the load-dependenceof the corresponding rate constant to be of the formω2(Fext) = ω2(0)exp[−Fext/(kBT )] where ω2(0) is thevalue of the rate constant in the absence of any externalforce. We’ll see some implications of these equations inseveral specific contexts later in this article.

D. Balance conditions for mechano-chemicalkinetics: cycles, and flux

On a discrete mechano-chemical state space, each stateis denoted by a vertex and the direct transition from onestate (denoted by, say, the vertex i) to another (denotedby, say, the vertex j) is represented by a directed edge

|ij >. The opposite transition from j to i is denotedby the directed edge |ji >. A transition flux can bedefined along any edge of this diagram. The forwardtransition flux J|ij> from the vertex i to the vertex j isgiven by WjiPi while the reverse flux, i.e., transition fluxJ|ji> from j to i is given by WijPj . Therefore, the nettransition flux in the direction from the vertex i to thevertex j is given by Jji = WjiPi −WijPj .A cycle in the kinetic diagram consists of at least three

vertices. Each cycle Cµ can be decomposed into two di-rected cycles (or, dicycles) [113] Cd

µ where d = ± corre-sponds the clockwise and counter-clockwise cycles. Thenet cycle flux J(Cµ) in the cycle Cµ, in the CW direction,is given by J(Cµ) = J(C+

µ )− J(C−µ ).

For each arbitrary dicycle Cdµ, let us define the dicycle

ratio

R(Cdµ) = Π<ij>ǫC+d

µWji/Π<ij>ǫC−d

µWij = Πµ,d

<ij>(Wji/Wij).

(11)where the superscript µ, d on the product sign denote aproduct evaluated over the directed edges of the cycle.So, by definition, R(C−µ ) = 1/R(C+

µ ).It has been proved rigorously [114] that, for detailed

balance to hold, the necessary and sufficient condition tobe satisfied by the transition probabilities is

R(Cdµ) = 1 for all dicycles Cd

µ (12)

For a non-equilibrium steady state (NESS), one candefine [113] the dicycle frequency Ωss(Cd

µ) which is the

number of dicycles Cdµ completed per unit time in the

NESS of the system. Then,

Ωss(C+µ )/Ωss(C−µ ) = Πµ,d

<ij>(Wji/Wij) = R(Cdµ) (13)

Clearly, R(Cdµ) 6= 1, in general, for any NESS.

Detailed balance is believed to be a property of systemsin equilibrium whereas the conditions under which molec-ular machines operate are far from equilibrium. Does itimply that detailed balance breaks down for molecularmachines? The correct answer this subtle question needsa careful analysis [115–117].If one naively assumes that the entire system returns to

its original initial state at the end of each cycle one wouldget the erroneous result that the detailed balance breaksdown. But, strictly speaking, the free energy of the fullsystem gets lowered by |δG| (e.g., because of the hydrol-ysis of ATP) in each cycle although the cyclic machineitself comes back to the same state. When the latter factis incorporated correctly in the analysis [115–117], onefinds that detailed balance is not violated by molecularmachines. This should not sound surprising- the transi-tion rates “do not know” whether or not the system hasbeen driven out of equilibrium by pumping energy intoit.

VI. SOLVING FORWARD PROBLEM BYSTOCHASTIC PROCESS MODELING: FROM

MODEL TO DATA

A. Average speed and load-velocity relation

For simplicity, let us consider the kinetic scheme shownin fig. (1(a)). In terms of the Fourier transform

Pµ(k, t) =

∞∑

j=−∞

Pµ(xj , t)e−ikxj (14)

of Pµ(xj , t), the master equations can be written as amatrix equation

∂P(k, t)

∂t= W(k)P(k, t) (15)

13

(a)

(b)

(c)

FIG. 1: Three examples of different types of networks of dis-crete mechano-chemical states. The bullets represent the dis-tinct states and the arrows denote the allowed transitions be-tween the two corresponding states. The scheme is (a) isunbrached whereas that in (b) has branched pathways con-necting the same pair of states. The mechanical step size isunique in both (a) and (b) whereas steps of more than onesize are possible in (c). (Adapted from fig.1 of ref.[120]).

where P is a column vector of M components (µ =1, ...M) and W(k) is the transition matrix in the k-space(i.e., Fourier space). Summing over the hidden chemicalstates, we get

P (k, t) =

M∑

µ=1

Pµ(k, t) =

M∑

µ=1

∞∑

j=−∞

Pµ(xj , t)e−ikxj . (16)

Taking derivatives of both sides of (16) with respect to qwe get [118, 119]

i

(

∂P (k, t)

∂k

)

k=0

= < x(t) >

−

(

∂2P

∂k2

)

k=0

+

(

∂P

∂k

)2

k=0

= < x2(t) > − < x(t) >2

(17)

Evaluating P (k, t), in principle, the stationary drift ve-locity (i.e., asymptotic mean velocity) V and the corre-sponsding diffusion constant D can be obtained from

V = limt→∞i∂

∂t

[(

∂P (k, t)

∂k

)

k=0

]

D = limt→∞1

2

∂

∂t

[(

−∂2P (k, t)

∂k2

)

k=0

+

(

∂P (k, t)

∂k

)2

k=0

]

(18)

It may be tempting to attempt a direct utilization ofthe general form

P (k, t) =∑

µ

Bµeλµ(k)t (19)

where the coefficients Bµ are fixed by the initial condi-tions and λµ(k) are the eigenvalues of W(k). However,for the practical implementation of this method analyti-cally the main hurdle would be to get all the eigenvaluesof W(k). Fortunately, only the smallest eigenvalue λmin,which dominates P (k) in the limit t→∞, is required forevaluating V and D [118, 119]:

V = i

(

∂λmin(k, t)

∂k

)

k=0

D = −1

2

(

∂2λmin(k, t)

∂k2

)

k=0

(20)

Even the forms (20) are not convenient for evaluatingV and D. Most convenient approach is based on thecharacteristic polynomial Q(k) associated with the ma-trix W(k), i.e., Q(k;λ) = det[λI −W(k)]. Therefore,λmin(k) is a root of the polynomial Q(k;λ), i.e., solutionof the equation

Q(k;λ) =

M∑

µ=0

Cµ(k)[λ(k)]µ = 0. (21)

Hence [118, 119]

V = −iC′0

C1(0)

D =C′′0 − 2iC′1(0)V − 2C2(0)V

2

2C1(0)(22)

where C′µ = [∂Cµ(k)/∂k]k=0.

14

For a postulated kinetic scheme, W is given. Thenthe expressions (22) are adequate for analytical deriva-tion of V and D for the given model. However, in orderto calculate the distributions of the dwell times of themotors, it is more convenient to work with the Fourier-Laplace transform, rather than the Fourier transform, ofthe probability densities. Therefore, we now derive al-ternative expressions for V and D in terms of the fullFourier-Laplace transform of the probability density.Taking Laplace transform of (14) with respect to time

Pµ(k, s) =

∫ ∞

0

Pµ(k, t)e−st, (23)

the matrix form of the master equation reduces toP(k, s) = R(k, s)−1P(0) where R(k, s) = sI −W(k)and P(0) is the column vector of initial probabilities.Now the characteristic polynomial is the determinant ofR(k, s) which can be expressed as [120]

|R(k, s)| =

M∑

µ=0

cµ(k)sµ = 0. (24)

Equation (24) is formally similar to (21). As expected,we get [120]

V = −ic′0

c1(0)

D =c′′0 − 2ic′1(0)V − 2c2(0)V

2

2c1(0)(25)

•Example: an M-step unbranched mechano-

chemical cycle

As an illustrative example, let us consider the un-branched mechano-chemical cycle [5] with M = 4, asshown in fig.2. This special value of M is motivated bythe typical example of a kinesin motor for which the fouressential steps in each cycle are as follows: (i) a substrate-binding step (e.g., binding of an ATP molecule), (ii) achemical reaction step (e.g., hydrolysis of ATP), (iii) aproduct-release step (e.g., release of ADP), and (iv) amechanical step (e.g., power stroke).

FIG. 2: An unbranched mechano-chemical cycle of the molec-ular motor with M = 4. The horizontal dashed line shows thelattice which represents the track; j and j + 1 represent twosuccessive binding sites of the motor. The circles labelled byintegers denote different “chemical” states in between j andj + 1. (Adapted from fig.7 of ref.[121]).

Suppose, The forward transitions take place at ratesuj whereas the backward transitions occur with the rates

wj . The average velocity V of the motor is given by [5]

V =1

RM

[

1−

M−1∏

j=0

(

wj

uj

)]

(26)

where

RM =

M−1∑

j=0

rj (27)

with

rj =

(

1

uj

)[

1 +

M−1∑

k=1

k∏

i=1

(

wj+i

uj+i

)]

(28)

while D is given by

D =

[

(V SM + dUM )

R2M

−(M + 2)V

2

]

d

M(29)

where

SM =

M−1∑

j=0

sj

M−1∑

k=0

(k + 1)rk+j+1 (30)

and

UM =

M−1∑

j=0

ujrjsj (31)

while,

sj =1

uj

(

1 +M−1∑

k=1

k∏

i=1

wj+1−i

uj−i

)

. (32)

For various extensions of this scheme see ref.[5].In the simpler case shown in (9), where M = 2, and

the second step is purely irreversible, using the step sizeℓ explicitly (to make the dimensions of the expressionsexplicitly clear), we get

V = ℓ

[

ω1ω2

ω1 + ω−1 + ω2

]

D =ℓ2

2

[

(ω1ω2)− 2(V/ℓ)2

ω1 + ω−1 + ω2

]

(33)

Note that if, in addition, ω−1 vanishes, i.e., if both thesteps are fully irreversible, then d/V = ω−11 + ω−12 , i.e.,the average time taken to move forward by one site is thesum of the mean residence time in the two steps of thecycle.•Effects of branched pathways and inhomo-

geneities of tracks

Microtubule-associated proteins and actin-related pro-teins can give rise to inhomogeneities of the tracks forcytoskeletal motors. The intrinsic sequence inhomogene-ity of DNA and RNA strands can nontrivial influences onthe motors that use nucleic acid strands as their tracks[122].

15

B. Jamming on a crowded track: flux-densityrelation

In the preceeding subsection we considered lone motormoving along a filamentous track. Now we consider atrack with heavy motor traffic. In principle, unless con-trolled by some regulatory signals, formation of “trafficjams” [123–125] on crowded tracks cannot be ruled out[126–129]. These phenomena are formally similar to sys-tems of sterically interacting self-propelled particles. Oneof the simplest theoretical models for such systems is thetotally asymmetric simple exclusion process (TASEP).In fact, TASEP was originally inspired by traffic of ri-bosomes on a mRNA track during translation [130–133].However, in recent years it has found application also inthe context of traffic-like collective movements of ribo-somes in more realistic models [134, 135] as well as manyother types of molecular motors, e.g., kinesins [103, 104],RNA polymerase [136, 137], growth of fungal hyphae[138], growth of bacterial flagella [139], crowding of de-polymerizing kinesins at the tip of a microtube [140], etc.

C. Information processing machines: fidelity versuspower and efficiency

Power and efficiency are most appropriate quantities tocharacterize the performance of porters, rowers, sliders,etc. But, not all machines fall in these categories. Forinformation processing machines, which we consider inthis subsection, fidelity of information transfer is moreimportant than power and efficiency.

The sequence of the monomeric subunits of a polynu-cleotide or that of a polypeptide is dictated by that ofthe corresponding template strand. During the polymer-ization process, after preliminary tentative selection, theselected monomeric subunit is subjected to several levelsof checks by the quality control system of the polymeriz-ing machinery. Only after a selected subunit is screenedby such a stringent test, it is covalently bonded to theelongating biopolymer.

Discrimination of monomers based merely on the dif-ferences of free-energy gains cannot account for the ob-served high fidelity of these processes. For example, “ki-netic proofreading” [141, 142] has been invoked to explainthe kinetic processes that contribute to the high level oftranslational fidelity. Following three features are essen-

tial for kinetic proofreading [143]:(i) formation of an initial enzyme-reactant complex,(ii) a strongly forward driven step that results in a high-energy intermediate complex, and(iii) one or more branched pathways along which dissocia-tion of the enzyme-reactant complex, and rejection of thereactant, is possible before the complex gets an opportu-nity to make the final transition to yield the product.

Let is consider the kinetic scheme shown below

Mj + Pn + S I1 → I2↓→Mj+1 + Pn+1

(34)

where Mj denotes the polymerase motor located at thediscrete position j on its template, Pn represents theelongating biopolymer consisting of n subunits and S isa single subunit. Note that the scheme shown in (34)is, at least, formally an extension of (9) where an extraintermediate state I2 and a branched path from I2 havebeen added. This scheme is one of the simplest possibleimplementations of kinetic proofreading.Kinetic proofreading leads to futile cycles in which at

least part of the input free energy is dissipated with-out elongating the nascent polypeptide by an amino acidmonomer. The effects of the consequent loose mechano-chemical coupling [144] of the translational machinery onthe rate of polypeptide synthesis has been investigated[135, 145, 146].Occasionally wrong monomers escape detection in

spite of elaborate selection procedure. In case of DNAreplication, the polymerase normally detects the errorimmediately and corrects it before elongating it further.For this purpose, the nascent DNA is transferred to an-other specific site on the same polymerase where thewrong monomer is cleaved before transferring the nascentDNA back to the site of elongation activity. The inter-play of the two contradictory activities of elongation andshortening of a nascent DNA exposes leads to the cou-pling of their corresponding rates in the overall rate ofDNA polymerization by a DdDP [147].

D. Beyond average: dwell time distribution (DTD)

Two motors with identical average velocities may ex-hibits widely different types of fluctuations. Suppose thesuccessive mechanical steps are taken by the motor attimes t1, t2, ..., tn−1, tn, tn+1, .... Then, the time of dwellbefore the k-th step is defined by τk = tk − tk−1. Inbetween successive steps, the motor may visit several“chemical” states and each state may be visited morethan once. But, the purely chemical transitions wouldnot be visible in a mechanical experimental set up thatrecords only its position. The number of visits to a givenstate and the duration of stay in a state in a given visitare random quantities.In order to appreciate the origin of the fluctions in the

dwell times, let us consider the simple N -step kinetics:

M1 M2 M3... Mj ... MN (35)

Suppose tµ,ν is the duration of stay of the motor in theµ-th state during its ν-th visit to this state. If τ is thedwell time, then

τ =

N∑

µ=1

nµ∑

ν=1

tµ,ν (36)

16

where nµ is the number of visits to the µ-th state. It isstraightforward to check that

< τ > =

N∑

µ=1

< nµ >< tµ >

(37)

where < nµ > is the avarege number of visitits to theµ-th state and < tµ > is the avarege time of dwell in theµ-th state is a single visit to it. More interestingly [148],

< τ2 > − < τ >2=

N∑

µ=1

(< t2µ > − < tµ >2) < nµ >

+

N∑

µ=1

(< n2µ > − < nµ >2) < tµ >2

+ 2∑

µ>ν

(< nµnν > − < nµ >< nν >) < tµ >< tν >

(38)

The first and second terms on the right hand side of (38)capture, respectively, the fluctuations in the lifetimes ofthe individual states and that in the number of visitsto a kinetic state. Note that the number of visits to aparticular state depends on the number of visits to theneighboring states on the kinetic diagram; this interstatecorrelation is captured by the third term on the righthand side of (38).Several different analytical and numerical techniques

have been developed for calculation of the dwell timedistribution [120, 149–151]. Since the dwell time is essen-tially a first passage time [152], an absorbing boundarymethod [150] has been used.

1. DTD for a motor that never steps backward

As an example, we consider again the simple scheme(9). In this case, the DTD is

f(t) =

(

ω1ω2

ω− − ω+

)

(e−ω+t − e−ω−t) (39)

where

ω± =ω1 + ω−1 + ω2

2±

[

√

(ω1 + ω−1 + ω2)2

4− ω1ω2

]

(40)In the special case ω−1 = 0, ω+ = ω1 and ω− = ω2 and,hence,

f(t) =

(

ω1ω2

ω2 − ω1

)

(e−ω1t − e−ω2t) (41)

Similar sum of exponentials for DTD have been derivedalso for machines with more complex mechano-chemicalkinetics (see, for example, refs.[121, 146, 153]).

It is possible to establish on general grounds that, for amotor with N mechano-chemical kinetic states like (35),the DTD is a sum of n exponentials of the form [148]

f(t) =

N∑

j=1

Cje−ωjt (42)

where N − 1 of the N coefficients Cj (1 ≤ j ≤ N) areindependent of each other because of the constraint im-posed by the normalization condition for the distributionf(t). Also note that the prefactors Cj can be both posi-tive or negative while ωj > 0 for all j.If we consider an even more special case of the scheme

(9) where ω−1 = 0, ω1 = ω2 = ω, i.e., all the steps arecompletely irreversible and take place at the same rateω, the DTD becomes the Gamma-distribution

f(t) =ωN tN−1e−ωt

Γ(N)(43)

where Γ(N) is the gamma function with N = 2. In-terestingly, for the Gamma-distribution, the randomnessparameter [154] (also called the Fano factor [155])

r = (< τ2 > − < τ >2)1/2/ < τ > (44)

is exactly given by r = 1/N . Can the experimentallymeasured DTD be used to determine the number of statesN? Unfortunately, for real motors, (i) not each step ofa cycle is fully irreversible, (ii) the rate constants for dif-ferent steps are not necessarily identical, (iii) branchedpathways are quite common. Consequently, 1/r may pro-vide just a bound on the rough estimate of N .Can one use the general form (42) of DTD to extract

all the rate constants for the kinetic model by fitting itwith the experimentally measured DTD? The answer is:NO. First, even if a good estimate of N is available, thenumber of parameters that can be extracted by fittingthe experimental DTD data to (42) is 2N − 1 (n valuesof ωj and N − 1 values of Cj). On the other hand, thenumber of possible rate constants may be much higher[148]. For example, if transitions from every kinetic stateto every other kinetic state is allowed, the total numberof rate constants would be N(N − 1). In other words, ingeneral, the kinetic rate constants are underspecified bythe DTD. Second, as the expression (39) for the DTD ofthe example (9) shows expplicitly, the ω’s that appear inthe exponentials may be combinations of the rate con-stants for the distinct transitions in the kinetic model.It is practically impossible to extract the individual rateconstants from the estimated ω’s unless any explicit rela-tion like (40) between the estimated ω’s and actual rateconstants is apriori available.

2. Conditional DTD for motors with both forward andbackward stepping

For motors which can step both forward and backward,more relevant information on the kinetics of a motor are

17

contained in the conditional dwell time distribution [156].If n+ and n− are the numbers of forward and backwardsteps, respectively, then for large n = n+ + n−, the cor-responding step splitting probabilities are Π+ = n+/nand Π− = n−/n. The dwell times before a forward stepand before a backward step can be measured separately.Hence, the prior dwell times τ+ and τ− can be obtainedby restricting the summations in

τ←± =1

n±

±∑

τk (45)

to just forward (+) or just backward (-) steps, respec-tively. In terms of splitting probabilities and prior dwelltimes, the mean dwell time 〈τ〉 can be expressed as

〈τ〉 = Π+τ←+ +Π−τ

←− (46)

Compared to the prior dwell times, more detailed in-formation on the stepping statistics is contained in thefour conditional dwell times, which are defined as follows:

τ++ = dwell time between a + step followed by a + step

τ+− = dwell time between a + step followed by a − step

τ−+ = dwell time between a − step followed by a + step

τ−− = dwell time between a − step followed by a − step

(47)

It is helpful to introduce pairwise step probabilities

Π++,Π+−,Π−+,Π−−. Note that Π++ + Π+− = 1, andΠ−+ +Π−− = 1. Neglecting finite time corrections,

Π++ = n++/(n++ + n+−), Π+− = n+−/(n++ + n+−)

Π−+ = n−+/(n−+ + n−−), Π−− = n−−/(n−+ + n−−)

(48)

Hence

τ←+ = Π++τ++ +Π+−τ−+

τ←− = Π−+τ+− +Π−−τ−− (49)

Defining the post dwell times τ→± in a fashion similar tothat used for defining the prior dwell times, we get

τ→+ = Π++τ++ +Π+−τ+−

τ→− = Π−+τ−+ +Π−−τ−− (50)

and

〈τ〉 = Π+τ→+ +Π−τ

→− (51)

This formalism has been applied, for example, single-headedkinesin KIF1A [157].

VII. A SUMMARY OF EXPERIMENTALTECHNIQUES: ENSEMBLE VERSUS

SINGLE-MACHINE

The experimental techniques for probing the struc-ture and dynamics of molecular machines can be di-vided broadly into two groups: (i) ensemble-averaged,(ii) single-machine.

X-ray diffraction and electron microscopy, two of themost powerful biophysical techniques, yield the structureof molecular machines averaged over an ensemble. Elec-tron microscopy is a powerful alternative for the deter-mination of structures of those macromolecules whosecrystals are not available. Cryo-electron microscopy[1, 158] combines the technique of electron microscopywith cryogenics-based sample preparation.The single-molecule techniques [159–164] can be fur-

ther classified into two groups: (i) methods of imaging,and (ii) methods of manipulation. These are not mutu-ally exclusive and can be probed by a single experimentalset up. The most direct single-molecule imaging is car-ried out by fluorescence-based optical microscopy. Tra-ditional diffraction-limited optical microscopy provides a“hazy” image of the machine. However, modern opti-cal nanoscopy has broken the resolution limit imposedby diffraction [165]. Fluorescence microscopy provides aglimpse (howsoever hazy) of single molecules. Imaginga fluorescently labelled molecular machine in real timeenables us to study its dynamics just as ecologists use“radio collars” to track individual animals.The techniques developed for the mechanical manipu-

lation of a single machine can be classified on the basisof tranducers used; the table below provides a summary.

Mechanical manipulation of a single biomolecule

ւ ց

Mechanical transducers Field-based t ransducersւ ց ւ ց

SFM Micro-needle EM-field Flow-fieldւ ց

Electric field Ma gnetic field

VIII. SOLVING INVERSE PROBLEM BYPROBABILISTIC REVERSE ENGINEERING:

FROM DATA TO MODEL

A discrete kinetic model of a molecular motor can beregarded as a network where each node represents a dis-tinct mechano-chemical state. The directed links denotethe allowed transitions. Therefore, such a model is unam-biguously specified in terms of the following parameters:(i) the total number N of the nodes, (ii) the N ×N ma-trix whose elements are the rates of the transitions amongthese states; a vanishing rate indicates a forbidden directtransition.In the preceeding sections we handled the “forward

problem” by starting with a model that is formulatedon the basis of apriori hypotheses which are, essen-tially, educated guess as to the mechano-chemical kinet-ics of the motor. Standard theoretical treatments of themodel yields data on various aspects on the modelledmotor; this approach is expressed below schematically.

Theoretical model → Experimental data

Consistency between theoretical prediction and experme-ntal data validates the model. However, any inconsis-

18

tency between the two indicates a need to modify themodel.In most real situations the numerical values of the rate

constants of the kinetic model are not known apriori. Inprinciple, these can be extracted by analyzing the exper-imental data in the light of the model. However, not onlythe rate constants but also the number of states and theoverall architecture of the mechano-chemical network aswell as the kinetic scheme postulated by the model maybe uncertain. In that case, the experimental data shouldbe utilized to “select” the most appropriate model fromamong the plausible ones. In fact, more than one model,based on different hypotheses, may appear to be consis-tent with the same set of experimental data within a levelof accuracy. The experimental data can be exploited atleast to “rank” the models in the order of their successin accounting for the same data set.The “inverse problem” of inferring the model from the

observed experimental data has to be based on the the-ory of probability. Such “statistical inference” [166] canbe drawn by following methods developed by statisti-cians over the last one century. Inferring the completenetwork of mechano-chemical states and kinetic schemeof a molecular machine from its observed properties isreminiscent of inferring the operational mechanism of agiven functioning macroscopic machine by “reverse en-gineering”. This inverse problem, which is the mainaim of this section, is expressed below schematically.

Theoretical model ← Experimental data

It is worth emphasizing that both the directions of in-vestigations, i.e. the forward problem and the inverseproblem are equally important and complementary toeach other [167].

A. Frequentist versus Bayesian approach

Suppose, ~m be a column vector whose M compo-nents are the M parameters of the model, i.e., the trans-pose of ~m is ~mT = (m1,m2, ...,mM ) Let the data ob-tained in N observations of this model are represented

by the N -component column vector ~d whose transpose

is ~dT = (d1, d2, ..., dN ). Our “inverse problem” is to in-

fer information on ~m from the observed ~d. The philoso-phy underlying the frequentist approach, i.e., approachesbased on maximum-likelihood (ML) estimation and theBayesian approach for extracting these information aredifferent in spirit, as we explain in the next two subsub-sections [168].For simplicity, let us assume that a device has only two

possible distinct states denoted by E1 and E2.

E1kf

kr

E2 (52)

Let us imagine that we are given the actual sequence ofthe states, over the time interval 0 ≤ t ≤ T , generated

by the Markovian kinetics of the device. But, the mag-nitudes of the rate constants kf and kr are not supplied.We’ll now formulate a method, based on ML analysis[169] to extract the numerical values of the parameterskf and kr.

Suppose t(1)j and t

(2)j denote the time interval of the j-

th residence of the device in states E1 and E2, respectively.Moreover, suppose that the device makes N1 and N2 vis-its to the states E1 and E2, respectively, and N = N1+N2

is the total number of states in the sequence. Therefore,

total time of dwell in the two states are T1 =∑N1

j=1 t(1)j

and T2 =∑N2

j=1 t(2)j where T1 + T2 = T .

Since the dwell times are exponentially distributed fora Poisson process, the likelihood of any state trajectoryS is the conditional probability

P (S|kf , kr) =

(

ΠN1

j=1kfe−kf t

(1)j

)

(

ΠN2

j=1kre−krt

(2)j

)

=

(

kN1

f e−kfT1

)(

kN2r e−krT2

)

(53)

1. Maximum-likelihood estimate

ML approcah [170] is based on finding the estimatesof the set of model parameters that corresponds to the

maximum of the likelihood P (~d|~m) for a fixed set of data~d.

For the kinetic scheme shown in equation (52), the theML estimates of kf and kr are obtained by using (53)in d[lnP (S|kf , kr)]/dkf = 0 = d[lnP (S|kf , kr)]/dkr. It

is straightforward to see [169] that these estimates arekf = N1/T1 and kr = N2/T2.

2. Bayesian estimate

For drawing statistical inference regarding a kineticmodel, the Bayesian approach has gained increasing pop-ularity in recent years [171–176]. The areas of researchwhere this has been applied successfully include variousbiological processes in, for example, genetics [177, 178],biochemistry [179], cognitive sciences [180], ecology [181],etc.

In the Bayesian method there is no logical distinc-tion between the model parameters and the experimentaldata; in fact, both are regarded as random. The only dis-tinction between these two types of random variables isthat the data are observed variables whereas the modelparameters are unobserved. The problem is to estimatethe probability distribution of the model parameters fromthe distributions of the observed data.

19

The Bayes theorem states that

P (~m|~d) =P (~d|~m)P (~m)

P (~d)(54)

where P (~d) can be expressed as

P (~d) =

∫

P (~d|~m)P (~m)d~m (55)

The likelihood P (~d|~m) is the conditional probability forthe observed data, given a set of particular values ofthe model parameters, that is predicted by the kineticmodel. However, implementation of this scheme also re-quires P (~m) as input. In Bayesian terminology P (~m)is called the prior because this probability is assumed

apriori by the analyzer before the outcomes of the exper-iments have been analyzed. In contrast, the left handside of equation (54) gives the posterior probability, i.e.,after analyzing the data.Thus, an experimenter learns from the Bayesian anal-

ysis of the data. Such a learning begins with an inputin the form of a prior probability; the choice of the priorcan be based on physical intuition, or general argumentsbased, for example, on symmetries. Prior choice can be-come simple if some experience have been gained fromprevious measurements. Often an uniform distributionof the model parameter(s) is assumed over its allowedrange if no additional information is available to bias itschoice. To summarize, Bayesian analysis needs not just

the likelihood P (~d|~m) but also the prior P (~m).For the kinetic scheme shown in equation (52), the

Bayes’ theorem (54) takes the form

P (kf , kr|S) =P (S|kf , kr)P (kf , kr)

P (S)

=P (S|kf , kr)P (kf , kr)

∑

k′

f,k′

rP (S|k′f , k

′r)P (k′f , k

′r)

(56)

We now assume a uniform prior, i.e., constant for positivekf and kr, but zero otherwise. Then, P (kf , kr|S) is pro-portional to the likelihood function P (S|kf , kr) (within

a normalization factor). Normalizing, we get

P (kf , kr|S) =

[

TN1+11

Γ(N1 + 1)kN1

f e−kfT1

][

TN2+12

Γ(N2 + 1)kN2r e−krT2

]

(57)The mean of kf is N1+1)/T1 whereas the most-likely es-timate is N1/T1. Similarly, the mean and most probableestimates of kr are obtained by replacing the subscripts1 by subscripts 2. Moreover, the variance of kf and krare (N1 + 1)/T 2

1 and (N2 + 1)/T 22 , respectively.

3. Hidden Markov Models

The actual sequence of states of the motor, generatedby the underlying Markovian kinetics, is not directly visi-ble. For example, a sequence of states that differ “chem-ically” but not mechanically do not appear as distinct

on the recording of the position of the motor in a singlemotor experiment. This problem is similar to an olderproblem in cell biology: ion-channel kinetics [182, 183].Current passes through the channel only when it is inthe “open” state. However, if the channel has more thanone distinct closed states, the recordings of the currentreveals neither the actual closed state in which the chan-nel was nor the transitions between those closed stateswhen no current was recorded.

Hidden Markov Model (HMM) [184–187] has beenapplied to analyze FRET trajectories [188, 189], step-ping recordings of molecular motors [190, 191], and acto-myosin contractile system [192] to extract kinetic infor-mation.

For a pedagogical presentation of the main ideas be-hind HMM, we start with the kinetic scheme shown in(54) and a simple, albeit unrealistic, situation and thenby gradually adding more and more realistic features, ex-plain the main concepts in a transparent manner [169].First, suppose that the actual sequence of states (trajec-tory) itself is visible; this case can be analyzed either bythe ML-analysis of by Bayesian approach both of whichwe have presented above. We now relax the strong as-sumption about the trajectory and proceed as below.

•If state trajectory is hidden and visible trajec-

tory is noise-free

The sequence of states of the device is, as before, gen-erated by a Markov processs which is hidden. Supposethe device emits photons from time to time that are de-tected by appropriate photo-detectors. For simplicity, weassume just two detection channels labelled by 1 and 2.For the time being, we also assume a perfect one-to-onecorrespondence between the state of the light emittingdevice and the channel that detects the photon. If thechannel 1 (2) clicks then the light emitting device wasin the state E1 (E2) at the time of emission. The inter-val ∆tj = tj+1 − tj between the arrival of the j-th andj + 1-th photons (1 ≤ j ≤ N) is random.

Thus, from the photo-detectors we get a visible se-quence of the channel index (a sequence made of a bi-nary alphabet) which we call “noiseless photon trajec-tory” [169]. The sequence of states in the noiseless phototrajectory is also another Markov chain that is conven-tionally referred to as the “random telegraph process”.Note that the photon detected by channel 1 (or, channel2) can take place at any instant during the dwell of thedevice in state 1 (or, state 2). Therefore, the sequenceof states in the noiseless photon trajectory does not re-veal the actual instants of transition from one state ofthe device to another.

Since the noiseless phton trajectory corresponds to arandom telegraph process, the transition probabilities for

20

this process are

P (E1|E1; kf , kr,∆tj) =kr

kf + kr+

kfkf + kr

e−(kf+kr)∆tj

P (E1|E2; kf , kr,∆tj) =kr

kf + kr[1− e−(kf+kr)∆tj ]

P (E2|E1; kf , kr,∆tj) =kf

kf + kr[1− e−(kf+kr)∆tj ]

P (E2|E2; kf , kr,∆tj) =kf

kf + kr+

krkf + kr

e−(kf+kr)∆tj

(58)

where P (Eµ|Eν ; kf , kr,∆tj) is the conditional probability

that state of the device is Eµ given that it was in the stateEν at a time ∆t earlier.The likelihood of the visible data sequence V is now

given by

P (V |kf , kr) = P (V1|kf , kr)ΠN−1j=1 P (Vj+1|Vj ; kf , kr,∆tj)

(59)where the first factor on the right hand side is the ini-tial probability (usually assumed to be the equilibriumprobability). The transition probabilities on the righthand side of equation (59) are the conditional probabili-ties given in equation (58). Unlike the previous simplercase, where the state sequence itself was visible, no ana-lytical closed-form solution is possible in this case. Nev-ertheless, analysis can be carried out numerically.•If state trajectory is hidden and visible trajec-

tory is noisy

In the preceeding case of a noise-free photon trajectory,we assumed that from the channel index we could getperfect knowledge of the state of the emitting device.More precisely, the conditional probabilities were

P (1|E1) = 1

P (1|E2) = 0

P (2|E1) = 0

P (2|E2) = 1

(60)

However, in reality, background noise is unavoidable.Therefore, if a photon is detected by the channel 1, itcould have been emitted by the device in its state E1(i.e., it is, indeed, a signal photon) or it could have comefrom the background (i.e., it is a noise photon). Supposeps is the probability that the detected photon is really asignal that has come from the emitting device. Supposepb1 is the probability of arrival of a background photon inthe channel 1. The probability that a background photonarrives in channel 2 is 1− pb1. Then [169],

E(1|E1) = ps + (1− ps)pb1

E(2|E1) = 1− P (1|E1) = (1− ps)(1 − pb1)

E(1|E2) = (1− ps)pb1

E(2|E2) = 1− P (1|E2) = ps + (1 − ps)(1− pb)

(61)

Thus, in this case, the relation between the states of thehidden and visible states is not one-to-one, but one-to-many.Therefore, given a hidden state of the device, a setof “emission probabilities” determine the probability ofeach possible observable state; these are listed in equa-tions (61) for the device (52).•HMM: formulation for a generic model of molec-

ular motor

On the basis of the simple example of a 2-state sys-tem presented above, we conclude that, for data analysisbased on a HMM four key ingredients have to be speci-fied:(i) the alphabet of the “visible” sequence µ (1 ≤ µ ≤N), i.e., N possible distinct visible states; (ii) the alpha-bet of the “hidden” Markov sequence j (1 ≤ j ≤ M),i.e., M allowed distinct hidden states, (iii) the hidden-to-hidden transition probabilities W (j → k), and (iv)hidden-to-visible emission probabilities E(j → µ). In ad-dition to the transition probabilities and emission proba-bilities, which are the parameters of the model, the HMMalso needs the initial state of the hidden variable(s) as in-put parameters.

Visible: V0 V1 ... Vt VT

⇑ ⇑ ⇑ ⇑

Hidden: H0 → H1 → ...→ Ht → HT

Suppose P (V |HMM, λ) denotes the probability

that an HMM with parameters λ generates a visiblesequence V . Then,

P (V |HMM, λ) =∑

H

P (V |H; λ)P (H|λ)

(62)where P (H|λ) is the conditional probability that the

HMM generates a hidden sequence H for the givenparameters λ and P (V |H; λ) is the conditional

probability that, given the hidden sequence H (for pa-rameters λ) the visible sequence V would be ob-tained.Once P (V |HMM, λ) is computed, the parame-

ter set λ are varied to maximize P (V |HMM, λ)

(for the convenience of numerical computation, oftenlnP (V |HMM, λ) is maximized. The total number

of possible hidden sequences of length T is TMN . Inorder to carry out the summation in equation (62) onehas to enumerate all possible hidden sequences and thecorresponding probabilities of occurrences. A successfulimplementation of the HMM requires use of an efficienctnumerical algorithm; the Viterbi algorithm is one suchcandidate.In case of a molecular motor, the “chemical states”

are not visible in a single molecule experiment. More-over, even its mechanical state that is “visible” in therecordings may not be its true position because of (a)measurement noise, and (b) steps missed by the detector.Let is denote the “visible” sequence by the recorded posi-tions Y whereas the hidden sequence is the compositemechano-chemical states X,C where X and C denote

21

the true position and chemical state, respectively. Thetransition probabilities are denoted by W (Xt−1, Ct−1 →Xt, Ct) One possible choice for the emission probabilitiesE is [190]

E(Xt → Yt) =

√

1

(2πσ2t )exp[−

(Yt −Xt)2

(2σ2t )

] (63)

In this case,

P (Y |HMM, λ) =∑

X,C

P (Y |X,C; λ)P (X,C|λ)

(64)where

P (X,C|λ) = PX0,C0W (X0, C0 → X1, C1)W (X1, C1 → X2, C2).....W (XT−1, CT−1 → XT , CT ) (65)

and

P (Y |X,C; λ) = E(X0 → Y0)E(X1 → Y1)...E(Xt → Yt)...E(XT → YT ) (66)

The usual strategy [188, 190] consists of the followingsteps: Step I: Initialization of the parameter values for it-eration. Step II: Iterative re-estimation of parameters formaximum likelihood: the parameters W (Xt−1, Ct−1 →Xt, Ct), E(Xt → Yt) and PX0,C0 are re-estimated iter-atively till P (Y |HMM, λ) saturates to a maximum.

Step III: Construction of “idealized” trace: using thefinal estimation of the model parameters, the position ofthe motor as a function of time is reconstructed; natu-rally, this trace is noise-free. Step IV: Extraction of thedistributions of step sizes and dwell times: from the ideal-ized trace, the distributions of the steps sizes dwell timesare obtained. These distributions can be compared withthe corresponding theoretical predictions.

B. Extracting FP-based model from data?

In this section so far we have discussed methods for ex-tracting the master-equation based models that describethe kinetics of motors in terms of discrete jumps on a fullydiscrete mechano-chemical state space. However, as wesummarized in section V, kinetics of molecular motorscan be formulated also in terms of FP equations whichtreat space as a continuous variable. Can one extractthe parameters of such FP-based models from the datacollected from single-molecule experiments?

One of the key ingredients of the FP-based approach isthe profile of the potential Vµ(x) felt by the motor in the“chemical” state µ. To my knowledge, it has not beenpossible to extract it from any experimental data. Now,let us define

dφ(x)/dx =1

∑

µ Pµ(x)

∑

µ

Pµ(x)[dVµ(x)/dx] (67)

to be the profile of the potential averaged over all the

chemical states. It can be shown [193] that

φ(x)

kBT= FxkBT − ln[

∑

µ

Pµ(x)] −J

D

∫ x

0

1∑

µ Pµ(y)dy

(68)Based on this observation, a prescription has been sug-gested [193] to extract φ(x) by analyzing the time se-ries of motor positions obtained from single-motor ex-periments.

IX. OVERLAPPING RESEARCH AREAS

A. Symmetry breaking: directed motility and cellpolarity

Energy is a scalar quantity whereas velocity is a vector.How does consumption of energy give rise to a non-zeroaverage velocity of a molecular motor? Moreover, a di-rected movement that a motor exhibits on the average,requires breaking the forward-backward symmetry on itstrack. What are the possible cause and effects of thisbroken symmetry at the molecular level?As far as the cause of this asymmetry is concerned, the

asymmetry of the tracks alone cannot explain the “di-rected” movement of the motors, because on the sametrack members of different families of motors can, onthe average, move in opposite directions. Obviously, thestructural design of the motors and their interactionswith the respective tracks also play crucial roles in deter-mining their direction of motion along a track. Further-more, this broken symmetry at the molecular level, e.g.,the “directed” movement of molecular motors, has im-portant effects on various biological phenomena, partic-ularly “vectorial” processes, at the sub-cellular and cel-lular levels. In general, a cell is not isotropic. Motors areessential in breaking the cellular symmetry [194]. Canwe establish a unique set of basic principle underlyingthe symmetry breaking [194, 195]?

22

Therefore, the cause and effects of broken symmetryof molecular motors can be examined in the broader con-text of the fundamental principles of symmetry breakingin physics and biology [196] (see also other articles inthe special “Perspectives on Symmetry Breaking in Biol-ogy” [197]). For macroscopic systems in thermodynamicequilibrium, symmetry breaking is explained in terms ofthe form of the free energy. However, since living cellsare far from thermodynamic equilibrium, the theory ofsymmetry breaking in those systems cannot be based onthermodynamic free energy. Kinetics cannot be ignoredin the study of symmetry breaking in living systems.

B. Self-organization and pattern formation:assembling machines and cellular morphogenesis

The interior of a living bacterial cell is far from ho-mogeneous; the intracellular space of eukaryotes are di-vided into separate compartments. Scaling is one of theinteresting properties of many physical quantities thatgives rise to some well known universalities. The scal-ing properties of a cell and the subcellular compartments[198, 199]. often depend on the machineries which as-semble them.The size, shape, location and number of intracellu-

lar compartments as well as modular intracellular ma-chineries are self-organized, rather than self-assembled.Dissipation takes place in “self-organization” and distin-guishes it from “self-assembly”; the latter corresponds tothe minimum of thermodynamic free energy whereas self-organized system does not attain thermodynamic equi-librium [195, 200–202]. Molecular motors and their fila-mentous tracks play important roles in the intracellularself-organization process [195, 200] and even in the cellu-lar morphogenesis which may be regarded as a problemof pattern formation far from equilibrium.

C. Dissipationless computation: polymerases as“tape-copying Turing machines”

The concept of information in the context of biolog-ical systems has been discussed at length in the past[203]. The operation of molecular machines involved ingenetic processes can be analyzed in terms of storage andtransmission of information. In fact, a particular class ofmachines carry out what may be viewed as data trans-mission whereas that of others may be viewed as digital-to-analog conversion [204] . For these obvious reasons,a broad class of molecular machines are interesting alsofrom the perspective of information theory, electronic en-gineering and computer science.Computation can be viewed as a transition from one

state to another. However, in a conventional digital com-puter an elementary operation is logically irreversible. Tounderstand the meaning of this term, consider the twosummations 3 + 1 = 4 and 2 + 2 = 4. If the computer

retains only the output, i.e., 4 and erases the input num-bers (i.e., 3 and 1, or 2 and 2, as the case may be) aftersumming the two input numbers, then, given only theoutput (i.e., 4) it is impssible to figure out whether theinput were 3, 1 or 2, 2. Note that erasing every bit leadsto loss of information which may also be interpreted asan increase of entropy. However, strategies for logically

reversible computation have been developed [205, 206].For example, the simplest strategy for logically reversiblecomputation is based on the prescription that neither theinitial input nor the data in any intermediate step shouldbe erased; instead, these should be retained in an auxil-iary register.In practice, computation is carried out with a device

that is governed strictly by the laws of physics that in-cludes thermodynamics. Since entropy increases in ev-ery irreversible physical process, it would be tempting toassociate the logical irreversibility of computation with aphysical irreversibility of the computational device. Sinceeach bit of a classical digital computer has only two pos-sible states, at first sight, one would expect dissipation ofkBT ln2 energy for every bit erased [207]. But, the pos-sibility of logically reversible computation raises a funda-mental question: is it possible for a physical computingdevice to carry out physically reversible (and, hence non-dissipative) computation?Operation of a polymerase (and that of a ribosome)

can be regarded as computation. More precisely, sucha computing machine can be viewed as a “tape-copyingTuring machine” that polymerizes its output tape, in-stead of merely writing on a pre-synthesized tape [205]Dissipationless operation of these machines is possibleonly if every step is error-free which, in turn, is achiev-able in the vanishingly snall speed, i.e., reversible limit[208].

D. Enzymatic processes: conformationalfluctuations, static and dynamic disorder

For a motor that doesn’t step backwards, the positionadvances in the forward direction one step at a time; how-ever, the time gap between the successive steps, i.e., dwelltime, is a random variable. Similarly, in single-moleculeenzymology, the population of the product molecules in-creases by one in each enzymatic cycle, the time gap be-tween the release of the successive product molecules israndom. The distribution of this time [209, 210] is analo-gous to that of the dwell times of molecular motors [120].Therefore, the research fields of single-motor mechanicsand single-enzyme reactions can enrich each other by ex-change of concepts and techniques [148].As a concrete example, consider the chemical reaction

E + Sω1

ω−1

I1ω2→E + P (69)

catalyzed by the enzyme E where S and P denote thesubstrate and product, respectively. The enzyme and

23

the substrate form, at a rate ω1, an intermediate com-plex I1 that can either dissociate into the free enzymeand the substrate at a rate ω−1, or get converted irre-versibly into the product and free enzyme at a rate ω2.In a bulk sample, where a large number of enzymes con-vert many substrates into products by catalyzing this re-action simultaneously, the measured rate of the reactionis actually an average over the ensemble. This rate wascalculated by Michaelis and Menten about 100 years agoand is given by the celebrated Michaelis-Menten equation[211].Formally, the scheme (69) is very similar to the scheme

(9) that we presented earlier as a very simple example ofthe mechano-chemical cycle of a molecular motor. Thetime taken by the chemical reaction (69) fluctuates fromone enzymatic cycle to another. One of the fundamen-tal questions is: what are the conditions under whichthe average rate would still satisfy the Michaelis-Mentenequation [209, 210]?Another topic that overlaps research on molecular mo-

tor kinetics and chemical kinetics is allosterism [212].A motor protein has separate sites for binding the fuelmolecule and the track. How do these two sites commu-nicate? How does the binding of ligand at one site affectthe binding affinity of the other? The mechanochemicalcycle of a motor can be analyzed [213] from the perspec-tive of allosterism which is one of the key mechanisms ofcooperativity in protein kinetics [214].

E. Applications in biomimetics andnano-technology

Initially, technology was synonymous with macro-technology. The first tools applied by primitive humanswere, perhaps, wooden sticks and stone blades. Later, asearly civilizations started using levers, pulleys and wheelsfor erecting enormous structures like pyramids. Untilnineteenth century, watch makers were, perhaps, the onlypeople working with small machines. Using magnifyingglasses, they worked with machines as small as 0.1mm.Micro-technology, dealing with machines at the lengthscale of micrometers, was driven, in the second half ofthe twentieth century, largely by the computer miniatur-ization.In 1959, Richard Feynman delivered a talk [215] at a

meeting of the American Physical Society. In this talk,entitled “There’s Plenty of Room at the Bottom”, Feyn-man drew attention of the scientific community to theunlimited possibilities of manupilating and controllingthings on the scale of nano-meters. This famous talk isnow regarded by the majority of physicists as the defin-ing moment of nano-technology [216]. In the same talk,in his characteristic style, Feynman noted that ”manyof the cells are very tiny, but they are very active, theymanufacture various substances, they walk around, theywiggle, and they do all kinds of wonderful things- all ona very small scale”.

From the perspective of applied research, the naturalmolecular machines opened up a new frontier of nano-technology [217–222]. The miniaturization of compo-nents for the fabrication of useful devices, which are es-sential for modern technology, is currently being pursuedby engineers following mostly a top-down (from largerto smaller) approach. On the other hand, an alternativeapproach, pursued mostly by chemists, is a bottom-up(from smaller to larger) approach.Unlike man-made machines these are products of Na-

ture’s evolutionary design over billions of years. In fact,cell has been compared to an “archeological excavationsite” [223], the oldest modules of functional devices arethe analogs of the most ancient layer of the exposed siteof excavation. We can benefit from Nature’s billion yearexperience in nano-technolgy. The term biomimetics hasalready become a popular buzzword [220, 221]; this fielddeals with the design of artificial systems utilizing theprinciples of natural biological systems.

X. CONCEPT OF BIOLOGICAL MACHINES:FROM ARISTOTLE TO ALBERTS

The concept of “living machine” evolved over manycenturies. Some of the greatest thinkers in human historymade significant contributions to this concept. It startedwith the man-machine (and, more generally, animal-machine) analogy. Aristotle [224] distinguished betweenthe “body” and the “soul” of an organism. However,after more than one and half millenia, an intellectu-ally provocative idea was put forward by Rene Descarteswhen he argued that animals were “living machines”.This idea took its extreme form in Julien Offray deLa Mettrie’s book [225] L’homme Machine (’man a ma-chine’). Leibnitz [226] wrote that the body of a livingbeing is a kind of “divine machine or natural automa-ton” and it “surpasses all artificial automata”, becausenot each part of a man-made machine is itself a machine.In contrast, he argued, living bodies are “machines intheir smallest parts ad infinitum”. Is this statement tobe interpreted as Leibnitz’s speculation for the existenceof machines within machines in a living organism? Thedebate over this interpretation continues [227, 228].All the great thinkers from Aristotle to Descartes and

Leibnitz compared the whole organism with a machine,the organs being the coordinated parts of that machine.Cell was unknown; even micro-organisms became visibleonly after the invention of the optical microscope in theseventeenth century. Marcelo Malpighi, father of micro-scopic anatomy, speculated in the 17th century about theexistence molecular machines in living systems. He wrote(as quoted in english by Marco Piccolino [229]) that theorganized bodies of animals and plants been constructedwith “ very large number of machines”. He went on tocharacterize these as “extremely minute parts so shapedand situated, such as to form a marvelous organ”. Un-fortunately, the molecular machines were invisible not

24

only to the naked eye, but even under the optical micro-scopes available in his time. In fact, individual molec-ular machines could be “caught in the act” only in thelast quarter of the 20th century. We highlight here theprogress made during the last three centuries when grad-ually the analogy with machine got extended to cover alllevels of biological organization- from the topmost levelof organisms down to cellular and subcellular levels. Notsurprisingly, muscles seem to have attracted maximumattention in the context of machinery of life.

Thomas Henry Huxley, dubbed as “Darwin’s bulldog”for his strong support for Darwinian ideas of evolution,delivered his famous lecture, titled “On the Physical Ba-sis of Life”, on 8th November, 1868 (see ref.[9] for thepublished version). Among the provocating statementsand insightful comments in this article, which receivedlot of hostile criticism at that time, I quote only a fewthat are directly relevant from the perspective of molec-ular machines. Huxley had the foresight to see that [9]“speech, gesture, and every other form of human actionare, in the long run, resolvable into mascular contrac-tion, and muscular contraction is but a transitory changein the relative positions of the parts of a muscle”- a re-markably insightful comment on contractility and motil-ity because the mechanism of muscle contraction was dis-covered almost 90 years later, one of the discoverers beinghis grandson!

David Ferrier, a pioneering neurologist and a youngercontemporary of Thomas Huxley, in his lecture [230] de-livered at the Middlesex Hospital Medical School, onOctober 5th, 1870, not only echoed similar ideas butstressed the role of physical energy, rather than any hy-pothetical vital action, for sustaining life of an organism.A few years later, Robert Henry Thurston, the first pres-ident of the ASME (American Society of Mechanical En-gineers) and a pioneer of modern engineering education,investigated what he called a “vital machine” (or “primemotor”) [231] from an engineer’s perspective.

In the initial stages, most of the visionaries comparedanimals with a machine. Although movements of plantsreceived much less attention, results of pioneering sys-tematic studies of these phenomena were reported al-ready in the nineteenth century by Charles Darwin ina classic book [232] which was co-authored by his son.Later, in the preface of his report on the classic investi-gations on the mechanical response of plants to stimuli,Jagadis Chandra Bose [233] wrote: “From the point ofview of its movements a plant may be regarded ...simplyas a machine, transforming the energy supplied to it, inways more or less capable of mechanical explanation”.Interestingly, the first chapter was titled “The plant as amachine” [233].

Based on the well known fact that all animals exhibitirritability and contractility [234], Thomas Henry Huxleyhad already speculated in ref.[9], that “it is more thanprobable, that when the vegetable world is thoroughlyexplored, we shall find all plants in possession of the samepowers..”. In support of this possibility he described a

phenomenon that is now known as cytoplasmic streaming[8]. His speculation that “the cause of these currentslie in contraction” was established a century later whencytoplasmic streaming was shown to be caused by anacto-myosin system.

“The story of the living machine” [235] narrated byHerbert William Conn, one of the founding members andthe third president of the Society of Americam Bacteriol-ogists (renamed, in 1960, American Society for Microbi-ology), is a critical overview of the understanding of thesemachines at the end of the nineteenth century. Becauseof his deep understanding of the basic principles of phys-ical sciences as well as evolutionary biology, his narrativeon the fundamental principles of living machines remainsas contemporary today as it was at the time of its publi-cation. Conn asked whether the operations of individualorganisms, i.e., the living machines, could be “reducedto the action of still smaller machines” [235]. Conn ar-gued that one can look upon each constituent cell of anorganism also “a little engine with admirably adaptedparts” [235]. His summary [235] that a living organism isa “series of machines one within the other” sounds verysimilar to Leibnitz’s philosophical idea. Conn went evenfurther: “As a whole it is machine, and its parts are sep-arate machines. Each part is further made up of stillsmaller machines until we reach the realm of the micro-scope. Here still we find the same story. Even the partsformerly called units, prove to be machines”. He spec-ulated [235] “we may find still further machines within”cells. He ended his summary with the statement “Andthus vital activity is reduced to a complex of machines,all acting in harmony with each other to produce togetherthe one result- life” [235].

Conn [235] not only discussed “the running of theliving machine”, but also “the origin of the living ma-chine”. In the latter context, while pointing out the dif-ferences in the principles of engineering design of man-made machines and evolutionary principles of nature’snano-machines, he wrote [235]: “It is something as ifsteam engine of Watt should be slowly changed by addingpiece after piece until there was finally produced the mod-ern quadruple expansion engine, but all this change beingmade upon the original engine without once stopping itsmotion.”

Jaques Loeb, a famous embryologist, delivered a se-ries of eight lectures at the Columbia university in 1902(see ref.[236] for a more complete published version). Inthe first lecture, Loeb started by saying “In these lec-tures we shall consider living organisms as chemical ma-chines, consisting essentially of colloidal material, whichpossess the peculiarities of automatically developing, pre-serving, and reproducing themselves” [236]. He empha-sized the crucial differences between natural and artificialmachines by the following statement: “The fact that themachines which can be created by man do not possessthe power of automatic development, self-preservation,and reproduction constitutes for the pesent a fundamen-tal difference between living machines and artificial ma-

25

chines”. However, just like some of his other visionarypredecessors, he also admitted that “nothing contradictsthe possibility that the artificial production of living mat-ter may one day be accomplished” [236].

The concept of “living machine” as a “transformer ofenergy”, from the level of a single cell to the level of amulti-cellular organism as complex as a man, found men-tion in many lectures and books of the leading biologistsin the late nineteenth and early twentieh centuries (see,for example, [237]). Research on molecular machines wasfocussed almost exclusively on the mechanism of musclecontraction during the first half of the 20th century andit was dominated by Archibald Hill and Otto Meyerhof[238, 239] who shared the Nobel prize in Physiology (ormedicine) in 1922.

By mid-twentieth century, physical sciences alreadywitnessed spectacular progress and life sciences was justembarking on its golden period. In his Guthrie lecture(also titled “The Physical Basis of Life”), delivered on21st November, 1947 (see ref.[240] for the full text), J.D. Bernal drew attention of the audience to the structureand dynamics of machines. Ten years later, in a reviewarticle, the title of which again had the words “physi-cal basis of life”, Schmitt [241] made even more concretereferences to molecular machines covering almost all thetypes of machines that we have sketched in this article.

The concept of molecular machines [242] was men-tioned on many occasions in the late twentieth century byleading molecular biologists who made outstanding con-tributions in elucitaing their mechanisms of operation.The strongest impact was made, however, by the influ-ential paper of Bruce Alberts [243], then the president ofthe National Academy of Sciencs (USA). He wrote that“the entire cell can be viewed as a factory that contains

an elaborate network of interlocking assembly lines, each

of which is composed of a set of large protein machines”[243].

Why did I start my story with Aristotle? In D’ArcyThompson’s words [244] “We know that the history ofbiology harks back to Aristotle by a road that is straightand clear, but that beyond him the road is broken andthe lights are dim”. Moreover, a section of the mod-ern enterprise in animal sciences is pursuing importantinvestigations on the efficiency of energy utilization byanimals [245] treating an entire animal as an a “combus-tion engine”, a modern and scientifically correct versionof the original philosophical idea developed by Aristotleand propagated by his followers.

Why am I closing my story with Alberts? Because ofa fortutious coincidence, Alberts’ vision for training thenext generation of biologists for studying natural nano-machines coincided with the explosive beginning of nano-technology that includes research programs on artificialnano-machines. Alberts’ article [243] was appeared at atime when the statistical physics of Brownian ratchetswas getting lot of attention [94]. The concept of molec-ular machine has matured fully into an area of inter-disciplinary research in the twenty-first century. A new

exciting era of research has just begun!

XI. SUMMARY AND OUTLOOK

So far as the current status of our understanding isconcerned, I would say that till the middle of the 20thcentury we had never seen or manipulated a single molec-ular machine although machine-like operation of a cell oran entire organism was fairly well established. In the last60 years this area has seen explosive growth in activity.The structures as well as the mechanisms of many ma-chines no longer appear any more mysterious than thoseof macroscopic machines.It is practically impossible to predict new ideas in any

field of research; molecular machines are no exception.The reason, as Peter Medawar [246] admitted in his pres-idential address at the Cambridge meeting of the BritishAssociation for the Advancement of Science, is as follows:“to predict an idea is to have an idea, and if we havean idea it can no longer be the subject of prediction”.Therefore, in this section, I do not propose any new ideabut merely mention a few systems and phenomena whichneed new ideas for their studies and understanding.Let me begin with solving the forward problem with

process modeling. Here we need new ideas to makeprogress in two opposite directions: (a) in-silicomodelingof single individual machines in terms of its coordinat-ing parts in an aqueous medium- aquatic nano-robotics;(b) integration of nano-machines and machine-assembliesinto a micro-factory, the living cell. So far as the point(a) is concerned, serious efforts have been made by de-veloping coarse-grained computational models [247–249]that can be regarded as the substitutes for full moleculardynamics based models. However, a technical (or algo-rithmic) breakthrough is needed to achieve the ultimategoal of “seeing” the operation of a machine in its natu-ral aqueous environment by carrying out an experimentsin-silico.Next let me explain the aim of developing integrated

models. It has been strongly argued at the dawn of thismillenium [250] that the machinery of life is modular.The machineries of transcription, translation, replication,chromosome segregation, etc. are all examples of mod-ules which perform specific tasks. The components ofsome of these modules are parts of a single machine, e.g.,all devices participating in translation function on a sin-gle platform provided by a ribosome. Signal transductionmachinery is an extreme example of the other types offunctional modules which are spatially distributed overa significantly large region of intracellular space withoutneed for direct physical contacts among the parts.The machines that we have considered here are all

more or less spatially-confined functional modules. Nor-mally, functional modules are not completely isolated andmust communicate and coordinate their function withother modules. For example, DNA replication and chro-mosome segregation machineries require proper coordi-

26

nation. To my knowledge, no attempt has been madeso far for quantitative stochastic process modeling incor-porating more than one functional module in a seamlessfashion. Such an enterprise may look like what is nowthe happening in systems biology: to integrate the op-erations of machineries for different functional moduleswithin a single theoretical framework.Finally, let me point out some of the standard prac-

tices of statistical inference that, to my knowledge, havenot been followed so far while reverse-engineering molec-ular machines. The experimental data for molecular ma-chines have been analyzed by several groups to extractan underlying kinetic model. However, most often theanalysis carried out during such reverse engineering isbased on a single working hypothesis. It would be desir-able to follow Platt’s [251] principle of “strong inference”which is an extension of Chamberlin’s [252] “method ofmultiple working hypothesis”. By multiple model I donot mean models hierarchically nested such that eachone is a special case of the model at the next level. Bythe term multiple model, I mean truly competing modelsthat may, however, overlap partially. The relative scoresof the competing models (and the corresponding underly-ing hypothese) would be a true reflection of their merits.For example, in case of a closely related systems in cell bi-ology and systems biology, experimental data have beenanalyzed recently within the framework of this method[253]. In the case of molecular machines, a method formodel selection has been developed [254]; it extracts thebest model by optimizing the maximum evidence, ratherthan maximum likelihood, that treats, for example, the

number of discrete states of the system as a variationalparameter. This is a step in the right direction. However,I am not aware of any work that ranks alternative modelsaccording to relative scores computed on the basis of theprinciple of strong inference.The molecular mechanism of muscle myosin and the

structure of double-stranded DNA were both discoveredin the 1950s. These discoveries set the stage for the re-search on cytoskeletal motors as well as on machines thatmake, break and manipulate nucleic acids and proteins.Last 50 years has seen enormous progress. I wanted totell you about the PIs and their contributions during thisglorious period. But, I have run out of space. So, I’ll nar-rate this fascinating story in detail elsewhere [255].Acknowledgements: I apologize to all those re-searchers whose work, although no less important, couldnot be included in this article because of space limita-tions. I thank all my students and colleagues with whomI have collaborated on molecular machines. I am also in-debted to many others (too many to be listed by name)for enlightening discussions and correspondences over thelast few years. I thank Ashok Garai for a critical read-ing of the manuscript. Active participation in the MBIannual program on “Stochastics in Biological Systems”during 2011-12 motivated me to look at molecular ma-chines from the perspectives of statisticians. I thankMarty Golubitsky, Director of MBI, for the hospitalityat OSU. This work has been supported at IIT Kanpurby the Dr. Jag Mohan Garg Chair professorship, and atOSU by the MBI and the National Science Foundationunder grant DMS 0931642.

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