arXiv:1203.2673v2 [physics.bio-ph] 14 Mar 2012 ∗ I.Mtrn navsosfud rmNwo oLangevin to Newton from fluid: viscous a in Motoring III. I aaou:tpcleape fmlclrmcie and machines molecular of examples typical catalogue: A II. Introduction 1. contents: of Table lcrncades [email protected] address: Electronic IE rcsiiy uyrto tl oc n wl time dwell and questions force general stall Fundamental ratio, II.F. duty Processivity, II.E. motors Rotary II.D. degradatio and manipulation synthesis, for Machines II.C. IB yokltlmotors machines Cytoskeletal molecular II.B. for Fuels II.A. I..Lnei qain tcatcdynamics stochastic equation: dynamics Langevin deterministic III.B. equation: Newton’s III.A. rbblsi ees niern n h I-fo Aristo from PIs- the and engineering reverse probabilistic IC1 ebaeascae ahnsfrmacromolecule for machines Membrane-associated II.C.1. IC4 nrpesaduzpeso akgdDA chromati DNA: packaged of unzippers and polymerization life Unwrappers template-dictated of II.C.4. for macromolecules Machines degrading II.C.3. for Machines II.C.2. nano-pist filaments: cytoskeletal of pull and Push II.B.4. and crossbridge tracks motor-filament filamentous Contractility: chipping II.B.3. for Machines II.B.2. Porters II.B.1. ace,de o vnhv n arsoi onepr.Bu counterpart. macroscopic any popular have even transduction, not energy of does mechanisms ratchet, the possible from the far thermodynami conditions of equilibrium isothermal of under law operate second machines, the operations. f appropriate on output for rests an system Carnot, into transmission energy a input coun to “transduces” macroscopic pling that their engine like an Just has molecules. RNA and protein ASnmes 71.c89.20.-a 87.16.Ac numbers: PACS technology. writ and is science di It of several from areas data. concepts mo experimental overlapping (b) of showing overvieperspective, analysis and non-technical from processes, drawn a kinetic ference give influentia mechano-chemical we very of establ a Here well modeling of a century. publication become twentieth the has the from It by evident years. pioneered as fifty machine, research subcel last molecular and the of in cellular idea vigorously the The at microscope. analogies li man- cal similar philosophers The to leading extended many ratchet. was which Brownian in as debate operating philosophical than rather stroke, ratchet Brownian with mous 2 I...Mtrn na‘sik’ eim ucl’ idea Purcell’s medium: ‘‘sticky’’ a in Motoring III.B.1. oeua ahnscnito ihrasnl rti rama a or protein single a either of consist machines Molecular ahmtclBocecsIsiue h hoSaeUniver State Ohio The Institute, Biosciences Mathematical ahnso ie aaou,sohsi rcs modeling, process stochastic catalogue, life: of Machines 1 rnlcto:epres motr n packers and importers exporters, translocation: rowers and sliders of dynamics collective eateto hsc,Ida nttt fTcnlg,Kan Technology, of Institute Indian Physics, of Department ag ubro oeua ahnsatal xct noi a execute actually machines molecular of number large a ; eahs Chowdhury Debashish Dtd uy2 2018) 2, July (Dated: eetbace fpyisadfo other from and physics of branches fferent h hoyo etegns inee by pioneered engines, heat of theory The mdnmceulbim oevr one Moreover, equilibrium. rmodynamic eAitteadDsatsparticipated, Descartes and Aristotle ke ,2, 1, zdb ena n aldBrownian called and Feynman by ized ua eesatrteivnino opti- of invention the after levels lular ae yAbrstwrsteedof end the towards Alberts by paper l se oi fcretinterdisciplinary current of topic ished e o o-xet n rmabroad a from and non-experts for ten eprs aho hs nano-machines these of each terparts, s oee,teegnso molecular of engines the However, cs. ftesrtge o a stochastic (a) for strategies the of w r hc ste tlzdb t cou- its by utilized then is which orm e eeto ae nsaitclin- statistical on based selection del acl apgi a enpursued been has Malpighi, Marcelo t, iy oubs H420 USA. 43210, OH Columbus, sity, ahn nlg,atpco intense of topic a analogy, machine ∗ rmlclrcmlxcmoe of composed complex cromolecular fuels oeua ahn sntsynony- not is machine molecular fmcooeue flife of macromolecules of n u 006 India 208016, pur eoelr n helicases and remodellers n n n nano-springs and ons l oAlberts to tle ypower sy 2

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 grade waste into products that are recycled as raw mate- rials for fresh synthesis. All the automated processes in this high-tech micro-city are run by nano-machines [1]. Cell is the structural and functional units of life. How This is not the plot of a science fiction, but a dramatized “active” is the interior of a living cell? Imagine an un- picture of the dynamic interior of a cell. der water “metro city” which is, however, only about 10µm long in each direction! In this city, there are “high- A molecular machine is either a single protein or a ways” and “railroad” tracks on which motorized “ve- macromolecular complex consisting of proteins and RNA hicles” transport cargo to various destinations. It has molecules [2]. Just like their macroscopic counterparts, an elaborate mechanism of preserving the integrity of these nano-machines take an input (most often, chemical the chemically encoded blueprint of the construction and energy) and it transduces input energy into an output. If maintenance of the city. The “factories” not only sup- the output is mechanical work the machine is usually re- ply their products for the construction and repair works, ferred to as a molecular motor [3–5]. Similarly, a cell can but also manufacture the components of the machines. also be regarded as a micron-size “energy transforming This eco-friendly city re-charges spent “chemical fuel” in device”. The concept of “machine” is not restricted only uniquely designed “power plants”. This city also uses to subcellular or cellular levels of biological organization. a few “alternative energy” sources in some operations. In ancient times, philosophers proposed the concept of Finally, it has special “waste-disposal plants” which de- “living machine” to describe a whole animal, including a 3 man; we’ll discuss the evolution of this concept towards II. A CATALOGUE: TYPICAL EXAMPLES OF the end of this article. MOLECULAR MACHINES AND FUELS Not only animals, but even plants also move in re- sponse to external stimuli [6, 7] and movements take Based on the mode of operation, the biomolecular ma- place in plants at all levels- from whole plant and plant chines can be divided broadly into two groups. Cyclic cell to the subcellular level. I cannot resist the tempta- machines operate in repetitive cycles in a manner that tion of quoting Thomas Huxley’s poetic description of cy- is very similar to that of the cyclic engines which run toplasmic streaming [8], in particular, and of intracellular our cars. In contrast, some other molecular machines are motor traffic, in general, in plants [9]: “..the wonderful one-shot machines that exhaust an internal source of free noonday silence of a tropical forest is, after all, due only energy in a single round. Force exerted by a compressed to the dulness of our hearing; and could our ears catch spring, upon its release, is a typical example of a one-shot the murmur of these tiny maelstroms, as they whirl in the machine. innumerable myriads of living cells which constitute each tree, we should be stunned, as with the roar of a great city”. A. Fuels for molecular machines In this article, however, we focus attention almost ex- clusively on molecular machines that drive subcellular The most common way of supplying energy to a nat- processes within living cells. The processes driven by ural nanomachine is to utilize the chemical energy (or, molecular machines include not only intracellular mo- more appropriately, free energy) released by a chemi- tor transport, but also manipulation, polymerization and cal reaction. Most of the machines use the so-called degradation of the bio-molecules [10, 11]. Explaining the high-energy compounds- particularly, nucleoside triphos- physical principles that govern these machine-driven pro- phates (NTPs)- as an energy source to generate the me- cesses will bring us closer to the ultimate goal of biologi- chanical energy required for their directed movement. cal physics: understanding “what is life” in terms of the The most common chemical reaction is the hydrolysis of laws of physics (and chemistry). ATP (ADP): AT P → ADP +Pi. Some other high-energy Biomolecular machines operate in a domain where the compounds can also supply input energy; one typical ex- appropriate units of length, time, force and energy are, ample being the hydrolysis of Guanosine Triphosphate nano-meter, milli-second, pico-Newton and kB T , respec- (GTP) to Guanosine Diphosphate (GDP). Under normal tively (kB being the Boltzmann constant and T is the ab- physiological conditions, hydrolysis of ATP is extremely solute temperature). Aren’t the operational mechanism slow. Most of the molecular motors fuelled by ATP func- of molecular machines similar to their macroscopic coun- tion also as ATPase enzyme (i.e., catalyzes hydrolysis of terparts except, perhaps, the difference of scale? NO. In ATP) thereby speeding up the energy-supplying reaction spite of the striking similarities, it is the differences be- by several orders of magnitude. tween molecular machines and their macroscopic coun- If a cyclic machine runs on a specific chemical fuel then terparts that makes the studies of these systems so inter- the spent fuel must be removed as waste products and esting from the perspective of physicists. fresh fuel must be supplied to the machine. Fortunately, This article is a brief guide for a beginner embark- normal cells have machineries for recycling waste prod- ing on an exploration of the exciting frontier of research ucts to manufacture fresh fuel, e.g., synthesizing ATP on molecular machines. It begins with a catalog of from ADP. This raises an important question: since ATP the known molecular machines and a list of fundamen- is a higher-energy compound than ADP, how are the tal questions on their operational mechanism that the ATP-synthesizing machines driven to perform this ener- explorer should try to address. Like any travel guide, getically “uphill” task? Fortunately, chemical fuel is not this article also cautions the explorer about the counter- the only means by which input energy can be supplied to intuitive phenomena that await him/her in this new ter- intracellular molecular machines. ritory of his/her exploration where he/she may need new A cell gets its energy from external sources. It has sophisticated technical tools that were not needed for special machines to convert the input energy into some handling macroscopic machines. For the explorer, this “energy currency”. For example, chemical energy sup- article also charts a tentative road map along with a sum- plied by the food we consume is converted into an electro- mary of the mathematical strategies that he/she might chemical potential ∆µ that not only can be used to syn- use to make progress in this frontier territory. Excursions thesize ATP, but can also directly run some other ma- to some of the listed border areas that overlap with areas chines. In plants similar proton-motive forces are gen- of research in other branches of science and engineering erated by machines which are driven by the input sun- could be enjoyable and intellectually rewarding for the light. Only hydrogen ion (H+), i.e., a proton, and sodium explorer. For any explorer, it is good to know the experi- ion (Na+) are used in the kingdom of life to create the ence of the pioneers. The final section, on the evolution electro-chemical potentials, i.e., it used either a proton- of the concept of biological machines, is likely to be en- motive force (PMF) or a sodium-motive force (SMF) as joyed as a dessert by physicists, and as a fresh food for an energy currency. thought by historians and philosophers of science. The advantages of using light, instead of chemical re- 4

Motor superfamily Filamentous track Minimum step size the motor itself! It is desirable that such a motor “walks” Myosin F-actin 36 nm for a significant distance on its track carrying the cargo; Kinesin Microtubule 8 nm for obvious reasons, such motors are referred to as porters Dynein Microtubule 8 nm [13]. Kinesin and dyneins attached simultaneously to the same “hard” cargo can get engaged in a tug-of-war lead- TABLE I: Superfamilies of motor proteins and the corre- sponding tracks. 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 kinesins action, as the input energy for a molecular motor are as and myosins which, however, walk on their respective follows: (i) light can be switched on and off easily and tracks. Alternatively, along its journey route, a cargo rapidly, (ii) usually, no waste product, which would re- may be transferred from the MT-based transport net- quire disposal or recycling, is generated. work, which dominate at the cell center, to the F-actin Thus, study of molecular machines deals with two com- based network that covers most of the cell periphery [16]. plementary aspects of bioenergetics: (a) conversion of A “soft” cargo pulled by many kinesins can get elongated energy input from the external sources into the energy into a tube [17]. currency of the cell, and (b) conversion of the energy currency to drive various active other active processes.

B. Cytoskeletal motors

2. Machines for chipping filamentous tracks 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 also A MT depolymerase is a kinesin motor that chips away form the networks of “highways” (or, “tracks”) on which its own track from one end [18]. Members of the kinesin- cytoskeletal motor proteins [3, 4] can move. Filamen- 13 family can reach either end of the MT diffusively tous actin (F-actin) and microtubules (MT) which serve (without ATP hydrolysis) and, then, start chipping the as tracks are “polar” in the sense that the structure and track from the end where it reaches. In contrast, mem- kinetics of the two ends of each filament are dissimilar. bers of the kinesin-8 family walk towards the plus end of The superfamilies of cytoskeletal motors and the cor- the MT track hydrolyzing ATP and after reaching that responding filamentous tracks are listed in table I. Every end starts chipping it from there [19, 20]. Chipping by superfamily can be further divided into families. Mem- both families of depolymerase kinesins are energized by bers of every family move always in a particular direc- ATP hydrolysis. tion on its track; for example, kinesin-1 and cytoplasmic dynein 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 hydrolyzes ATP (so-called ATPase site). Both these sites are lo- 3. Contractility: motor-filament crossbridge and collective cated, for example, in the head domain of myosins and dynamics of sliders and rowers kinesins both of which walk on their heads! The motor- binding sites on the tracks are equispaced; the actual step size of a motor can be, in principle, an integral multiple Some motors are capable of sliding two different fila- of the minimum step size which is the separation between ments with respect to each other by stepping simulate- two neighboring motor-binding sites on the correspond- nously on these two filaments [21]. Some sliders work in ing track. groups and each detaches from the filament after every single stroke; these are often referred to as rowers because of the analogy with rowing with oars [13]. Contractility, 1. Porters rather than motility, at the subcellular and cellular level are driven by the sliders and rowers [22, 23]. Some exam- Some linear motors are cargo transporters [12]; how- ples of this category are listed in the table II; the details ever, the size of the cargoes are usually much larger than can be found in the cited review articles. 5

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 as well 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-pistons Membrane Polymer and nano-springs Nuclear envelope RNA/Protein [43, 47] Membrane of endoplasmic reticulum Protein [48] Membranes of mitochondria/chloroplasts Protein [49, 50] Elongation of filamentous biopolymers that presses Membrane of peroxisome Protein [51] against a light object (e.g., a membrane) can result in a “push” [32]. Similarly, a depolymerizing tubular fila- TABLE IV: Membrane-bound translocases. ment can “pull” a light ring-like object by inserting its hook-like outwardly curled depolymerizing tip into the ring [33]. A flexible filament, upon compression by in- put energy, can store energy that can perform mechan- from three sources: (a) bending of stiff DNA molecule ical work when the filament springs back to its original inside the capsid; (b) strong electrostatic repulsion be- relaxed shape [34]. Some typical examples are given in tween the negatively charged strands of the DNA; (c) tableIII. loss of entropy caused by the packaging [46]. The pack- aging motor has to be powerful enough to overcome such a high pressure. C. Machines for synthesis, manipulation and degradation of macromolecules of life

1. Membrane-associated machines for macromolecule translocation: exporters, importers and packers 2. Machines for degrading macromolecules of life In many situations, the motor remains immobile and pulls a macromolecule; the latter are often called translo- case. Some translocases export (or, import) either a pro- Restriction-modification (RM) enzyme defend bacte- tein [41] or a nucleic acid strand [42, 43] across the plasma rial hosts against bacteriophage infection by cleaving membrane of the cell or, in case of eukaryotes, across in- the phage genome while the DNA of the host bacteria ternal membranes. A list is provided in table IV. are not cleaved. Exosome and proteasome are machines The genome of many viruses are packaged into a pre- that shred RNA and proteins into their basic subunits, fabricated empty container, called viral capsid, by a pow- namely, nucleotides and amino acids, respectively. Sim- erful motor attached to the entrance of the capsid. As ilarly, there are machines for degrading polysachharides, the capsid gets filled, The pressure inside the capsid in- e.g., cellulosome (a cellulose degrading machine), starch creases which opposes further filling [44, 45]. The effec- degrading enzymes, chitin degrading enzyme (chitinase), tive force, which opposes packaging, gets contributions etc. These machines are listed in table V. 6

Polymer Examples of Machines a RNA secondary structure. During DNA replication, DNA (polynucleotide) RM enzyme [52] a helicase moves ahead of the polymerase, like a mine RNA (polynucleotide) Exosome [53] sweeper, unzipping the duplex DNA and dislodging other Protein (polypeptide) Proteasome [54] DNA-bound proteins. However, the transcriptional and Cellulose (polysachharide) Cellulosome [56] translational machineries do not need assistance of any Starch (polysachharide) Starch degrad. enzyme [55] helicase because these are capable of unzipping DNA and Chitin (polysachharide) Chitinase [57] unwinding RNA, respectively, on their own. A helicase TABLE V: Machines for degradation of macromolecules of can be monomeric, or dimeric or hexameric. life.

Machine Template Product Function D. Rotary motors DdDP DNA DNA DNA replication [58] DdRP DNA RNA Transcription [59–61] Rotary molecular motors are, at least superficially, RdDP RNA DNA Reverse transcription [62] very similar to the motor of a hair dryer. Two rotary RdRP RNA RNA RNA replication [63] motors have been studied most extensively. (i) A rotary Ribosome mRNA Protein Translation [64–66] motor embedded in the membrane of bacteria drive the bacterial flagella [70, 71] which, the bacteria use for their TABLE VI: Types of polymerizing machines, the templates they use and the corresponding product of polymerization. swimming in aqueous media. (ii) A rotary motor, called ATP synthase [72, 73], is embedded on the membrane of mitochondria, the powerhouses of a cell. A synthase drives a chemical reaction, typically the synthesis of some 3. Machines for template-dictated polymerization product; the ATP synthase produces ATP, the “energy currency” of the cell, from ADP. Two classes of biopolymers, namely, polynucleotides and polypeptides perform wide range of important func- tions in a living cell. DNA and RNA are examples of E. Processivity, duty ratio, stall force and dwell polynucleotides while proteins are polypeptides. Both time polynucleotides and polypeptides are made from a lim- ited number of different species of monomeric building One can define processivity of a motor in three different blocks, namely, nucleotides and amino acids,respectively. ways: The sequence of the monomeric subunits to be used for (i) Average number of chemical cycles in between attach- synthesis of each of these are dictated by that of the ment and the next detachment from the track; corresponding template. These polymers are elongated, (ii) mean time of a single run, i.e., in between an attach- step-by-step, during their birth by successive addition of ment and the next detachment of the motor from the monomers, one at a time. The template itself also serves track; as the track for the polymerizer machine that takes chem- (iii) mean distance spanned by the motor on the track in ical energy as input to polymerize the biopolymer as well a single run. as for its own forward movement. Therefore, these ma- Since the definitions (ii) and (iii) are directly accessible chines are also referred to as motors. to experiments, these are more useful than the definition Depending on the nature of the template and prod- (i). Another related, but distinct, concept is that of duty uct nucleic acid strands, polymerases can be classified ratio which is defined as the average fraction of the time as DNA-dependent DNA polymerase (DdDP), DNA- that each head spends remaining attached to its track dependent RNA polymerase (DdRP), etc. as listed in during one cycle. the table VI. To translocate processively, a motor may utilize one of the three following strategies: Strategy I: the motor may have more than one track- 4. Unwrappers and unzippers of packaged DNA: chromatin binding domain (oligomeric structure can give rise to remodellers and Helicases such a possibility quite naturally). Most of the cytoskele- tal motors like conventional two-headed kinesin use such In an eukaryotic cell DNA is packaged in a hierarchi- a strategy [74]. One of the track-binding sites remains cal structure called chromatin. In order to use a sin- bound to the track while the other searches for its next gle strand of the DNA as a template for transcription binding site. or replication, it has to be unpackaged either locally or Strategy II: A motor may possess non-motor extra do- globally. ATP-dependent chromatin remodelers [67] are mains or some accessory protein(s) bound to it which motors that perform this unpackaging. However, only can interact with the track even when none of the motor one of the strands of the unpackaged duplex DNA serves domains of the motor is attached to the track. Dynein as a template; the duplex DNA is unzipped by a DNA seems to exploit dynactin [75] to enhance its own proces- helicase [68, 69]. Similarly, a RNA helicase unwinds sivity. 7

Strategy III: it can use a “clamp-like” device to remain which dominate the dynamics of molecular machines have attached to the track; opening of the clamp will be re- negligible effect on macroscopic machines. quired before the motor detaches from the track. For example, DdDP utilizes this strategy [76]. An external force directed opposite to the natural di- A. Newton’s equation: deterministic dynamics rection of walk of a motor is called a load force. Average velocity of a motor decreases with increasing load force. For simplicity, let us first consider a hypothetical sce- The magnitude of the load force at which the average nario where neither the tracks nor the fuel molecules are velocity of the motor vanishes, is called the stall force. present in the aqueous medium in which there is a free The force-velocity relation is one of the most fundamen- (i.e., not bound to any other molecule) motor protein. tal characteristic properties of a motor. Its status in bio- Suppose the medium consists of only N “particle-like” physics is comparable, for example, to that of the I-V molecules and the center of mass of the motor protein characteristics of a device in semiconductor physics. is also represented by a “particle”. Then, the exact tra- Two motors with identical average velocities may ex- jectories of all these N + 1 “particles” can be obtained hibits widely different types of fluctuations. Therefore, by solving the corresponding coupled Newton’s equations as we’ll show in detail, a deeper understanding of the op- that describe the dynamics of the N + 1 particles. In erational mechanism of a motor can be gained from the reality, this approach is impractical because, even with distributions of their “dwells” at the successive spatial the largest and fastest computers available at present, positions on the track. we cannot get the trajectories over time intervals of the order of 1s which is relevant for majority of the motor proteins as long as N is large (N ∼ 1023). F. Fundamental general questions

B. Langevin equation: stochastic dynamics (i) Single-molecule mechanism [77]: How do the in- teractions among the component structural units of an A more pragmatic approach would be to solve only the individual motor, motor-track interactions, motor-ligand equations of motion for the motor protein by treating the (fuel) interactions and the mechano-chemical kinetics of N particles of the medium as merely the constituents of the system determine, for example, (a) the directional- a reservoir. In other words, one monitors the motion in a ity, (b) processivity, (c) dwell-time distribution, (d) force- 6-dimensional subspace of the full 6(N + 1)-dimensional velocity relation, and (e) efficiency of transduction? Does phase space of the system. The price one has to pay for a given dimeric motor walk hand-over-hand or crawl like this simplification is that instead of the original Newton’s an inchworm, and why? Do the ATPase domains of a equation for the motor protein, one has to now solve a hexameric motor “fire” (a) in series, or (b) in parallel, Langevin equation that describes the “Brownian” motion or, (c) in random sequence? of the motor protein. Since we are soon going to extend Multi-motor coordination in a “workshop” (ii) : The the discussion in the presence of filamentous tracks which in-vivo motors do not work in isolation . What are the are essentially one-dimensional, we write the Langevin mechanisms and consequences of the spatio-temporal co- equation for the “Brownian” motion of the motor protein ordination of the motors? For example, a single mitotic in one-dimension m(d2x/dt2) = F + ξ where m is the spindle consists of many polymerizing and depolymeriz- d mass of the protein, Fd = −γv is the viscous drag and and ing MTs, several types of cytoskeletal motors, including ξ is the random Brownian force. Note that the Langevin depolymerases [28]. A replisome, the workshop for DNA equation is neither deterministic nor symmetric under replication, has to coordinate the operations of clamps time-reversal (t →−t). and clamp loaders with those of primases, polymerases, etc. [58]. Similarly, a ribosome is a mobile platform on which the operations of several devices have to be coordi- 1. Motoring in a “sticky” medium: Purcell’s idea nated properly during protein synthesis [64]. Well known co-directional as well as head-on collisions of polymerases and those of ribosomes can create traffic jam under some Using the Stokes formula γ =6πηr for a globular pro- circumstances whereas a collision can restart stalled traf- tein of radius r ∼ 10nm moving with an average ve- fic in other circumstances [78]. locity v ∼ 1m/sec (corresponding to 1nm/ns) in the aqueous medium of viscosity η ∼ 10−3Pas we get an estimate Fd ∼ 200pN. Surprisingly, this viscous drag force is about 200 times larger than the elastic force it III. MOTORING IN A VISCOUS FLUID: FROM experiences when stretched by 1nm ! Consequently, the NEWTON TO LANGEVIN Reynold’s number, which is the ratio of the inertial and viscous forces, is at most of the order O(10−2). The Force is one of the most fundamental quantities in Reynold’s number will be of the order of 10−2 if you ever physics. As we’ll argue in this section, some of the forces try (not recommended) to swim in honey! 8

Now let us supply the tracks and fuel molecules to the high deformability of molecular motors for its biological motor protein in the same medium. The Langevin equa- function. But, this difference is minor compared to the tion will now include additional terms Fext which rep- others and will not be elaborated further here. resents the net force arising from the interaction of the motor protein with its track and the ligand; the ligand could be the fuel molecule or the molecules produced by A. Molecular machines are run by isothermal its “burning” (e.g., ATP or ADP). Besides, since the dy- engines namics of motor proteins is expected to be dominated by hydrodynamics at low Reynold’s number [79], one gener- This is in sharp contrast to the macroscopic thermal ally drops the inertial term. In this “overdamped” regime engines which require at least two thermal reservoirs at different temperatures and which convert part of the in- γv = Fext + ξ (1) 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 intracellular 2. Motoring under random impacts from surroundings: heat engine, let us create a temperature gradient on a Brownian force length scale ℓ ∼ 10nm. An elementary analysis of heat diffusion equation is adequate to show that a tempera- The energy released by the hydrolysis of a single ATP ture gradient on a length scale ℓ relaxes within a time −21 2 molecule is about 10 J. Interestingly, the mean ther- interval τtemp ∼ chℓ /κ where ch is the specific heat per mal energy kBT associated with a molecule at a temper- unit volume and κ is the thermal conductivity. Using −21 ature of the order of T ∼ 100K, is also kBT ∼ 10 J. the typical characteristic values of ch and κ for water, −6 −8 Moreover, equating this thermal energy with the work one finds [81] τtemp ≃ 10 − 10 s. Thus, temperature done by the thermal force Ft in causing a displacement gradients cannot be maintained for the entire duration of of 1nm we get Ft ∼ 1pN. This is comparable to the elas- even one single cycle of a cyclic molecular machine which tic force experienced by a typical motor protein when is, typically, few orders of magnitude longer than τtemp. stretched by 1nm. Thus, a motor protein that gets bom- In other words, for all practical purposes, natural nano- barded from all sides by random thermal forces is similar machines operate isothermally. Because of the condition to a tiny creature is a very strong hurricane! Therefore, T = constant, the molecular machines operate as free in contrast to the deterministic dynamics of the macro- energy transducers. scopic machines, the dynamics of molecular motors is stochastic (i.e., probabilistic). Already in the first half of the twentieth century B. Inadequacy of equilibrium thermodynamics D’Arcy Thompson, father of modern bio-mechanics, real- ized the importance of viscous drag and Brownian forces A majority of the molecular motors are chemo- at the cellular (and subcellular) level. He pointed out mechanical machines for which input and output are that [80] that in this microscopic world “gravitation is chemical energy and mechanical work, respectively. Re- forgotten, and the viscosity of the liquid,..., the molecu- call that for Carnot’s thermo-mechanical machine, the lar shocks of the Brownian movement, .... the electric thermodynamic efficiency is ηeq−th =1 − (TL/TH ) where charges of the ionized medium, make up the physical en- TH and TL are the temperatures of the two reservoirs vironment ... predominant factors are no longer those of at high and low temperatures, respectively. Similarly, our scale”. for an isothermal chemo-mechanical engine, the thermal reservoirs would be replaced by chemical reservoirs at fixed chemical potentials µH and µL (µH > µL). Con- IV. ENERGY TRANSDUCTION BY sequently, the heat flow of the thermal engine would be MOLECULAR MACHINES: FROM CARNOT TO replaced by particle flow in the chemical engine. Just FEYNMAN as the difference of heat input and output in the heat engine is converted to mechanical work, the difference Molecular motors generate force by transducing en- µH − µL would be converted into output work in the ergy. Interestingly, one of the two mechanisms of en- chemical engine. Obviously, the corresponding thermo- ergy transduction that I describe here, does not have dynamic efficiency would be [82] ηeq−ch =1 − (µL/µH ). any analogous macroscopic counterpart. I explain in con- Note that, because of the quasi-static nature of each step siderable detail why thermodynamic formalisms, which in the equilibrium thermodynamic theory of these cyclic have been successfully utilized to calculate the common engines, each cycle takes infinite time. Naturally, the performance characteristics of macroscopic machines, are power output Pout = 0 for both the thermal and chemi- inadequate for natural nano-machines. cal Carnot engines. Molecular motors are made of soft matter whereas Can we apply the theory of such isothermal Carnot en- macroscopic motors are normally made of hard matter gines to a chemo-mechanical molecular machine by iden- to withstand wear and tear. Nature seems to exploit the tifying, for example, µH and µL (µH >µL) as the chem- 9 ical potentials of ATP and ADP, respectively? The an- D. Domain of non-equilibrium statistical mechanics swer is: NO. The cycle time of the real molecular ma- chines is finite and their power output is non-zero. More- In general, molecular motors operate far beyond the over, motors are examples of open systems [83] which linear response regime and, therefore, the formalism continue to run in repetitive cycles as long as energy is of non-equilibrium thermodynamics [87] for coupled pumped in. Such a system cannot be in thermodynamics mechano-chemical processes is not applicable. More- equilibrium. over, thermodynamic formalisms developed for macro- scopically large systems do not account for the sponta- neous fluctuations. On the other hand, because of the small size of a molecular motor and because of the low concentrations of the molecules involved in its operation, C. Inadequacy of endo-reversible thermodynamics fluctuations of positions, conformations as well as the cycle times are intrinsic features of their kinetics. There- fore, one has to use the more sophisticated toolbox of For macroscopic engines with finite cycle time, the stochastic processes and non-equilibrium statistical me- characteristics of performance are often reliably calcu- chanics for theoretical treatment of molecular machines. lated within the theoretical framework of endo-reversible Interestingly, as we argue below, noise need not be a nui- thermodynamics [84]. In this case, the ratio of the rates sance for a motor; instead, a motor can move forward by of output and input energies defines the efficiency of gainfully exploiting this noise! transduction. The rate of output work is called power output of the engine. E. Defining efficiency: from Carnot to Stokes The formalism of endo-reversible thermodyamics for heat engines is based on the assumption that the work- ing substance of the engine is coupled to the two thermal The performance of macroscopic motors are charac- reservoirs by heat conductors of finite thermal conduc- terized by a combination of its efficiency, power output, tivity (non-zero thermal resistance). Entire dissipation maximum force or torque that it can generate. Just like is assumed to take place in the irreversible process of the performance of their macroscopic counterparts with heat conduction through these thermal resistors whereas finite cycle time, that of molecular motors [88] have also all the processes in the working material of the engine been characterized in terms of efficiency at maximum are assumed to be fully reversible (i.e., no entropy is gen- power, rather than maximum efficiency. However, the erated internally). In this scenario, the corresponding efficiency of molecular motors can be defined in several thermal conductances in the Carnot engine are infinite. different ways [89]. The efficiency of a motor, with finite cycle time, is Efficiency at maximum power output η(Pmax), rather generally defined by than maximum efficiency itself, is the most appropriate quantitative measure of the performance of such engines. η = Pout/Pin (2) The upper-bound of η(Pmax) is given by the Curzon- 1/2 where Pin and Pout are the input and output powers, Ahlborn expression [85] ηCA(Pmax)=1 − (TL/TH ) . Similarly, within the framework of the endoreversible respectively. The usual definition of thermodynamic effi- thermodynamics, a phenomenological theory for chem- ciency ηT is based on the assumption that, like its macro- ical engines can also be formulated if the cycle time is scopic counterpart, a molecular motor has an output finite [86]. Can’t we use this approach for molecular ma- power [90] chines which are also chemical engines with finite cycle Pout = −FextV. (3) times? Recall that endo-reversible thermodynamics is based where Fext is the externally applied opposing (load) force. on the assumption that dissipation takes place only in Although this definition is unambiguous, it is unsatisfac- the process of transfer of matter between reservoirs and tory for practical use in characterizing the performance the working substance whereas processes that the work- of molecular motors. As explained earlier, a molecular ing substance goes through in each cycle are perfectly motor has to work against the omnipresent viscous drag reversible (and, therefore, non-dissipative). This assump- in the intracelluar medium even when no other force op- tion is valid if the relaxations in the working substance poses its movement (i.e., even if Fext = 0). of the engine are very rapid compared to the processes A generalized efficiency ηG is also represented by the involving the coupling of the working substance with the same expression (2) where, instead of (3), the output reservoirs. The validity of this assumption requires clear power is assumed to be [91] separation of time scales of relaxation in the working 2 Pout = FextV + γV . (4) substance and in the working substance-reservoir cou- pling. But, such separation of time scales does not hold This definition treats the load force and viscous drag on in molecular machines which consist of macromolecules. equal footing. In contrast, the “Stokes efficiency” ηS for a 10 molecular motor driven by a chemical reaction is defined it imply that these motors are perpetual motors of the as [92] second kind that violate the second law of thermody-

2 namics? In that case, does it have any similarity with γV the Maxwell’s demon [101]? A short answer is: No, a ηS = (5) (∆G)hri + FextV Brownian ratchet is not a Maxwell’s demon because it is an open system far from the state of equilibrium whereas where hri is the average rate of the chemical reaction and the second law of thermodynamics is strictly valid only ∆G is the chemical free energy consumed in each reaction for systems in stable thermodynamic equilibrium. cycle. This efficiency is named after Stokes because the viscous drag is calculated from Stokes law. As we show in the next subsection, the directional movement of some motors arises from the rectification of random thermal noise. For such motors, an alternative measure of performance is the rectification efficiency [93]. G. Physical realizations of Brownian ratchet in Thus, different definitions of the efficiency of a molecular molecular motors motor may arise from different interpretations of their tasks or may characterize distinct aspects of their oper- ation. Nevertheless, for any definition of efficiency, it is Is there any real biomolecular motor which can be re- essential to ensure that its allowed values lie between 0 garded as a true physical realization of Brownian ratch- and 1. ets? The answer is an emphatic “yes” and we list a few typical examples here. KIF1A, a single-headed kinesin, is an example of cy- F. Noisy power stroke and Brownian ratchet toskeletal motor whose operational mechanism can be in- terpreted as a Brownian ratchet [102–104]. The original If the input energy directly causes a conformational acto-myosin crossbridge model suggested by Huxley [105] change of the protein machinery which manifests itself can be interpreted as Brownian ratchet [106] although as a mechanical stroke of the motor, the operation of the Huxley himself did not use this terminology. Brownian motor is said to be driven by a “power stroke” mecha- ratchet can also account for the force generation by a nism. This is also the mechanism used by all man made polymerizing cytoskeletal filament [107]. macroscopic machines. However, in case of molecular motors, the power stroke is always “noisy” because of The crossing of a membrane during the export/import the Brownian forces acting on it. of a protein of length L (amino acid monomers) takes Let us contrast this with the following alternative sce- place at a faster rate by the Brownian ratchet mechanism nario: suppose, the machine exhibits both “forward” and compared to that by pure translational diffusion [108]. “backward” movements because of spontaneous thermal It has been claimed that the translocation of a mRNA fluctuations. If now energy input is utilized to prevent molecule across the nuclear membrane of an eukaryotic “backward” movements, but allow the “forward” move- cell takes place also by a similar Brownian ratchet mech- ments, the system will exhibit directed, albeit noisy, anism through the nuclear pore complex [109]. The elon- movement in the “forward” direction. Note that the for- gation of a mRNA during transcription by a RNA poly- ward movements in this case are caused directly by the merase can also be a physical realization of the Brownian spontaneous thermal fluctuations, the input energy rec- ratchet mechanism [110, 111]. Translocation, one of the tifies the “backward” movements. This alternative sce- crucial steps of the mechano-chemical cycle of a ribosome nario is called the Brownian ratchet mechanism [94, 95]. is a physical realization of Brownian ratchet [65]. The concept of Brownian ratchet [96] was popularized by Richard Feynman with his ratchet-and-pawl device [97]. However, in the context of molecular motors, the Brownin ratchet mechanism was proposed by several groups in the 1990s [94, 98]. V. MATHEMATICAL DESCRIPTION OF Thus, in principle, there are two idealized scenarios for MECHANO-CHEMICAL KINETICS: a transition from a conformation A to a conformation B- CONTINUOUS LANDSCAPES VS. DISCRETE one is by a pure power stroke and the other by a purely NETWORKS Brownian ratchet mechanism [99]. However, for a real molecular motor, it is difficult to unambiguously distin- guish between a power stroke and a Brownian ratchet We combine the fundamental principles of (stochas- [100]; the actual mechanism may be a combination of tic) nano-mechanics and (stochastic) chemical kinetics to the two idealized extremes. formulate the general theoretical framework for a quan- •Are Brownian motors Maxwell’s demon? titative description of the mechano-chemistry or chemo- Brownian ratchet mechanism transduces random ther- mechanics of molecular motors. We mention a few alter- mal energy of the reservoir into mechanical work. Does native formalisms. 11

A. Motor kinetics as wandering in a of the potential Vµ(x) is governed by the master equation time-independent mechano-chemical free-energy landscape ∂Pµ(x, t) = Pµ′ (x, t)Wµ′ →µ(x) − Pµ(x, t)Wµ→µ′ (x) ∂t ′ ′ Xµ Xµ This approach is useful for an intuitive physical ex- (6) planation of the coupled mechano-chemical kinetics of where P (x, t) is the probability that the motor protein molecular motors. At least one of the independent coor- µ “sees” the landscape V (x) at time t and W ′ (x) is the dinate axes of this landscape represents the position of µ µ→µ transition probability per unit time for a transition from the motor in real space (i.e., its “mechanical coordinate”) the chemical state µ to the chemical state µ′ at position while at least another represents its chemical state (.e., x. “chemical coordinate”). Therefore, the minimum dimen- Instead of the Langevin equation, an equivalent sion of this “land” is 2 and the free energy can be rep- Fokker-Planck equation can be used to describe the wan- resented by the hight at each point on the “land”. Both dering of the motor protein in the potential energy land- the position and chemical state variables are assumed scape V (x). The advantage of this alternative formu- to be continuous. The profiles of the free energy along µ lation is that the Fokker-Planck equation can be com- both the position and chemical coordinates, obtained by bined naturally with the master equation that describes taking appropriate cross sections of the free energy land- the time-evolution of the potential energy landscape. In scape, are periodic with the same periodicity in both the this hybrid formulation, P (x, t) represents the probabil- directions. But, the profile is tiled forward along the µ ity that, at time t, the motor is located at x, while “see- chemical direction so that the bottom of the minima are ing” the potential energy landscape V (x). The equation located deeper in the forward direction along the “chem- µ governing the time dependence of P (x, t) ical” coordinate; this tilt accounts for the lowering of free µ energy caused, for example, by ATP hydrolysis. ∂P (x, t) 1 ∂ µ = {V ′(x) − F }P (x, t) ∂t η ∂x µ µ   k T ∂2P (x, t) + B µ η ∂x2   + Pµ′ (x, t)Wµ′ →µ(x) B. Motor kinetics as wandering in the ′ time-dependent real-space potential landscape Xµ − Pµ(x, t)Wµ→µ′ (x) (7) µ′ Consider those special situations where the chemical X states of the motor are long lived and change in fast dis- where η is a phenomenological coefficient that captures crete jumps so that the position of the motor can con- the effect of viscous drag. Note that there is no term tinue to change without alteration in its chemical state, in this equation which would correspond to a mixed except during chemical transitions when position remains mechano-chemical transition. frozen. Thus, no mixed mechano-chemical transition is allowed in this scenario. In such situations, we can as- sume that the potential landscape in real space remains C. Motor kinetics as a jump process in a fully unchanged for a while, during which the wanderings of discrete mechano-chemical network the motor in this landscape (i.e., the positional dynamics of the motor) is governed by the “frozen” spatial profile In this formulation, both the positions and “internal” of the potential. The profile of the potential switches se- (or “chemical”) states of the motors are assumed to be quentially from one form to another and the sequence of discrete. Let Pµ(i,t) be the probability of finding the m profiles is repeated in each cycle although the switch- motor at the discrete position labelled by i and in the ing times are random. Different profiles correspond to “chemical” state µ at time t. Then, the master equation different motor-track interactions which is dependent on for Pµ(i,t) is given by the nature of the ligand (if any) bound to the motor. ∂P (i,t) µ = [ P (j, t)k (j → i) − P (i,t)k (i → j)] In this formulation we assume that the allowed po- ∂t µ µ µ µ sitions of the motor on its track form a continuum x. Xj6=i Xj6=i The discrete index µ = 1, 2, ..., M denote the M dis- + [ Pµ′ (i,t)Wµ′→µ(i) − Pµ(i,t)Wµ→µ′ (i)] tinct spatial profiles of the potential Vµ(x) that are pos- µ′ µ′ tulated apriori. At any instant of time t, the state X X ′ ′ of the motor is given by (x, µ). In the overdamped + [ Pµ (j, t)ωµ →µ(j → i) ′ regime, the time-evolution of the position x(t) of the mo- Xj6=i Xµ tor obeys the Langevin equation (1) with [112] Fext = − Pµ(i,t)ωµ→µ′ (i → j)] (8) −dVµ(x)/dx + Fload. The time-dependence of the profile ′ Xj6=i Xµ 12 where the terms enclosed by the three different brackets ratio [.] correspond to the purely mechanical, purely chemical d µ,d R(C ) = Π +d W /Π −d W = Π (W /W ). and mechano-chemical transitions, respectively. µ ǫCµ ji ǫCµ ij ji ij As a concrete example, which will be used also for on (11) several other occasions later in this review, consider a 2- where the superscript µ, d on the product sign denote a state motor that, at any site j, can exist in one of the product evaluated over the directed edges of the cycle. − + only two possible chemical states labelled by the symbols So, by definition, R(Cµ )=1/R(Cµ ). 1j and 2j. We assume the mechano-chemical cycle of this It has been proved rigorously [114] that, for detailed motor to be balance to hold, the necessary and sufficient condition to be satisfied by the transition probabilities is ω1 ω2 1j ⇋ 2j → 1j+1 (9) ω− d d 1 R(Cµ) = 1 for all dicycles Cµ (12) where the rates of the allowed transitions are shown ex- For a non-equilibrium steady state (NESS), one can ss d plicitly above or below the corresponding arrow. Note define [113] the dicycle frequency Ω (Cµ) which is the that the transition 1 ⇋ 2 is purely chemical whereas d j j number of dicycles Cµ completed per unit time in the the transition 2j → 1j+1 is a mixed mechano-chemical NESS of the system. Then, transition. The corresponding master equations are given ss + ss − µ,d d by Ω (Cµ )/Ω (Cµ ) = Π(Wji/Wij )= R(Cµ) (13)

d dP1(i) Clearly, R(C ) 6= 1, in general, for any NESS. = ω P (i − 1)+ ω P (i) − ω P (i) µ dt 2 2 −1 2 1 1 Detailed balance is believed to be a property of systems dP (i) in equilibrium whereas the conditions under which molec- 2 = ω P (i) − ω P (i) − ω P (i) dt 1 1 −1 2 2 2 ular machines operate are far from equilibrium. Does it (10) imply that detailed balance breaks down for molecular machines? The correct answer this subtle question needs

Suppose a load force Fext opposes the meacho-chemical a careful analysis [115–117]. assumes transition 2j → 1j+1. The effect of the load force If one naively that the entire system returns to can be incorporated by assuming the load-dependence its original initial state at the end of each cycle one would of the corresponding rate constant to be of the form get the erroneous result that the detailed balance breaks ω2(Fext) = ω2(0)exp[−Fext/(kBT )] where ω2(0) is the down. But, strictly speaking, the free energy of the full value of the rate constant in the absence of any external system gets lowered by |δG| (e.g., because of the hydrol- force. We’ll see some implications of these equations in ysis of ATP) in each cycle although the cyclic machine several specific contexts later in this article. itself comes back to the same state. When the latter fact is incorporated correctly in the analysis [115–117], one finds that detailed balance is not violated by molecular machines. This should not sound surprising- the transi- D. Balance conditions for mechano-chemical kinetics: cycles, and flux tion rates “do not know” whether or not the system has been driven out of equilibrium by pumping energy into it. On a discrete mechano-chemical state space, each state is denoted by a vertex and the direct transition from one state (denoted by, say, the vertex i) to another (denoted VI. SOLVING FORWARD PROBLEM BY by, say, the vertex j) is represented by a directed edge STOCHASTIC PROCESS MODELING: FROM |ij >. The opposite transition from j to i is denoted MODEL TO DATA by the directed edge |ji >. A transition flux can be defined along any edge of this diagram. The forward A. Average speed and load-velocity relation transition flux J|ij> from the vertex i to the vertex j is given by WjiPi while the reverse flux, i.e., transition flux For simplicity, let us consider the kinetic scheme shown J|ji> from j to i is given by Wij Pj . Therefore, the net in fig. (1(a)). In terms of the Fourier transform transition flux in the direction from the vertex i to the vertex j is given by Jji = WjiPi − Wij Pj . ∞ ¯ −ikxj A cycle in the kinetic diagram consists of at least three Pµ(k,t)= Pµ(xj ,t)e (14) vertices. Each cycle Cµ can be decomposed into two di- j=−∞ d X rected cycles (or, dicycles) [113] Cµ where d = ± corre- sponds the clockwise and counter-clockwise cycles. The of Pµ(xj ,t), the master equations can be written as a matrix equation net cycle flux J(Cµ) in the cycle Cµ, in the CW direction, + − is given by J(Cµ)= J(Cµ ) − J(Cµ ). ∂P¯(k,t) d = W(k)P¯(k,t) (15) For each arbitrary dicycle Cµ, let us define the dicycle ∂t 13

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

∂P¯(k,t) i = < x(t) > ∂k  k=0 ∂2P¯ ∂P 2 − + = < x2(t) > − < x(t) >2 ∂k2 ∂k  k=0  k=0 (17)

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

∂ ∂P¯(k,t) V = lim i t→∞ ∂t ∂k  k=0 1 ∂ ∂2P¯(k,t) ∂P¯(k,t) 2 D = lim − + t→∞ 2 ∂t ∂k2 ∂k  k=0  k=0 (18)

It may be tempting to attempt a direct utilization of the general form (c)

λµ(k)t P¯(k,t)= Bµe (19) µ X

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 eigenvalues of W(k). Fortunately, only the smallest eigenvalue λmin, which dominates P¯(k) in the limit t → ∞, is required for evaluating V and D [118, 119]:

∂λ (k,t) V = i min FIG. 1: Three examples of different types of networks of dis- ∂k crete mechano-chemical states. The bullets represent the dis-  k=0 1 ∂2λ (k,t) tinct states and the arrows denote the allowed transitions be- D = − min (20) tween the two corresponding states. The scheme is (a) is 2 2 ∂k k=0 unbrached whereas that in (b) has branched pathways con-   necting the same pair of states. The mechanical step size is Even the forms (20) are not convenient for evaluating unique in both (a) and (b) whereas steps of more than one V and D. Most convenient approach is based on the size are possible in (c). (Adapted from fig.1 of ref.[120]). characteristic 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., solution of the equation

M µ where P¯ is a column vector of M components (µ = Q(k; λ)= Cµ(k)[λ(k)] =0. (21) µ=0 1, ...M) and W(k) is the transition matrix in the k-space X (i.e., Fourier space). Summing over the hidden chemical states, we get Hence [118, 119] C′ V = −i 0 C1(0) C′′ − 2iC′ (0)V − 2C (0)V 2 D = 0 1 2 (22) M M ∞ 2C1(0) −ikxj P¯(k,t)= P¯µ(k,t)= Pµ(xj ,t)e . (16) ′ µ=1 µ=1 j=−∞ where C = [∂Cµ(k)/∂k]k=0. X X X µ 14

For a postulated kinetic scheme, W is given. Then wj . The average velocity V of the motor is given by [5] the expressions (22) are adequate for analytical deriva- M−1 tion of V and D for the given model. However, in order 1 w V = 1 − j (26) to calculate the distributions of the dwell times of the R u M j=0 j motors, it is more convenient to work with the Fourier-  Y   Laplace transform, rather than the Fourier transform, of where the probability densities. Therefore, we now derive al- ternative expressions for V and D in terms of the full M−1 Fourier-Laplace transform of the probability density. RM = rj (27) j=0 Taking Laplace transform of (14) with respect to time X ∞ with ˜ ¯ −st Pµ(k,s)= Pµ(k,t)e , (23) M−1 k 0 1 wj+i Z rj = 1+ (28) uj uj+i the matrix form of the master equation reduces to k=1 i=1   X Y  P˜ (k,s) = R(k,s)−1P(0) where R(k,s) = sI − W(k) while D is given by and P(0) is the column vector of initial probabilities. Now the characteristic polynomial is the determinant of (VS + dU ) (M + 2)V d D = M M − (29) R(k,s) which can be expressed as [120] R2 2 M  M  M where µ |R(k,s)| = cµ(k)s =0. (24) M−1 M−1 µ=0 X SM = sj (k + 1)rk+j+1 (30) j=0 Equation (24) is formally similar to (21). As expected, X kX=0 we get [120] and c′ M−1 V = −i 0 c1(0) UM = uj rj sj (31) ′′ ′ 2 j=0 c − 2ic (0)V − 2c2(0)V X D = 0 1 (25) 2c1(0) while, Example: an M-step unbranched mechano- M−1 k • 1 wj+1−i chemical cycle sj = 1+ . (32) u u j i=1 j−i As an illustrative example, let us consider the un-  Xk=1 Y  branched mechano-chemical cycle [5] with M = 4, as For various extensions of this scheme see ref.[5]. shown in fig.2. This special value of M is motivated by In the simpler case shown in (9), where M = 2, and the typical example of a kinesin motor for which the four the second step is purely irreversible, using the step size essential steps in each cycle are as follows: (i) a substrate- ℓ explicitly (to make the dimensions of the expressions binding step (e.g., binding of an ATP molecule), (ii) a explicitly clear), we get chemical reaction step (e.g., hydrolysis of ATP), (iii) a product-release step (e.g., release of ADP), and (iv) a ω ω V = ℓ 1 2 mechanical step (e.g., power stroke). ω + ω + ω  1 −1 2  ℓ2 (ω ω ) − 2(V/ℓ)2 D = 1 2 (33) 2 ω + ω + ω  1 −1 2  Note that if, in addition, ω−1 vanishes, i.e., if both the −1 −1 steps are fully irreversible, then d/V = ω1 + ω2 , i.e., the average time taken to move forward by one site is the sum of the mean residence time in the two steps of the FIG. 2: An unbranched mechano-chemical cycle of the molec- cycle. ular motor with M = 4. The horizontal dashed line shows the •Effects of branched pathways and inhomo- lattice which represents the track; j and j + 1 represent two geneities of tracks successive binding sites of the motor. The circles labelled by Microtubule-associated proteins and actin-related pro- integers denote different “chemical” states in between j and teins can give rise to inhomogeneities of the tracks for j + 1. (Adapted from fig.7 of ref.[121]). cytoskeletal motors. The intrinsic sequence inhomogene- ity of DNA and RNA strands can nontrivial influences on Suppose, The forward transitions take place at rates the motors that use nucleic acid strands as their tracks uj whereas the backward transitions occur with the rates [122]. 15

B. Jamming on a crowded track: flux-density Let is consider the kinetic scheme shown below relation Mj + Pn + S ⇋ I1 → I2 → Mj+1 + Pn+1 ↓ In the preceeding subsection we considered lone motor (34) moving along a filamentous track. Now we consider a where M denotes the polymerase motor located at the track with heavy motor traffic. In principle, unless con- j discrete position j on its template, P represents the trolled by some regulatory signals, formation of “traffic n elongating biopolymer consisting of n subunits and S is jams” [123–125] on crowded tracks cannot be ruled out a single subunit. Note that the scheme shown in (34) [126–129]. These phenomena are formally similar to sys- is, at least, formally an extension of (9) where an extra tems of sterically interacting self-propelled particles. One intermediate state I and a branched path from I have of the simplest theoretical models for such systems is the 2 2 been added. This scheme is one of the simplest possible totally asymmetric simple exclusion process (TASEP). implementations of kinetic proofreading. In fact, TASEP was originally inspired by traffic of ri- Kinetic proofreading leads to futile cycles in which at bosomes on a mRNA track during translation [130–133]. least part of the input free energy is dissipated with- However, in recent years it has found application also in out elongating the nascent polypeptide by an amino acid the context of traffic-like collective movements of ribo- monomer. The effects of the consequent loose mechano- somes in more realistic models [134, 135] as well as many chemical coupling [144] of the translational machinery on other types of molecular motors, e.g., kinesins [103, 104], the rate of polypeptide synthesis has been investigated RNA polymerase [136, 137], growth of fungal hyphae [135, 145, 146]. [138], growth of bacterial flagella [139], crowding of de- Occasionally wrong monomers escape detection in polymerizing kinesins at the tip of a microtube [140], etc. spite of elaborate selection procedure. In case of DNA replication, the polymerase normally detects the error immediately and corrects it before elongating it further. For this purpose, the nascent DNA is transferred to an- C. Information processing machines: fidelity versus other specific site on the same polymerase where the power and efficiency wrong monomer is cleaved before transferring the nascent DNA back to the site of elongation activity. The inter- play of the two contradictory activities of elongation and Power and efficiency are most appropriate quantities to shortening of a nascent DNA exposes leads to the cou- characterize the performance of porters, rowers, sliders, pling of their corresponding rates in the overall rate of etc. But, not all machines fall in these categories. For DNA polymerization by a DdDP [147]. information processing machines, which we consider in this subsection, fidelity of information transfer is more important than power and efficiency. D. Beyond average: dwell time distribution (DTD) The sequence of the monomeric subunits of a polynu- cleotide or that of a polypeptide is dictated by that of Two motors with identical average velocities may ex- the corresponding template strand. During the polymer- hibits widely different types of fluctuations. Suppose the ization process, after preliminary tentative selection, the successive mechanical steps are taken by the motor at selected monomeric subunit is subjected to several levels times t1,t2, ..., tn−1,tn,tn+1, .... Then, the time of dwell of checks by the quality control system of the polymeriz- before the k-th step is defined by τk = tk − tk−1. In ing machinery. Only after a selected subunit is screened between successive steps, the motor may visit several by such a stringent test, it is covalently bonded to the “chemical” states and each state may be visited more elongating biopolymer. than once. But, the purely chemical transitions would not be visible in a mechanical experimental set up that Discrimination of monomers based merely on the dif- ferences of free-energy gains cannot account for the ob- records only its position. The number of visits to a given state and the duration of stay in a state in a given visit served high fidelity of these processes. For example, “ki- are random quantities. netic proofreading” [141, 142] has been invoked to explain In order to appreciate the origin of the fluctions in the the kinetic processes that contribute to the high level of dwell times, let us consider the simple N-step kinetics: translational fidelity. Following three features are essen- tial for kinetic proofreading [143]: M1 ⇋ M2 ⇋ M3... ⇋ Mj ⇋ ... ⇋ MN (35) (i) formation of an initial enzyme-reactant complex, (ii) a strongly forward driven step that results in a high- Suppose tµ,ν is the duration of stay of the motor in the energy intermediate complex, and µ-th state during its ν-th visit to this state. If τ is the (iii) one or more branched pathways along which dissocia- dwell time, then tion of the enzyme-reactant complex, and rejection of the N nµ reactant, is possible before the complex gets an opportu- τ = tµ,ν (36) nity to make the final transition to yield the product. µ=1 ν=1 X X 16 where nµ is the number of visits to the µ-th state. It is It is possible to establish on general grounds that, for a straightforward to check that motor with N mechano-chemical kinetic states like (35), the DTD is a sum of n exponentials of the form [148] N N <τ > = < tµ > −ωj t µ=1 f(t)= Cj e (42) X j=1 (37) X where N − 1 of the N coefficients Cj (1 ≤ j ≤ N) are where < nµ > is the avarege number of visitits to the independent of each other because of the constraint im- µ-th state and is the avarege time of dwell in the posed by the normalization condition for the distribution µ-th state is a single visit to it. More interestingly [148], f(t). Also note that the prefactors Cj can be both posi- N tive or negative while ωj > 0 for all j. < τ 2 > − <τ >2= ( − 2) If we consider an even more special case of the scheme µ µ µ (9) where ω = 0,ω = ω = ω, i.e., all the steps are µ=1 −1 1 2 X completely irreversible and take place at the same rate N 2 2 2 ω, the DTD becomes the Gamma-distribution + ( − ) N N−1 −ωt µ=1 ω t e X f(t)= (43) Γ(N) + 2 ( − < nν >) < tν > µ>ν X where Γ(N) is the gamma function with N = 2. In- (38) terestingly, for the Gamma-distribution, the randomness parameter [154] (also called the Fano factor [155]) The first and second terms on the right hand side of (38) capture, respectively, the fluctuations in the lifetimes of r = (< τ 2 > − <τ >2)1/2/<τ> (44) the individual states and that in the number of visits is exactly given by r = 1/N. Can the experimentally to a kinetic state. Note that the number of visits to a measured DTD be used to determine the number of states particular state depends on the number of visits to the N? Unfortunately, for real motors, (i) not each step of neighboring states on the kinetic diagram; this interstate a cycle is fully irreversible, (ii) the rate constants for dif- correlation is captured by the third term on the right ferent steps are not necessarily identical, (iii) branched hand side of (38). pathways are quite common. Consequently, 1/r may pro- Several different analytical and numerical techniques vide just a bound on the rough estimate of N. have been developed for calculation of the dwell time Can one use the general form (42) of DTD to extract distribution [120, 149–151]. Since the dwell time is essen- all the rate constants for the kinetic model by fitting it tially a first passage time [152], an absorbing boundary with the experimentally measured DTD? The answer is: method [150] has been used. NO. First, even if a good estimate of N is available, the number of parameters that can be extracted by fitting 1. DTD for a motor that never steps backward the experimental DTD data to (42) is 2N − 1 (n values of ωj and N − 1 values of Cj ). On the other hand, the number of possible rate constants may be much higher As an example, we consider again the simple scheme [148]. For example, if transitions from every kinetic state (9). In this case, the DTD is to every other kinetic state is allowed, the total number of rate constants would be N(N − 1). In other words, in ω1ω2 f(t)= (e−ω+t − e−ω−t) (39) general, the kinetic rate constants are underspecified by ω − ω  − +  the DTD. Second, as the expression (39) for the DTD of where the example (9) shows expplicitly, the ω’s that appear in the exponentials may be combinations of the rate con- ω + ω + ω (ω + ω + ω )2 stants for the distinct transitions in the kinetic model. ω = 1 −1 2 ± 1 −1 2 − ω ω ± 2 4 1 2 It is practically impossible to extract the individual rate r  (40) constants from the estimated ω’s unless any explicit rela- tion like (40) between the estimated ω’s and actual rate In the special case ω−1 = 0, ω+ = ω1 and ω− = ω2 and, hence, constants is apriori available.

ω1ω2 f(t)= (e−ω1t − e−ω2t) (41) ω − ω 2. Conditional DTD for motors with both forward and  2 1  backward stepping Similar sum of exponentials for DTD have been derived also for machines with more complex mechano-chemical For motors which can step both forward and backward, kinetics (see, for example, refs.[121, 146, 153]). more relevant information on the kinetics of a motor are 17 contained in the conditional dwell time distribution [156]. X-ray diffraction and electron microscopy, two of the If n+ and n− are the numbers of forward and backward most powerful biophysical techniques, yield the structure steps, respectively, then for large n = n+ + n−, the cor- of molecular machines averaged over an ensemble. Elec- responding step splitting probabilities are Π+ = n+/n tron microscopy is a powerful alternative for the deter- and Π− = n−/n. The dwell times before a forward step mination of structures of those macromolecules whose and before a backward step can be measured separately. crystals are not available. Cryo-electron microscopy Hence, the prior dwell times τ+ and τ− can be obtained [1, 158] combines the technique of electron microscopy by restricting the summations in with cryogenics-based sample preparation. ± The single-molecule techniques [159–164] can be fur- ← 1 ther classified into two groups: (i) methods of imaging, τ± = τk (45) n± and (ii) methods of manipulation. These are not mutu- X to just forward (+) or just backward (-) steps, respec- ally exclusive and can be probed by a single experimental tively. In terms of splitting probabilities and prior dwell set up. The most direct single-molecule imaging is car- times, the mean dwell time hτi can be expressed as ried out by fluorescence-based optical microscopy. Tra- ditional diffraction-limited optical microscopy provides a ← ← hτi = Π+τ+ + Π−τ− (46) “hazy” image of the machine. However, modern opti- Compared to the prior dwell times, more detailed in- cal nanoscopy has broken the resolution limit imposed formation on the stepping statistics is contained in the by diffraction [165]. Fluorescence microscopy provides a four conditional dwell times, which are defined as follows: glimpse (howsoever hazy) of single molecules. Imaging a fluorescently labelled molecular machine in real time τ++ = dwell time between a + step followed by a + stepenables us to study its dynamics just as ecologists use τ+− = dwell time between a + step followed by a − step“radio collars” to track individual animals. τ = dwell time between a − step followed by a + step The techniques developed for the mechanical manipu- −+ lation of a single machine can be classified on the basis τ−− = dwell time between a − step followed by a − stepof tranducers used; the table below provides a summary. (47) Mechanical manipulation of a single biomolecule It is helpful to introduce pairwise step probabilities ւ ց Π++, Π+−, Π−+, Π−−. Note that Π++ + Π+− = 1, and Mechanical transducers Field-based t ransducers Π−+ + Π−− = 1. Neglecting finite time corrections, ւց ւց SFM Micro-needle EM-field Flow-field Π++ = n++/(n++ + n+−), Π+− = n+−/(n++ + n+−) ւ ց Π = n /(n + n ), Π = n /(n + n ) −+ −+ −+ −− −− −− −+ −− Electric field Ma gnetic field (48) Hence τ ← = Π τ + Π τ VIII. SOLVING INVERSE PROBLEM BY + ++ ++ +− −+ PROBABILISTIC REVERSE ENGINEERING: ← τ− = Π−+τ+− + Π−−τ−− (49) FROM DATA TO MODEL → Defining the post dwell times τ± in a fashion similar to that used for defining the prior dwell times, we get A discrete kinetic model of a molecular motor can be → regarded as a network where each node represents a dis- τ+ = Π++τ++ + Π+−τ+− tinct mechano-chemical state. The directed links denote → τ− = Π−+τ−+ + Π−−τ−− (50) the allowed transitions. Therefore, such a model is unam- and biguously specified in terms of the following parameters: (i) the total number N of the nodes, (ii) the N × N ma- → → hτi = Π+τ+ + Π−τ− (51) trix whose elements are the rates of the transitions among these states; a vanishing rate indicates a forbidden direct This formalism has been applied, for example, single- transition. headedkinesin KIF1A [157]. In the preceeding sections we handled the “forward problem” by starting with a model that is formulated VII. A SUMMARY OF EXPERIMENTAL on the basis of apriori hypotheses which are, essen- TECHNIQUES: ENSEMBLE VERSUS tially, educated guess as to the mechano-chemical kinet- SINGLE-MACHINE ics of the motor. Standard theoretical treatments of the model yields data on various aspects on the modelled The experimental techniques for probing the struc- motor; this approach is expressed below schematically. ture and dynamics of molecular machines can be di- Theoretical model → Experimental data vided broadly into two groups: (i) ensemble-averaged, Consistency between theoretical prediction and experme- (ii) single-machine. ntal data validates the model. However, any inconsis- 18 tency between the two indicates a need to modify the by the Markovian kinetics of the device. But, the mag- model. nitudes of the rate constants kf and kr are not supplied. In most real situations the numerical values of the rate We’ll now formulate a method, based on ML analysis constants of the kinetic model are not known apriori. In [169] to extract the numerical values of the parameters principle, these can be extracted by analyzing the exper- kf and kr. imental data in the light of the model. However, not only Suppose t(1) and t(2) denote the time interval of the j- the rate constants but also the number of states and the j j th residence of the device in states E1 and E2, respectively. overall architecture of the mechano-chemical network as Moreover, suppose that the device makes N1 and N2 vis- well as the kinetic scheme postulated by the model may its to the states E1 and E2, respectively, and N = N1 +N2 be uncertain. In that case, the experimental data should is the total number of states in the sequence. Therefore, be utilized to “select” the most appropriate model from N1 (1) total time of dwell in the two states are T = t among the plausible ones. In fact, more than one model, 1 j=1 j N2 (2) based on different hypotheses, may appear to be consis- and T2 = j=1 tj where T1 + T2 = T . P tent with the same set of experimental data within a level Since the dwell times are exponentially distributed for P of accuracy. The experimental data can be exploited at a Poisson process, the likelihood of any state trajectory least to “rank” the models in the order of their success S is the conditional probability in accounting for the same data set. The “inverse problem” of inferring the model from the N −k t(1) P (S|k , k ) = Π 1 k e f j observed experimental data has to be based on the the- f r j=1 f   ory of probability. Such “statistical inference” [166] can (2) N2 −krtj be drawn by following methods developed by statisti- Πj=1kre cians over the last one century. Inferring the complete   network of mechano-chemical states and kinetic scheme = kN1 e−kf T1 kN2 e−krT2 (53) of a molecular machine from its observed properties is f r    reminiscent of inferring the operational mechanism of a given functioning macroscopic machine by “reverse en- gineering”. This inverse problem, which is the main 1. Maximum-likelihood estimate aim of this section, is expressed below schematically. Theoretical model ← Experimental data ML approcah [170] is based on finding the estimates It is worth emphasizing that both the directions of in- of the set of model parameters that corresponds to the vestigations, i.e. the forward problem and the inverse maximum of the likelihood P (d~|~m) for a fixed set of data problem are equally important and complementary to ~ each other [167]. d. For the kinetic scheme shown in equation (52), the the ML estimates of kf and kr are obtained by using (53) in d[lnP (S|k , k )]/dk =0= d[lnP (S|k , k )]/dk . It A. Frequentist versus Bayesian approach f r f f r r is straightforward to see [169] that these estimates are k = N /T and k = N /T . Suppose, ~m be a column vector whose M compo- f 1 1 r 2 2 nents are the M parameters of the model, i.e., the trans- T pose of ~m is ~m = (m1,m2, ..., mM ) Let the data ob- tained in N observations of this model are represented 2. Bayesian estimate by the N-component column vector d~ whose transpose T is d~ = (d1, d2, ..., dN ). Our “inverse problem” is to in- For drawing statistical inference regarding a kinetic fer information on ~m from the observed d~. The philoso- model, the Bayesian approach has gained increasing pop- phy underlying the frequentist approach, i.e., approaches ularity in recent years [171–176]. The areas of research based on maximum-likelihood (ML) estimation and the where this has been applied successfully include various Bayesian approach for extracting these information are biological processes in, for example, genetics [177, 178], different in spirit, as we explain in the next two subsub- biochemistry [179], cognitive sciences [180], ecology [181], sections [168]. etc. For simplicity, let us assume that a device has only two In the Bayesian method there is no logical distinc- possible distinct states denoted by E1 and E2. tion between the model parameters and the experimental data; in fact, both are regarded as random. The only dis- kf tinction between these two types of random variables is E1 ⇋ E2 (52) kr that the data are observed variables whereas the model parameters are unobserved. The problem is to estimate Let us imagine that we are given the actual sequence of the probability distribution of the model parameters from the states, over the time interval 0 ≤ t ≤ T , generated the distributions of the observed data. 19

The Bayes theorem states that on the recording of the position of the motor in a single motor experiment. This problem is similar to an older P (d~|~m)P (~m) P (~m|d~)= (54) problem in cell biology: ion-channel kinetics [182, 183]. P (d~) Current passes through the channel only when it is in the “open” state. However, if the channel has more than where P (d~) can be expressed as one distinct closed states, the recordings of the current P (d~)= P (d~|~m)P (~m)d~m (55) reveals neither the actual closed state in which the chan- nel was nor the transitions between those closed states Z when no current was recorded. The likelihood P (d~|~m) is the conditional probability for the observed data, given a set of particular values of the model parameters, that is predicted by the kinetic Hidden Markov Model (HMM) [184–187] has been applied to analyze FRET trajectories [188, 189], step- model. However, implementation of this scheme also re- quires P (~m) as input. In Bayesian terminology P (~m) ping recordings of molecular motors [190, 191], and acto- myosin contractile system [192] to extract kinetic infor- is called the prior because this probability is assumed apriori by the analyzer before the outcomes of the exper- mation. iments have been analyzed. In contrast, the left hand side of equation (54) gives the posterior probability, i.e., For a pedagogical presentation of the main ideas be- after analyzing the data. hind HMM, we start with the kinetic scheme shown in Thus, an experimenter learns from the Bayesian anal- (54) and a simple, albeit unrealistic, situation and then ysis of the data. Such a learning begins with an input by gradually adding more and more realistic features, ex- in the form of a prior probability; the choice of the prior plain the main concepts in a transparent manner [169]. can be based on physical intuition, or general arguments First, suppose that the actual sequence of states (trajec- based, for example, on symmetries. Prior choice can be- tory) itself is visible; this case can be analyzed either by come simple if some experience have been gained from the ML-analysis of by Bayesian approach both of which previous measurements. Often an uniform distribution we have presented above. We now relax the strong as- of the model parameter(s) is assumed over its allowed sumption about the trajectory and proceed as below. range if no additional information is available to bias its choice. To summarize, Bayesian analysis needs not just •If state trajectory is hidden and visible trajec- the likelihood P (d~|~m) but also the prior P (~m). tory is noise-free For the kinetic scheme shown in equation (52), the Bayes’ theorem (54) takes the form The sequence of states of the device is, as before, gen- erated by a Markov processs which is hidden. Suppose P (S|kf , kr)P (kf , kr) P (k , k |S) = the device emits photons from time to time that are de- f r P (S) tected by appropriate photo-detectors. For simplicity, we

P (S|kf , kr)P (kf , kr) assume just two detection channels labelled by 1 and 2. = ′ ′ ′ ′ (56) For the time being, we also assume a perfect one-to-one ′ ′ P (S|k , kr)P (k , kr) kf ,kr f f correspondence between the state of the light emitting We now assume a uniformP prior, i.e., constant for positive device and the channel that detects the photon. If the channel 1 (2) clicks then the light emitting device was kf and kr, but zero otherwise. Then, P (kf , kr|S) is pro- in the state E (E ) at the time of emission. The inter- portional to the likelihood function P (S|kf , kr) (within 1 2 val ∆t = t − t between the arrival of the j-th and a normalization factor). Normalizing, we get j j+1 j j + 1-th photons (1 ≤ j ≤ N) is random. T N1+1 T N2+1 P (k , k |S)= 1 kN1 e−kf T1 2 kN2 e−krT2 f r Γ(N + 1) f Γ(N + 1) r Thus, from the photo-detectors we get a visible se-  1  2  (57) quence of the channel index (a sequence made of a bi- The mean of kf is N1 +1)/T1 whereas the most-likely es- nary alphabet) which we call “noiseless photon trajec- timate is N1/T1. Similarly, the mean and most probable tory” [169]. The sequence of states in the noiseless photo estimates of kr are obtained by replacing the subscripts trajectory is also another Markov chain that is conven- 1 by subscripts 2. Moreover, the variance of kf and kr tionally referred to as the “random telegraph process”. 2 2 are (N1 + 1)/T1 and (N2 + 1)/T2 , respectively. Note that the photon detected by channel 1 (or, channel 2) can take place at any instant during the dwell of the device in state 1 (or, state 2). Therefore, the sequence 3. Hidden Markov Models of states in the noiseless photon trajectory does not re- veal the actual instants of transition from one state of The actual sequence of states of the motor, generated the device to another. by the underlying Markovian kinetics, is not directly visi- ble. For example, a sequence of states that differ “chem- Since the noiseless phton trajectory corresponds to a ically” but not mechanically do not appear as distinct random telegraph process, the transition probabilities for 20 this process are Thus, in this case, the relation between the states of the hidden and visible states is not one-to-one, but one-to- kr kf −(kf +kr )∆tj P (E1|E1; kf , kr, ∆tj ) = + e many.Therefore, given a hidden state of the device, a set kf + kr kf + kr of “emission probabilities” determine the probability of

kr −(kf +kr )∆tj each possible observable state; these are listed in equa- P (E1|E2; kf , kr, ∆tj ) = [1 − e ] kf + kr tions (61) for the device (52). •HMM: formulation for a generic model of molec- kf −(kf +kr )∆tj P (E2|E1; kf , kr, ∆tj ) = [1 − e ] ular motor k + k f r On the basis of the simple example of a 2-state sys- kf kr −(kf +kr )∆tj P (E2|E2; kf , kr, ∆tj ) = + e tem presented above, we conclude that, for data analysis kf + kr kf + kr based on a HMM four key ingredients have to be speci- (58) fied: (i) the alphabet of the “visible” sequence {µ} (1 ≤ µ ≤ where P (Eµ|Eν ; kf , kr, ∆tj ) is the conditional probability N), i.e., N possible distinct visible states; (ii) the alpha- that state of the device is Eµ given that it was in the state bet of the “hidden” Markov sequence {j} (1 ≤ j ≤ M), Eν at a time ∆t earlier. i.e., M allowed distinct hidden states, (iii) the hidden- The likelihood of the visible data sequence {V } is now to-hidden transition probabilities W (j → k), and (iv) given by hidden-to-visible emission probabilities E(j → µ). In ad- dition to the transition probabilities and emission proba- P ({V }|k , k )= P (V |k , k )ΠN−1P (V |V ; k , k , ∆t ) f r 1 f r j=1 j+1 j f r j bilities, which are the parameters of the model, the HMM (59) also needs the initial state of the hidden variable(s) as in- where the first factor on the right hand side is the ini- put parameters. tial probability (usually assumed to be the equilibrium probability). The transition probabilities on the right Visible: V0 V1 ... Vt VT hand side of equation (59) are the conditional probabili- ⇑ ⇑ ⇑ ⇑ ties given in equation (58). Unlike the previous simpler Hidden: H0 → H1 → ...→ Ht → HT case, where the state sequence itself was visible, no ana- Suppose P ({V }|HMM, {λ}) denotes the probability lytical closed-form solution is possible in this case. Nev- that an HMM with parameters {λ} generates a visible ertheless, analysis can be carried out numerically. sequence {V }. Then, •If state trajectory is hidden and visible trajec- tory is noisy P ({V }|HMM, {λ})= P ({V }|{H}; {λ})P ({H}|{λ}) In the preceeding case of a noise-free photon trajectory, {XH} we assumed that from the channel index we could get (62) perfect knowledge of the state of the emitting device. where P ({H}|{λ}) is the conditional probability that the More precisely, the conditional probabilities were HMM generates a hidden sequence {H} for the given

P (1|E1) = 1 parameters {λ} and P ({V }|{H}; {λ}) is the conditional probability that, given the hidden sequence {H} (for pa- P (1|E2) = 0 rameters {λ}) the visible sequence {V } would be ob- P (2|E1) = 0 tained. P (2|E2) = 1 Once P ({V }|HMM, {λ}) is computed, the parame- (60) ter set {λ} are varied to maximize P ({V }|HMM, {λ}) (for the convenience of numerical computation, often However, in reality, background noise is unavoidable. lnP ({V }|HMM, {λ}) is maximized. The total number Therefore, if a photon is detected by the channel 1, it of possible hidden sequences of length T is T MN . In could have been emitted by the device in its state E1 (i.e., it is, indeed, a signal photon) or it could have come order to carry out the summation in equation (62) one from the background (i.e., it is a noise photon). Suppose has to enumerate all possible hidden sequences and the corresponding probabilities of occurrences. A successful ps is the probability that the detected photon is really a signal that has come from the emitting device. Suppose implementation of the HMM requires use of an efficienct numerical algorithm; the Viterbi algorithm is one such pb1 is the probability of arrival of a background photon in the channel 1. The probability that a background photon candidate. In case of a molecular motor, the “chemical states” arrives in channel 2 is 1 − pb1. Then [169], are not visible in a single molecule experiment. More- E(1|E1) = ps + (1 − ps)pb1 over, even its mechanical state that is “visible” in the

E(2|E1) = 1 − P (1|E1)=(1 − ps)(1 − pb1) recordings may not be its true position because of (a) measurement noise, and (b) steps missed by the detector. E(1|E ) = (1 − p )p 2 s b1 Let is denote the “visible” sequence by the recorded posi- E(2|E2) = 1 − P (1|E2)= ps + (1 − ps)(1 − pb) tions {Y } whereas the hidden sequence is the composite (61) mechano-chemical states {X, C} where X and C denote 21 the true position and chemical state, respectively. The In this case, transition probabilities are denoted by W (Xt−1, Ct−1 → Xt, Ct) One possible choice for the emission probabilities E is [190] P ({Y }|HMM, {λ})= P ({Y }|{X, C}; {λ})P ({X, C}|{λ}) {X,CX} 2 (64) 1 (Yt − Xt) E(Xt → Yt)= 2 exp[− 2 ] (63) where s(2πσt ) (2σt )

P ({X, C}|{λ})= PX0,C0 W (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 following chemical states. It can be shown [193] that steps: Step I: Initialization of the parameter values for it- eration. Step II: Iterative re-estimation of parameters for φ(x) J x 1 = F xkB T − ln[ Pµ(x)] − dy maximum likelihood: the parameters {W (Xt−1, Ct−1 → k T D P (y) B µ 0 µ µ X , C )}, {E(X → Y )} and P are re-estimated iter- X Z t t t t X0 ,C0 (68) atively till P ({Y }|HMM, {λ}) saturates to a maximum. P Based on this observation, a prescription has been sug- Step III: Construction of “idealized” trace: using the gested [193] to extract φ(x) by analyzing the time se- final estimation of the model parameters, the position of ries of motor positions obtained from single-motor ex- the motor as a function of time is reconstructed; natu- periments. rally, this trace is noise-free. Step IV: Extraction of the distributions of step sizes and dwell times: from the ideal- ized trace, the distributions of the steps sizes dwell times IX. OVERLAPPING RESEARCH AREAS are obtained. These distributions can be compared with the corresponding theoretical predictions. A. Symmetry breaking: directed motility and cell polarity

Energy is a scalar quantity whereas velocity is a vector. B. Extracting FP-based model from data? How does consumption of energy give rise to a non-zero average velocity of a molecular motor? Moreover, a di- In this section so far we have discussed methods for ex- rected movement that a motor exhibits on the average, tracting the master-equation based models that describe requires breaking the forward-backward symmetry on its the kinetics of motors in terms of discrete jumps on a fully track. What are the possible cause and effects of this discrete mechano-chemical state space. However, as we broken symmetry at the molecular level? summarized in section V, kinetics of molecular motors As far as the cause of this asymmetry is concerned, the can be formulated also in terms of FP equations which asymmetry of the tracks alone cannot explain the “di- treat space as a continuous variable. Can one extract rected” movement of the motors, because on the same the parameters of such FP-based models from the data track members of different families of motors can, on collected from single-molecule experiments? the average, move in opposite directions. Obviously, the One of the key ingredients of the FP-based approach is structural design of the motors and their interactions the profile of the potential Vµ(x) felt by the motor in the with the respective tracks also play crucial roles in deter- “chemical” state µ. To my knowledge, it has not been mining their direction of motion along a track. Further- possible to extract it from any experimental data. Now, more, this broken symmetry at the molecular level, e.g., let us define the “directed” movement of molecular motors, has im- portant effects on various biological phenomena, partic- 1 ularly “vectorial” processes, at the sub-cellular and cel- dφ(x)/dx = P (x)[dV (x)/dx] (67) P (x) µ µ lular levels. In general, a cell is not isotropic. Motors are µ µ µ X essential in breaking the cellular symmetry [194]. Can P we establish a unique set of basic principle underlying to be the profile of the potential averaged over all the the symmetry breaking [194, 195]? 22

Therefore, the cause and effects of broken symmetry retains only the output, i.e., 4 and erases the input num- of molecular motors can be examined in the broader con- bers (i.e., 3 and 1, or 2 and 2, as the case may be) after text of the fundamental principles of symmetry breaking summing the two input numbers, then, given only the in physics and biology [196] (see also other articles in output (i.e., 4) it is impssible to figure out whether the the special “Perspectives on Symmetry Breaking in Biol- input were 3, 1 or 2, 2. Note that erasing every bit leads ogy” [197]). For macroscopic systems in thermodynamic to loss of information which may also be interpreted as equilibrium, symmetry breaking is explained in terms of an increase of entropy. However, strategies for logically the form of the free energy. However, since living cells reversible computation have been developed [205, 206]. are far from thermodynamic equilibrium, the theory of For example, the simplest strategy for logically reversible symmetry breaking in those systems cannot be based on computation is based on the prescription that neither the thermodynamic free energy. Kinetics cannot be ignored initial input nor the data in any intermediate step should in the study of symmetry breaking in living systems. be erased; instead, these should be retained in an auxil- iary register. In practice, computation is carried out with a device B. Self-organization and pattern formation: that is governed strictly by the laws of physics that in- assembling machines and cellular morphogenesis cludes thermodynamics. Since entropy increases in ev- ery irreversible physical process, it would be tempting to The interior of a living bacterial cell is far from ho- associate the logical irreversibility of computation with a mogeneous; the intracellular space of eukaryotes are di- physical irreversibility of the computational device. Since vided into separate compartments. Scaling is one of the each bit of a classical digital computer has only two pos- interesting properties of many physical quantities that sible states, at first sight, one would expect dissipation of gives rise to some well known universalities. The scal- kBT ln2 energy for every bit erased [207]. But, the pos- ing properties of a cell and the subcellular compartments sibility of logically reversible computation raises a funda- [198, 199]. often depend on the machineries which as- mental question: is it possible for a physical computing semble them. device to carry out physically reversible (and, hence non- The size, shape, location and number of intracellu- dissipative) computation? lar compartments as well as modular intracellular ma- Operation of a polymerase (and that of a ribosome) chineries are self-organized, rather than self-assembled. can be regarded as computation. More precisely, such Dissipation takes place in “self-organization” and distin- a computing machine can be viewed as a “tape-copying guishes it from “self-assembly”; the latter corresponds to Turing machine” that polymerizes its output tape, in- the minimum of thermodynamic free energy whereas self- stead of merely writing on a pre-synthesized tape [205] organized system does not attain thermodynamic equi- Dissipationless operation of these machines is possible librium [195, 200–202]. Molecular motors and their fila- only if every step is error-free which, in turn, is achiev- mentous tracks play important roles in the intracellular able in the vanishingly snall speed, i.e., reversible limit self-organization process [195, 200] and even in the cellu- [208]. lar morphogenesis which may be regarded as a problem of pattern formation far from equilibrium. D. Enzymatic processes: conformational fluctuations, static and dynamic disorder C. Dissipationless computation: polymerases as “tape-copying Turing machines” For a motor that doesn’t step backwards, the position advances in the forward direction one step at a time; how- ever, the time gap between the successive steps, i.e., dwell The concept of information in the context of biolog- time, is a random variable. Similarly, in single-molecule ical systems has been discussed at length in the past enzymology, the population of the product molecules in- [203]. The operation of molecular machines involved in creases by one in each enzymatic cycle, the time gap be- genetic processes can be analyzed in terms of storage and tween the release of the successive product molecules is transmission of information. In fact, a particular class of random. The distribution of this time [209, 210] is analo- machines carry out what may be viewed as data trans- gous to that of the dwell times of molecular motors [120]. mission whereas that of others may be viewed as digital- Therefore, the research fields of single-motor mechanics to-analog conversion [204] . For these obvious reasons, and single-enzyme reactions can enrich each other by ex- a broad class of molecular machines are interesting also change of concepts and techniques [148]. from the perspective of information theory, electronic en- As a concrete example, consider the chemical reaction gineering and computer science.

Computation can be viewed as a transition from one ω1 ω2 E + S ⇋ I1 → E + P (69) state to another. However, in a conventional digital com- ω−1 puter an elementary operation is logically irreversible. To understand the meaning of this term, consider the two catalyzed by the enzyme E where S and P denote the summations 3 + 1 = 4 and 2 + 2 = 4. If the computer substrate and product, respectively. The enzyme and 23 the substrate form, at a rate ω1, an intermediate com- From the perspective of applied research, the natural plex I1 that can either dissociate into the free enzyme molecular machines opened up a new frontier of nano- and the substrate at a rate ω−1, or get converted irre- technology [217–222]. The miniaturization of compo- versibly into the product and free enzyme at a rate ω2. nents for the fabrication of useful devices, which are es- In a bulk sample, where a large number of enzymes con- sential for modern technology, is currently being pursued vert many substrates into products by catalyzing this re- by engineers following mostly a top-down (from larger action simultaneously, the measured rate of the reaction to smaller) approach. On the other hand, an alternative is actually an average over the ensemble. This rate was approach, pursued mostly by chemists, is a bottom-up calculated by Michaelis and Menten about 100 years ago (from smaller to larger) approach. and is given by the celebrated Michaelis-Menten equation Unlike man-made machines these are products of Na- [211]. ture’s evolutionary design over billions of years. In fact, Formally, the scheme (69) is very similar to the scheme cell has been compared to an “archeological excavation (9) that we presented earlier as a very simple example of site” [223], the oldest modules of functional devices are the mechano-chemical cycle of a molecular motor. The the analogs of the most ancient layer of the exposed site time taken by the chemical reaction (69) fluctuates from of excavation. We can benefit from Nature’s billion year one enzymatic cycle to another. One of the fundamen- experience in nano-technolgy. The term biomimetics has tal questions is: what are the conditions under which already become a popular buzzword [220, 221]; this field the average rate would still satisfy the Michaelis-Menten deals with the design of artificial systems utilizing the equation [209, 210]? principles of natural biological systems. 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 fuel X. CONCEPT OF BIOLOGICAL MACHINES: molecule and the track. How do these two sites commu- FROM ARISTOTLE TO ALBERTS nicate? How does the binding of ligand at one site affect the binding affinity of the other? The mechanochemical The concept of “living machine” evolved over many cycle of a motor can be analyzed [213] from the perspec- centuries. Some of the greatest thinkers in human history tive of allosterism which is one of the key mechanisms of made significant contributions to this concept. It started cooperativity in protein kinetics [214]. with the man-machine (and, more generally, animal- machine) analogy. Aristotle [224] distinguished between the “body” and the “soul” of an organism. However, E. Applications in biomimetics and after more than one and half millenia, an intellectu- nano-technology ally provocative idea was put forward by Rene Descartes when he argued that animals were “living machines”. Initially, technology was synonymous with macro- This idea took its extreme form in Julien Offray de technology. The first tools applied by primitive humans La Mettrie’s book [225] L’homme Machine (’man a ma- were, perhaps, wooden sticks and stone blades. Later, as chine’). Leibnitz [226] wrote that the body of a living early civilizations started using levers, pulleys and wheels being is a kind of “divine machine or natural automa- for erecting enormous structures like pyramids. Until ton” and it “surpasses all artificial automata”, because nineteenth century, watch makers were, perhaps, the only not each part of a man-made machine is itself a machine. people working with small machines. Using magnifying In contrast, he argued, living bodies are “machines in glasses, they worked with machines as small as 0.1mm. their smallest parts ad infinitum”. Is this statement to Micro-technology, dealing with machines at the length be interpreted as Leibnitz’s speculation for the existence scale of micrometers, was driven, in the second half of of machines within machines in a living organism? The the twentieth century, largely by the computer miniatur- debate over this interpretation continues [227, 228]. ization. All the great thinkers from Aristotle to Descartes and In 1959, Richard Feynman delivered a talk [215] at a Leibnitz compared the whole organism with a machine, meeting of the American Physical Society. In this talk, the organs being the coordinated parts of that machine. entitled “There’s Plenty of Room at the Bottom”, Feyn- Cell was unknown; even micro-organisms became visible man drew attention of the scientific community to the only after the invention of the optical microscope in the unlimited possibilities of manupilating and controlling seventeenth century. Marcelo Malpighi, father of micro- things on the scale of nano-meters. This famous talk is scopic anatomy, speculated in the 17th century about the now regarded by the majority of physicists as the defin- existence molecular machines in living systems. He wrote ing moment of nano-technology [216]. In the same talk, (as quoted in english by Marco Piccolino [229]) that the in his characteristic style, Feynman noted that ”many organized bodies of animals and plants been constructed of the cells are very tiny, but they are very active, they with “ very large number of machines”. He went on to manufacture various substances, they walk around, they characterize these as “extremely minute parts so shaped wiggle, and they do all kinds of wonderful things- all on and situated, such as to form a marvelous organ”. Un- a very small scale”. fortunately, the molecular machines were invisible not 24 only to the naked eye, but even under the optical micro- phenomenon that is now known as cytoplasmic streaming scopes available in his time. In fact, individual molec- [8]. His speculation that “the cause of these currents ular machines could be “caught in the act” only in the lie in contraction” was established a century later when last quarter of the 20th century. We highlight here the cytoplasmic streaming was shown to be caused by an progress made during the last three centuries when grad- acto-myosin system. ually the analogy with machine got extended to cover all “The story of the living machine” [235] narrated by levels of biological organization- from the topmost level Herbert William Conn, one of the founding members and of organisms down to cellular and subcellular levels. Not the third president of the Society of Americam Bacteriol- surprisingly, muscles seem to have attracted maximum ogists (renamed, in 1960, American Society for Microbi- attention in the context of machinery of life. ology), is a critical overview of the understanding of these Thomas Henry Huxley, dubbed as “Darwin’s bulldog” machines at the end of the nineteenth century. Because for his strong support for Darwinian ideas of evolution, of his deep understanding of the basic principles of phys- delivered his famous lecture, titled “On the Physical Ba- ical sciences as well as evolutionary biology, his narrative sis of Life”, on 8th November, 1868 (see ref.[9] for the on the fundamental principles of living machines remains published version). Among the provocating statements as contemporary today as it was at the time of its publi- and insightful comments in this article, which received cation. Conn asked whether the operations of individual lot of hostile criticism at that time, I quote only a few organisms, i.e., the living machines, could be “reduced that are directly relevant from the perspective of molec- to the action of still smaller machines” [235]. Conn ar- ular machines. Huxley had the foresight to see that [9] gued that one can look upon each constituent cell of an “speech, gesture, and every other form of human action organism also “a little engine with admirably adapted are, in the long run, resolvable into mascular contrac- parts” [235]. His summary [235] that a living organism is tion, and muscular contraction is but a transitory change a “series of machines one within the other” sounds very in the relative positions of the parts of a muscle”- a re- similar to Leibnitz’s philosophical idea. Conn went even markably insightful comment on contractility and motil- further: “As a whole it is machine, and its parts are sep- ity because the mechanism of muscle contraction was dis- arate machines. Each part is further made up of still covered almost 90 years later, one of the discoverers being smaller machines until we reach the realm of the micro- his grandson! scope. Here still we find the same story. Even the parts David Ferrier, a pioneering neurologist and a younger formerly called units, prove to be machines”. He spec- contemporary of Thomas Huxley, in his lecture [230] de- ulated [235] “we may find still further machines within” livered at the Middlesex Hospital Medical School, on cells. He ended his summary with the statement “And October 5th, 1870, not only echoed similar ideas but thus vital activity is reduced to a complex of machines, stressed the role of physical energy, rather than any hy- all acting in harmony with each other to produce together pothetical vital action, for sustaining life of an organism. the one result- life” [235]. A few years later, Robert Henry Thurston, the first pres- Conn [235] not only discussed “the running of the ident of the ASME (American Society of Mechanical En- living machine”, but also “the origin of the living ma- gineers) and a pioneer of modern engineering education, chine”. In the latter context, while pointing out the dif- investigated what he called a “vital machine” (or “prime ferences in the principles of engineering design of man- motor”) [231] from an engineer’s perspective. made machines and evolutionary principles of nature’s In the initial stages, most of the visionaries compared nano-machines, he wrote [235]: “It is something as if animals with a machine. Although movements of plants steam engine of Watt should be slowly changed by adding received much less attention, results of pioneering sys- piece after piece until there was finally produced the mod- tematic studies of these phenomena were reported al- ern quadruple expansion engine, but all this change being ready in the nineteenth century by Charles Darwin in made upon the original engine without once stopping its a classic book [232] which was co-authored by his son. motion.” Later, in the preface of his report on the classic investi- Jaques Loeb, a famous embryologist, delivered a se- gations on the mechanical response of plants to stimuli, ries of eight lectures at the Columbia university in 1902 Jagadis Chandra Bose [233] wrote: “From the point of (see ref.[236] for a more complete published version). In view of its movements a plant may be regarded ...simply the first lecture, Loeb started by saying “In these lec- as a machine, transforming the energy supplied to it, in tures we shall consider living organisms as chemical ma- ways more or less capable of mechanical explanation”. chines, consisting essentially of colloidal material, which Interestingly, the first chapter was titled “The plant as a possess the peculiarities of automatically developing, pre- machine” [233]. serving, and reproducing themselves” [236]. He empha- Based on the well known fact that all animals exhibit sized the crucial differences between natural and artificial irritability and contractility [234], Thomas Henry Huxley machines by the following statement: “The fact that the had already speculated in ref.[9], that “it is more than machines which can be created by man do not possess probable, that when the vegetable world is thoroughly the power of automatic development, self-preservation, explored, we shall find all plants in possession of the same and reproduction constitutes for the pesent a fundamen- powers..”. In support of this possibility he described a tal difference between living machines and artificial ma- 25 chines”. However, just like some of his other visionary exciting era of research has just begun! predecessors, he also admitted that “nothing contradicts the possibility that the artificial production of living mat- ter may one day be accomplished” [236]. XI. SUMMARY AND OUTLOOK The concept of “living machine” as a “transformer of energy”, from the level of a single cell to the level of a So far as the current status of our understanding is multi-cellular organism as complex as a man, found men- concerned, I would say that till the middle of the 20th tion in many lectures and books of the leading biologists century we had never seen or manipulated a single molec- in the late nineteenth and early twentieh centuries (see, ular machine although machine-like operation of a cell or for example, [237]). Research on molecular machines was an entire organism was fairly well established. In the last focussed almost exclusively on the mechanism of muscle 60 years this area has seen explosive growth in activity. contraction during the first half of the 20th century and The structures as well as the mechanisms of many ma- it was dominated by Archibald Hill and Otto Meyerhof chines no longer appear any more mysterious than those [238, 239] who shared the Nobel prize in Physiology (or of macroscopic machines. medicine) in 1922. It is practically impossible to predict new ideas in any By mid-twentieth century, physical sciences already field of research; molecular machines are no exception. witnessed spectacular progress and life sciences was just The reason, as Peter Medawar [246] admitted in his pres- embarking on its golden period. In his Guthrie lecture idential address at the Cambridge meeting of the British (also titled “The Physical Basis of Life”), delivered on Association for the Advancement of Science, is as follows: 21st November, 1947 (see ref.[240] for the full text), J. “to predict an idea is to have an idea, and if we have D. Bernal drew attention of the audience to the structure an idea it can no longer be the subject of prediction”. and dynamics of machines. Ten years later, in a review Therefore, in this section, I do not propose any new idea article, the title of which again had the words “physi- but merely mention a few systems and phenomena which cal basis of life”, Schmitt [241] made even more concrete need new ideas for their studies and understanding. references to molecular machines covering almost all the Let me begin with solving the forward problem with types of machines that we have sketched in this article. process modeling. Here we need new ideas to make The concept of molecular machines [242] was men- progress in two opposite directions: (a) in-silico modeling tioned on many occasions in the late twentieth century by of single individual machines in terms of its coordinat- leading molecular biologists who made outstanding con- ing parts in an aqueous medium- aquatic nano-robotics; tributions in elucitaing their mechanisms of operation. (b) integration of nano-machines and machine-assemblies The strongest impact was made, however, by the influ- into a micro-factory, the living cell. So far as the point ential paper of Bruce Alberts [243], then the president of (a) is concerned, serious efforts have been made by de- the National Academy of Sciencs (USA). He wrote that veloping coarse-grained computational models [247–249] “the entire cell can be viewed as a factory that contains that can be regarded as the substitutes for full molecular an elaborate network of interlocking assembly lines, each dynamics based models. However, a technical (or algo- of which is composed of a set of large protein machines” rithmic) breakthrough is needed to achieve the ultimate [243]. goal of “seeing” the operation of a machine in its natu- Why did I start my story with Aristotle? In D’Arcy ral aqueous environment by carrying out an experiments Thompson’s words [244] “We know that the history of in-silico. biology harks back to Aristotle by a road that is straight Next let me explain the aim of developing integrated and clear, but that beyond him the road is broken and models. It has been strongly argued at the dawn of this the lights are dim”. Moreover, a section of the mod- millenium [250] that the machinery of life is modular. ern enterprise in animal sciences is pursuing important The machineries of transcription, translation, replication, investigations on the efficiency of energy utilization by chromosome segregation, etc. are all examples of mod- animals [245] treating an entire animal as an a “combus- ules which perform specific tasks. The components of tion engine”, a modern and scientifically correct version some of these modules are parts of a single machine, e.g., of the original philosophical idea developed by Aristotle all devices participating in translation function on a sin- and propagated by his followers. gle platform provided by a ribosome. Signal transduction Why am I closing my story with Alberts? Because of machinery is an extreme example of the other types of a fortutious coincidence, Alberts’ vision for training the functional modules which are spatially distributed over next generation of biologists for studying natural nano- a significantly large region of intracellular space without machines coincided with the explosive beginning of nano- need for direct physical contacts among the parts. technology that includes research programs on artificial The machines that we have considered here are all nano-machines. Alberts’ article [243] was appeared at a more or less spatially-confined functional modules. Nor- time when the statistical physics of Brownian ratchets mally, functional modules are not completely isolated and was getting lot of attention [94]. The concept of molec- must communicate and coordinate their function with ular machine has matured fully into an area of inter- other modules. For example, DNA replication and chro- disciplinary research in the twenty-first century. A new mosome segregation machineries require proper coordi- 26 nation. To my knowledge, no attempt has been made number of discrete states of the system as a variational so far for quantitative stochastic process modeling incor- parameter. This is a step in the right direction. However, porating more than one functional module in a seamless I am not aware of any work that ranks alternative models fashion. Such an enterprise may look like what is now according to relative scores computed on the basis of the the happening in systems biology: to integrate the op- principle of strong inference. erations of machineries for different functional modules The molecular mechanism of muscle myosin and the within a single theoretical framework. structure of double-stranded DNA were both discovered Finally, let me point out some of the standard prac- in the 1950s. These discoveries set the stage for the re- tices of statistical inference that, to my knowledge, have search on cytoskeletal motors as well as on machines that not been followed so far while reverse-engineering molec- make, break and manipulate nucleic acids and proteins. ular machines. The experimental data for molecular ma- Last 50 years has seen enormous progress. I wanted to chines have been analyzed by several groups to extract tell you about the PIs and their contributions during this an underlying kinetic model. However, most often the glorious period. But, I have run out of space. So, I’ll nar- analysis carried out during such reverse engineering is rate this fascinating story in detail elsewhere [255]. based on a single working hypothesis. It would be desir- Acknowledgements: I apologize to all those re- able to follow Platt’s [251] principle of “strong inference” searchers whose work, although no less important, could which is an extension of Chamberlin’s [252] “method of not be included in this article because of space limita- multiple working hypothesis”. By multiple model I do tions. I thank all my students and colleagues with whom not mean models hierarchically nested such that each I have collaborated on molecular machines. I am also in- one is a special case of the model at the next level. By debted to many others (too many to be listed by name) the term multiple model, I mean truly competing models for enlightening discussions and correspondences over the that may, however, overlap partially. The relative scores last few years. I thank Ashok Garai for a critical read- of the competing models (and the corresponding underly- ing of the manuscript. Active participation in the MBI ing hypothese) would be a true reflection of their merits. annual program on “Stochastics in Biological Systems” For example, in case of a closely related systems in cell bi- during 2011-12 motivated me to look at molecular ma- ology and systems biology, experimental data have been chines from the perspectives of statisticians. I thank analyzed recently within the framework of this method Marty Golubitsky, Director of MBI, for the hospitality [253]. In the case of molecular machines, a method for at OSU. This work has been supported at IIT Kanpur model selection has been developed [254]; it extracts the by the Dr. Jag Mohan Garg Chair professorship, and at best model by optimizing the maximum evidence, rather OSU by the MBI and the National Science Foundation than maximum likelihood, that treats, for example, the under grant DMS 0931642.

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