Exact Solutions for Kinetic Models of Macromolecular Dynamics

The linear framework II: using graph theory to analyse the transient regime of Markov processes

The linear framework uses finite, directed graphs with labelled edges to model biomolecular systems. Graph vertices represent chemical species or molecular states, edges represent reactions or transitions and edge labels represent rates that also describe how the system is interacting with its environment. The present paper is a sequel to a recent review of the framework that focussed on how graph-theoretic methods give insight into steady states as rational algebraic functions of the edge labels. Here, we focus on the transient regime for systems that correspond to continuous-time Markov processes. In this case, the graph specifies the infinitesimal generator of the process. We show how the moments of the first-passage time distribution, and related quantities, such as splitting probabilities and conditional first-passage times, can also be expressed as rational algebraic functions of the labels. This capability is timely, as new experimental methods are finally giving access to the transient dynamic regime and revealing the computations and information processing that occur before a steady state is reached. We illustrate the concepts, methods and formulas through examples and show how the results may be used to illuminate previous findings in the literature.

Discrete RNA-DNA hybrid cleavage by the EXD2 exonuclease pinpoints two rate-limiting steps

EXD2 is a recently identified exonuclease that cleaves RNA and DNA in double-stranded (ds) forms. It thus serves as a model system for investigating the similarities and discrepancies between exoribonuclease and exodeoxyribonuclease activities and for understanding the nucleic acid (NA) unwinding-degradation coordination of an exonuclease. Here, using a single-molecule fluorescence resonance energy transfer (smFRET) approach, we show that despite stable binding to both substrates, EXD2 barely cleaves dsDNA and yet displays both exoribonuclease and exodeoxyribonuclease activities toward RNA-DNA hybrids with a cleavage preference for RNA. Unexpectedly, EXD2-mediated hybrid cleavage proceeds in a discrete stepwise pattern, wherein a sudden 4-bp duplex unwinding increment and the subsequent dwell constitute a complete hydrolysis cycle. The relatively weak exodeoxyribonuclease activity of EXD2 partially originates from frequent hybrid rewinding. Importantly, kinetic analysis and comparison of the dwell times under varied conditions reveal two rate-limiting steps of hybrid unwinding and nucleotide excision. Overall, our findings help better understand the cellular functions of EXD2, and the cyclic coupling between duplex unwinding and exonucleolytic degradation may be generalizable to other exonucleases.

Nanopore tweezers measurements of RecQ conformational changes reveal the energy landscape of helicase motion

Helicases are essential for nearly all nucleic acid processes across the tree of life, yet detailed understanding of how they couple ATP hydrolysis to translocation and unwinding remains incomplete because their small (∼300 picometer), fast (∼1 ms) steps are difficult to resolve. Here, we use Nanopore Tweezers to observe single Escherichia coli RecQ helicases as they translocate on and unwind DNA at ultrahigh spatiotemporal resolution. Nanopore Tweezers simultaneously resolve individual steps of RecQ along the DNA and conformational changes of the helicase associated with stepping. Our data reveal the mechanochemical coupling between physical domain motions and chemical reactions that together produce directed motion of the helicase along DNA. Nanopore Tweezers measurements are performed under either assisting or opposing force applied directly on RecQ, shedding light on how RecQ responds to such forces in vivo. Determining the rates of translocation and physical conformational changes under a wide range of assisting and opposing forces reveals the underlying dynamic energy landscape that drives RecQ motion. We show that RecQ has a highly asymmetric energy landscape that enables RecQ to maintain velocity when encountering molecular roadblocks such as bound proteins and DNA secondary structures. This energy landscape also provides a mechanistic basis making RecQ an 'active helicase,' capable of unwinding dsDNA as fast as it translocates on ssDNA. Such an energy landscape may be a general strategy for molecular motors to maintain consistent velocity despite opposing loads or roadblocks.

Bounds on skewness and kurtosis of steady-state currents

Current fluctuations are a powerful tool to unravel the underlying physics of the observed transport process. This work discusses some general properties of the third and the fourth current cumulant (skewness and kurtosis) related to dynamics and thermodynamics of a transport setup. Specifically, several distinct bounds on these quantities are either analytically derived or numerically conjectured, which are applicable to (1) noninteracting fermionic systems, (2) noninteracting bosonic systems, (3) thermally driven classical Markovian systems, and (4) unicyclic Markovian networks. Finally, it is demonstrated that violation of the obtained inequalities can provide a broad spectrum of information about the physics of the analyzed system; e.g., it can enable one to infer the presence of interactions or unitary dynamics, unravel the topology of the Markovian network, or characterize the nature of thermodynamic forces driving the system. In particular, relevant information about the microscopic dynamics can be gained even at equilibrium when the current variance-a standard measure of current fluctuations-is determined mostly by the thermal noise.

Computational Aspects of Single-Molecule Kinetics for Coupled Catalytic Cycles: A Spectral Analysis

Catalysis from single active sites is analyzed using methods developed from single-molecule kinetics. Using a stochastic Markov-state description, the observable properties of general catalytic networks of reactions are expressed using an eigenvalue decomposition of the transition matrix for the Markov process. By the use of a sensitivity analysis, the necessary eigenvalues and eigenvectors are related to the energies of controlling barriers and wells located along the reaction routes. A generalization of the energetic span theory allows the eigenvalues to be computed from several activation energies corresponding to distinct barrier-well pairings. The formalism is demonstrated for model problems and for a physically realistic mechanism for an alkene hydrogenation reaction on a single-atom catalyst. The spectral analysis permits a hierarchy of timescales to be identified from the single-molecule signal, which correspond to specific relaxation modes in the network.

Kinetic and structural mechanism for DNA unwinding by a non-hexameric helicase

UvrD, a model for non-hexameric Superfamily 1 helicases, utilizes ATP hydrolysis to translocate stepwise along single-stranded DNA and unwind the duplex. Previous estimates of its step size have been indirect, and a consensus on its stepping mechanism is lacking. To dissect the mechanism underlying DNA unwinding, we use optical tweezers to measure directly the stepping behavior of UvrD as it processes a DNA hairpin and show that UvrD exhibits a variable step size averaging ~3 base pairs. Analyzing stepping kinetics across ATP reveals the type and number of catalytic events that occur with different step sizes. These single-molecule data reveal a mechanism in which UvrD moves one base pair at a time but sequesters the nascent single strands, releasing them non-uniformly after a variable number of catalytic cycles. Molecular dynamics simulations point to a structural basis for this behavior, identifying the protein-DNA interactions responsible for strand sequestration. Based on structural and sequence alignment data, we propose that this stepping mechanism may be conserved among other non-hexameric helicases.

DNA replication: In vitro single-molecule manipulation data analysis and models

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Data analysis allows to extract information from the noisy single-molecule data.

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Models provide insight in the underlying biochemical processes.

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Ligands can activate or inhibit DNA replication and DNA unwinding.

DNA replication is a key biochemical process of the cell cycle. In the last years, analysis of in vitro single-molecule DNA replication events has provided new information that cannot be obtained with ensembles studies. Here, we introduce crucial techniques for the proper analysis and modelling of DNA replication in vitro single-molecule manipulation data. Specifically, we review some of the main methods to analyze and model the real-time kinetics of the two main molecular motors of the replisome: DNA polymerase and DNA helicase. Our goal is to facilitate access to and understanding of these techniques to promotetheir use in the study of DNA replication at the single-molecule level. A proper analysis of single-molecule data is crucial to obtain a detailed picture of, among others, the kinetics rates, equilibrium contants and conformational changes of the system under study. The techniques presented here have been used or can be adapted to study the operation of other proteins involved in nucleic acids metabolism.

Modelling single-molecule kinetics of helicase translocation using high-resolution nanopore tweezers (SPRNT)

Single-molecule picometer resolution nanopore tweezers (SPRNT) is a technique for monitoring the motion of individual enzymes along a nucleic acid template at unprecedented spatiotemporal resolution. We review the development of SPRNT and the application of single-molecule kinetics theory to SPRNT data to develop a detailed model of helicase motion along a single-stranded DNA substrate. In this review, we present three examples of questions SPRNT can answer in the context of the Superfamily 2 helicase Hel308. With Hel308, SPRNT's spatiotemporal resolution enables resolution of two distinct enzymatic substates, one which is dependent upon ATP concentration and one which is ATP independent. By analyzing dwell-time distributions and helicase back-stepping, we show, in detail, how SPRNT can be used to determine the nature of these observed steps. We use dwell-time distributions to discern between three different possible models of helicase backstepping. We conclude by using SPRNT's ability to discern an enzyme's nucleotide-specific location along a DNA strand to understand the nature of sequence-specific enzyme kinetics and show that the sequence within the helicase itself affects both step dwell-time and backstepping probability while translocating on single-stranded DNA.

Stochastic models for single-cell data: Current challenges and the way forward

Although the quantity and quality of single-cell data have progressed rapidly, making quantitative predictions with single-cell stochastic models remains challenging. The stochastic nature of cellular processes leads to at least three challenges in building models with single-cell data: (a) because variability in single-cell data can be attributed to multiple different sources, it is difficult to rule out conflicting mechanistic models that explain the same data equally well; (b) the distinction between interesting biological variability and experimental variability is sometimes ambiguous; (c) the nonstandard distributions of single-cell data can lead to violations of the assumption of symmetric errors in least-squares fitting. In this review, we first discuss recent studies that overcome some of the challenges or set up a promising direction and then introduce some powerful statistical approaches utilized in these studies. We conclude that applying and developing statistical approaches could lead to further progress in building stochastic models for single-cell data.

Processivity and Velocity for Motors Stepping on Periodic Tracks

Processive molecular motors enable cargo transportation by assembling into dimers capable of taking several consecutive steps along a cytoskeletal filament. In the well-accepted hand-over-hand stepping mechanism, the trailing motor detaches from the track and binds the filament again in the leading position. This requires fuel consumption in the form of ATP hydrolysis and coordination of the catalytic cycles between the leading and the trailing heads. Alternate stepping pathways also exist, including inchworm-like movements, backward steps, and foot stomps. Whether all the pathways are coupled to ATP hydrolysis remains to be determined. Here, to establish the principles governing the dynamics of processive movement, we present a theoretical framework that includes all of the alternative stepping mechanisms. Our theory bridges the gap between the elemental rates describing the biochemical and structural transitions in each head and the experimentally measurable quantities such as velocity, processivity, and probability of backward stepping. Our results, obtained under the assumption that the track is periodic and infinite, provide expressions that hold regardless of the topology of the network connecting the intermediate states, and are therefore capable of describing the function of any molecular motor. We apply the theory to myosin VI, a motor that takes frequent backward steps and moves forward with a combination of hand-over-hand and inchworm-like steps. Our model quantitatively reproduces various observables of myosin VI motility reported by four experimental groups. The theory is used to predict the gating mechanism, the pathway for backward stepping, and the energy consumption as a function of ATP concentration.

Exact distributions of currents and frenesy for Markov bridges

We consider discrete-time Markov bridges, chains whose initial and final states coincide. We derive exact finite-time formulae for the joint probability distributions of additive functionals of trajectories. We apply our theory to time-integrated currents and frenesy of enzymatic reactions, which may include absolutely irreversible transitions. We discuss the information that frenesy carries about the currents and show that bridges may violate known uncertainty relations in certain cases. Numerical simulations are in perfect agreement with our theory.

How kinesin waits for ATP affects the nucleotide and load dependence of the stepping kinetics

Conventional kinesin, responsible for directional transport of cellular vesicles, takes multiple nearly uniform 8.2-nm steps by consuming one ATP molecule per step as it walks toward the plus end of the microtubule (MT). Despite decades of intensive experimental and theoretical studies, there are gaps in the elucidation of key steps in the catalytic cycle of kinesin. How the motor waits for ATP to bind to the leading head is controversial. Two experiments using a similar protocol have arrived at different conclusions. One asserts that kinesin waits for ATP in a state with both the heads bound to the MT, whereas the other shows that ATP binds to the leading head after the trailing head detaches. To discriminate between the 2 scenarios, we developed a minimal model, which analytically predicts the outcomes of a number of experimental observable quantities (the distribution of run length, the distribution of velocity [[Formula: see text]], and the randomness parameter) as a function of an external resistive force (F) and ATP concentration ([T]). The differences in the predicted bimodality in [Formula: see text] as a function of F between the 2 models may be amenable to experimental testing. Most importantly, we predict that the F and [T] dependence of the randomness parameters differ qualitatively depending on the waiting states. The randomness parameters as a function of F and [T] can be quantitatively measured from stepping trajectories with very little prejudice in data analysis. Therefore, an accurate measurement of the randomness parameter and the velocity distribution as a function of load and nucleotide concentration could resolve the apparent controversy.

Theory of Nonequilibrium Free Energy Transduction by Molecular Machines

Biomolecular machines are protein complexes that convert between different forms of free energy. They are utilized in nature to accomplish many cellular tasks. As isothermal nonequilibrium stochastic objects at low Reynolds number, they face a distinct set of challenges compared with more familiar human-engineered macroscopic machines. Here we review central questions in their performance as free energy transducers, outline theoretical and modeling approaches to understand these questions, identify both physical limits on their operational characteristics and design principles for improving performance, and discuss emerging areas of research.

Structural conditions on complex networks for the Michaelis-Menten input-output response

The Michaelis-Menten (MM) fundamental formula describes how the rate of enzyme catalysis depends on substrate concentration. The familiar hyperbolic relationship was derived by timescale separation for a network of three reactions. The same formula has subsequently been found to describe steady-state input-output responses in many biological contexts, including single-molecule enzyme kinetics, gene regulation, transcription, translation, and force generation. Previous attempts to explain its ubiquity have been limited to networks with regular structure or simplifying parametric assumptions. Here, we exploit the graph-based linear framework for timescale separation to derive general structural conditions under which the MM formula arises. The conditions require a partition of the graph into two parts, akin to a "coarse graining" into the original MM graph, and constraints on where and how the input variable occurs. Other features of the graph, including the numerical values of parameters, can remain arbitrary, thereby explaining the formula's ubiquity. For systems at thermodynamic equilibrium, we derive a necessary and sufficient condition. For systems away from thermodynamic equilibrium, especially those with irreversible reactions, distinct structural conditions arise and a general characterization remains open. Nevertheless, our results accommodate, in much greater generality, all examples known to us in the literature.

Phase transition in thermodynamically consistent biochemical oscillators

Biochemical oscillations are ubiquitous in living organisms. In an autonomous system, not influenced by an external signal, they can only occur out of equilibrium. We show that they emerge through a generic nonequilibrium phase transition, with a characteristic qualitative behavior at criticality. The control parameter is the thermodynamic force which must be above a certain threshold for the onset of biochemical oscillations. This critical behavior is characterized by the thermodynamic flux associated with the thermodynamic force, its diffusion coefficient, and the stationary distribution of the oscillating chemical species. We discuss metrics for the precision of biochemical oscillations by comparing two observables, the Fano factor associated with the thermodynamic flux and the number of coherent oscillations. Since the Fano factor can be small even when there are no biochemical oscillations, we argue that the number of coherent oscillations is more appropriate to quantify the precision of biochemical oscillations. Our results are obtained with three thermodynamically consistent versions of known models: the Brusselator, the activator-inhibitor model, and a model for KaiC oscillations.

Quantifying fluctuations in reversible enzymatic cycles and clocks

Biochemical reactions are fundamentally noisy at a molecular scale. This limits the precision of reaction networks, but it also allows fluctuation measurements that may reveal the structure and dynamics of the underlying biochemical network. Here, we study nonequilibrium reaction cycles, such as the mechanochemical cycle of molecular motors, the phosphorylation cycle of circadian clock proteins, or the transition state cycle of enzymes. Fluctuations in such cycles may be measured using either of two classical definitions of the randomness parameter, which we show to be equivalent in general microscopically reversible cycles. We define a stochastic period for reversible cycles and present analytical solutions for its moments. Furthermore, we associate the two forms of the randomness parameter with the thermodynamic uncertainty relation, which sets limits on the timing precision of the cycle in terms of thermodynamic quantities. Our results should prove useful also for the study of temporal fluctuations in more general networks.

Generic Schemes for Single-Molecule Kinetics. 3: Self-Consistent Pathway Solutions for Nonrenewal Processes

We recently developed a pathway analysis framework (paper 1) for describing single-molecule kinetics for renewal (i.e., memoryless) processes based on the decomposition of a kinetic scheme into generic structures. In our approach, waiting time distribution functions corresponding to such structures are expressed in terms of self-consistent pathway solutions and concatenated to form measurable probability distribution functions (PDFs), affording a simple way to decompose and recombine a network. Here, we extend this framework to nonrenewal processes, which involve correlations between events, and employ it to formulate waiting time PDFs, including the first-passage time PDF, for a general kinetic network model. Our technique does not require the assumption of Poissonian kinetics, permitting a more general kinetic description than the usual rate approach, with minimal topological restrictiveness. To demonstrate the usefulness of this technique, we provide explicit calculations for our general model, which we adapt to two generic schemes for single-enzyme turnover with conformational interconversion. For each generic scheme, wherein the intermediate state(s) need not undergo Poissonian decay, the functional dependence of the mean first-passage time on the concentration of an external substrate is analyzed. When conformational detailed balance is satisfied, the enzyme turnover rate (related to the mean first-passage time) reduces to the celebrated Michaelis-Menten functional form, consistent with our previous work involving a similar scheme with all rate processes, thereby establishing further generality to this intriguing result. Our framework affords a general and intuitive approach for evaluating measurable waiting time PDFs and their moments, making it a potentially useful kinetic tool for a wide variety of single-molecule processes.

First-passage times in renewal and nonrenewal systems

Fluctuations in stochastic systems are usually characterized by full counting statistics, which analyzes the distribution of the number of events taking place in the fixed time interval. In an alternative approach, the distribution of the first-passage times, i.e., the time delays after which the counting variable reaches a certain threshold value, is studied. This paper presents the approach to calculate the first-passage time distribution in systems in which the analyzed current is associated with an arbitrary set of transitions within the Markovian network. Using this approach, it is shown that when the subsequent first-passage times are uncorrelated, there exist strict relations between the cumulants of the full counting statistics and the first-passage time distribution. On the other hand, when the correlations of the first-passage times are present, their distribution may provide additional information about the internal dynamics of the system in comparison to the full counting statistics; for example, it may reveal the switching between different dynamical states of the system. Additionally, I show that breaking of the fluctuation theorem for first-passage times may reveal the multicyclic nature of the Markovian network.

Full molecular trajectories of RNA polymerase at single base-pair resolution

Optical tweezers enable scientists to follow the dynamics of molecular motors at high resolution. The ability to discern a motor’s discrete steps reveals important insights on its operation. Some motors operate at the scale of angstroms, rendering the observation of their steps extremely challenging. In some cases, such small steps have been observed sporadically; however, the full molecular trajectories of steps and intervals between steps remain elusive due to instrumental noise. Here, we eliminate the main source of noise of most high-resolution dual-trap optical tweezers and developed both a single-molecule assay and a self-learning algorithm to uncover the full trajectories of such a motor: RNA polymerase. Using this method, a whole new set of experiments becomes possible.

In recent years, highly stable optical tweezers systems have enabled the characterization of the dynamics of molecular motors at very high resolution. However, the motion of many motors with angstrom-scale dynamics cannot be consistently resolved due to poor signal-to-noise ratio. Using an acousto-optic deflector to generate a “time-shared” dual-optical trap, we decreased low-frequency noise by more than one order of magnitude compared with conventional dual-trap optical tweezers. Using this instrument, we implemented a protocol that synthesizes single base-pair trajectories, which are used to test a Large State Space Hidden Markov Model algorithm to recover their individual steps. We then used this algorithm on real transcription data obtained in the same instrument to fully uncover the molecular trajectories of Escherichia coli RNA polymerase. We applied this procedure to reveal the effect of pyrophosphate on the distribution of dwell times between consecutive polymerase steps.

Revealing dynamics of helicase translocation on single-stranded DNA using high-resolution nanopore tweezers

Enzymes that operate on DNA or RNA perform the core functions of replication and expression in all of biology. To gain high-resolution access to the detailed mechanistic behavior of these enzymes, we developed single-molecule picometer-resolution nanopore tweezers (SPRNT), a single-molecule technique in which the motion of polynucleotides through an enzyme is measured by a nanopore. SPRNT reveals two mechanical substates of the ATP hydrolysis cycle of the superfamily 2 helicase Hel308 during translocation on single-stranded DNA (ssDNA). By analyzing these substates at millisecond resolution, we derive a detailed kinetic model for Hel308 translocation along ssDNA that sheds light on how superfamily 1 and 2 helicases turn ATP hydrolysis into motion along DNA. Surprisingly, we find that the DNA sequence within Hel308 affects the kinetics of helicase translocation.

Intraflagellar transport velocity is governed by the number of active KIF17 and KIF3AB motors and their motility properties under load

Homodimeric KIF17 and heterotrimeric KIF3AB are processive, kinesin-2 family motors that act jointly to carry out anterograde intraflagellar transport (IFT), ferrying cargo along microtubules (MTs) toward the tips of cilia. How IFT trains attain speeds that exceed the unloaded rate of the slower, KIF3AB motor remains unknown. By characterizing the motility properties of kinesin-2 motors as a function of load we find that the increase in KIF3AB velocity, elicited by forward loads from KIF17 motors, cannot alone account for the speed of IFT trains in vivo. Instead, higher IFT velocities arise from an increased likelihood that KIF3AB motors dissociate from the MT, resulting in transport by KIF17 motors alone, unencumbered by opposition from KIF3AB. The rate of transport is therefore set by an equilibrium between a faster state, where only KIF17 motors move the train, and a slower state, where at least one KIF3AB motor on the train remains active in transport. The more frequently the faster state is accessed, the higher the overall velocity of the IFT train. We conclude that IFT velocity is governed by (i) the absolute numbers of each motor type on a given train, (ii) how prone KIF3AB is to dissociation from MTs relative to KIF17, and (iii) how prone both motors are to dissociation relative to binding MTs.

Local discretization method for overdamped Brownian motion on a potential with multiple deep wells

We present a general method for transforming the continuous diffusion equation describing overdamped Brownian motion on a time-independent potential with multiple deep wells to a discrete master equation. The method is based on an expansion in localized basis states of local metastable potentials that match the full potential in the region of each potential well. Unlike previous basis methods for discretizing Brownian motion on a potential, this approach is valid for periodic potentials with varying multiple deep wells per period and can also be applied to nonperiodic systems. We apply the method to a range of potentials and find that potential wells that are deep compared to five times the thermal energy can be associated with a discrete localized state while shallower wells are better incorporated into the local metastable potentials of neighboring deep potential wells.

Biophysical comparison of ATP synthesis mechanisms shows a kinetic advantage for the rotary process

The ATP synthase (F-ATPase) is a highly complex rotary machine that synthesizes ATP, powered by a proton electrochemical gradient. Why did evolution select such an elaborate mechanism over arguably simpler alternating-access processes that can be reversed to perform ATP synthesis? We studied a systematic enumeration of alternative mechanisms, using numerical and theoretical means. When the alternative models are optimized subject to fundamental thermodynamic constraints, they fail to match the kinetic ability of the rotary mechanism over a wide range of conditions, particularly under low-energy conditions. We used a physically interpretable, closed-form solution for the steady-state rate for an arbitrary chemical cycle, which clarifies kinetic effects of complex free-energy landscapes. Our analysis also yields insights into the debated "kinetic equivalence" of ATP synthesis driven by transmembrane pH and potential difference. Overall, our study suggests that the complexity of the F-ATPase may have resulted from positive selection for its kinetic advantage.

Tight-binding approach to overdamped Brownian motion on a bichromatic periodic potential

We present a theoretical treatment of overdamped Brownian motion on a time-independent bichromatic periodic potential with spatially fast- and slow-changing components. In our approach, we generalize the Wannier basis commonly used in the analysis of periodic systems to define a basis of S states that are localized at local minima of the potential. We demonstrate that the S states are orthonormal and complete on the length scale of the periodicity of the fast-changing potential, and we use the S-state basis to transform the continuous Smoluchowski equation for the system to a discrete master equation describing hopping between local minima. We identify the parameter regime where the master equation description is valid and show that the interwell hopping rates are well approximated by Kramers' escape rate in the limit of deep potential minima. Finally, we use the master equation to explore the system dynamics and determine the drift and diffusion for the system.

Skewness and Kurtosis in Statistical Kinetics

We obtain lower and upper bounds on the skewness and kurtosis associated with the cycle completion time of unicyclic enzymatic reaction schemes. Analogous to a well-known lower bound on the randomness parameter, the lower bounds on skewness and kurtosis are related to the number of intermediate states in the underlying chemical reaction network. Our results demonstrate that evaluating these higher order moments with single molecule data can lead to information about the enzymatic scheme that is not contained in the randomness parameter.

Fluctuating currents in stochastic thermodynamics. II. Energy conversion and nonequilibrium response in kinesin models

Unlike macroscopic engines, the molecular machinery of living cells is strongly affected by fluctuations. Stochastic thermodynamics uses Markovian jump processes to model the random transitions between the chemical and configurational states of these biological macromolecules. A recently developed theoretical framework [A. Wachtel, J. Vollmer, and B. Altaner, Phys. Rev. E 92, 042132 (2015)] provides a simple algorithm for the determination of macroscopic currents and correlation integrals of arbitrary fluctuating currents. Here we use it to discuss energy conversion and nonequilibrium response in different models for the molecular motor kinesin. Methodologically, our results demonstrate the effectiveness of the algorithm in dealing with parameter-dependent stochastic models. For the concrete biophysical problem our results reveal two interesting features in experimentally accessible parameter regions: the validity of a nonequilibrium Green-Kubo relation at mechanical stalling as well as a negative differential mobility for superstalling forces.

Dispersion for two classes of random variables: general theory and application to inference of an external ligand concentration by a cell

We derive expressions for the dispersion for two classes of random variables in Markov processes. Random variables such as current and activity pertain to the first class, which is composed of random variables that change whenever a jump in the stochastic trajectory occurs. The second class corresponds to the time the trajectory spends in a state (or cluster of states). While the expression for the first class follows straightforwardly from known results in the literature, we show that a similar formalism can be used to derive an expression for the second class. As an application, we use this formalism to analyze a cellular two-component network estimating an external ligand concentration. The uncertainty related to this external concentration is calculated by monitoring different random variables related to an internal protein. We show that, inter alia, monitoring the time spent in the phosphorylated state of the protein leads to a finite uncertainty only if there is dissipation, whereas the uncertainty obtained from the activity of the transitions of the internal protein can reach the Berg-Purcell limit even in equilibrium.

Examining kinesin processivity within a general gating framework

Kinesin-1 is a dimeric motor that transports cargo along microtubules, taking 8.2-nm steps in a hand-over-hand fashion. The ATP hydrolysis cycles of its two heads are maintained out of phase by a series of gating mechanisms, which lead to processive runs averaging ∼1 μm. A key structural element for inter-head coordination is the neck linker (NL), which connects the heads to the stalk. To examine the role of the NL in regulating stepping, we investigated NL mutants of various lengths using single-molecule optical trapping and bulk fluorescence approaches in the context of a general framework for gating. Our results show that, although inter-head tension enhances motor velocity, it is crucial neither for inter-head coordination nor for rapid rear-head release. Furthermore, cysteine-light mutants do not produce wild-type motility under load. We conclude that kinesin-1 is primarily front-head gated, and that NL length is tuned to enhance unidirectional processivity and velocity.

DOI:http://dx.doi.org/10.7554/eLife.07403.001

In cells, molecules are moved from one location to another by motor proteins. Kinesins are a large family of such motors that transport their cargos along long filaments known as microtubules. Most kinesin molecules are formed from two identical protein chains. Each chain has a motor region at one end (called the head) that can attach to microtubules. The other end of each protein chain wraps around its partner to form a common stalk region (called the tail) that links to the cargo being carried.

The two kinesin heads are connected to the tail via a ‘neck linker’ region, and they advance along the microtubule in strict alternation, similar to the way our legs move when walking. During each step, the front head remains tightly associated with the filament as the trailing head releases itself, advances beyond the front head, and reattaches to become the new leading head. The two heads need to coordinate their activities, so that at any given time, they're not at the same stage in the process. For example, if both heads remained bound to the microtubule at the same time, the motor would not be able to advance. If they both released, the motor would fall off the filament and diffuse away. However, the process by which the heads coordinate is not fully understood, and different models for how this process works have been proposed.

Now, Andreasson, Milic et al. have examined the role played by the neck linker in coordinating the motor's movement using a technique known as ‘optical trapping’. The experiments involved attaching microscopic beads to the motor proteins, which serve as markers that can be tracked. The beads can also be used to exert controlled forces on the kinesin molecules, to see how they respond to different loads.

Andreasson, Milic et al. extended the length of neck linker by inserting extra amino acids (which are the building blocks of proteins) into this region of the protein. It was found that kinesins can still walk even when each neck linker was extended by up to six additional amino acids. However, introducing even a single amino acid into the linker relaxed the normal tension that exists between the heads when these are both bound to the filament. This resulted in slowed speeds, shorter distances of travel, and less ability to sustain loads. The experimental results suggest that the length of the neck linker in naturally occurring kinesins may be optimized to support maximum movement. Based on their data, Andreasson, Milic et al. propose a general framework for understanding the communication that needs to take place between the heads in order to walk in a coordinated manner. Further work is required to understand if motor proteins other than kinesins can also be understood with this same framework.

DOI:http://dx.doi.org/10.7554/eLife.07403.002

The Mechanochemical Cycle of Mammalian Kinesin-2 KIF3A/B under Load

The response of motor proteins to external loads underlies their ability to work in teams and determines the net speed and directionality of cargo transport. The mammalian kinesin-2, KIF3A/B, is a heterotrimeric motor involved in intraflagellar transport and vesicle motility in neurons. Bidirectional cargo transport is known to result from the opposing activities of KIF3A/B and dynein bound to the same cargo, but the load-dependent properties of kinesin-2 are poorly understood. We used a feedback-controlled optical trap to probe the velocity, run length, and unbinding kinetics of mouse KIF3A/B under various loads and nucleotide conditions. The kinesin-2 motor velocity is less sensitive than kinesin-1 to external forces, but its processivity diminishes steeply with load, and the motor was observed occasionally to slip and reattach. Each motor domain was characterized by studying homodimeric constructs, and a global fit to the data resulted in a comprehensive pathway that quantifies the principal force-dependent kinetic transitions. The properties of the KIF3A/B heterodimer are intermediate between the two homodimers, and the distinct load-dependent behavior is attributable to the properties of the motor domains and not to the neck linkers or the coiled-coil stalk. We conclude that the force-dependent movement of KIF3A/B differs significantly from conventional kinesin-1. Against opposing dynein forces, KIF3A/B motors are predicted to rapidly unbind and rebind, resulting in qualitatively different transport behavior from kinesin-1.

Mechanisms and topology determination of complex chemical and biological network systems from first-passage theoretical approach

The majority of chemical and biological processes can be viewed as complex networks of states connected by dynamic transitions. It is fundamentally important to determine the structure of these networks in order to fully understand the mechanisms of underlying processes. A new theoretical method of obtaining topologies and dynamic properties of complex networks, which utilizes a first-passage analysis, is developed. Our approach is based on a hypothesis that full temporal distributions of events between two arbitrary states contain full information on number of intermediate states, pathways, and transitions that lie between initial and final states. Several types of network systems are analyzed analytically and numerically. It is found that the approach is successful in determining structural and dynamic properties, providing a direct way of getting topology and mechanisms of general chemical network systems. The application of the method is illustrated on two examples of experimental studies of motor protein systems.

Extracting signal from noise: kinetic mechanisms from a Michaelis-Menten-like expression for enzymatic fluctuations

Enzyme-catalyzed reactions are naturally stochastic, and precision measurements of these fluctuations, made possible by single-molecule methods, promise to provide fundamentally new constraints on the possible mechanisms underlying these reactions. We review some aspects of statistical kinetics: a new field with the goal of extracting mechanistic information from statistical measures of fluctuations in chemical reactions. We focus on a widespread and important statistical measure known as the randomness parameter. This parameter is remarkably simple in that it is the squared coefficient of variation of the cycle completion times, although it places significant limits on the minimal complexity of possible enzymatic mechanisms. Recently, a general expression has been introduced for the substrate dependence of the randomness parameter that is for rate fluctuations what the Michaelis-Menten expression is for the mean rate of product generation. We discuss the information provided by the new kinetic parameters introduced by this expression and demonstrate that this expression can simplify the vast majority of published models.

First-passage problems in DNA replication: effects of template tension on stepping and exonuclease activities of a DNA polymerase motor

A DNA polymerase (DNAP) replicates a template DNA strand. It also exploits the template as the track for its own motor-like mechanical movement. In the polymerase mode it elongates the nascent DNA by one nucleotide in each step. However, whenever it commits an error by misincorporating an incorrect nucleotide, it can switch to an exonuclease mode. In the latter mode it excises the wrong nucleotide before switching back to its polymerase mode. We develop a stochastic kinetic model of DNA replication that mimics an in vitro experiment where single-stranded DNA, subjected to a mechanical tension F, is converted to double-stranded DNA by a single DNAP. The F-dependence of the average rate of replication, which depends on the rates of both polymerase and exonuclease activities of the DNAP, is in good qualitative agreement with the corresponding experimental results. We introduce nine novel distinct conditional dwell times of a DNAP. Using the method of first-passage times, we also derive the exact analytical expressions for the probability distributions of these conditional dwell times. The predicted F-dependences of these distributions are, in principle, accessible to single-molecule experiments.

Statistical analysis of molecular signal recording

A molecular device that records time-varying signals would enable new approaches in neuroscience. We have recently proposed such a device, termed a “molecular ticker tape”, in which an engineered DNA polymerase (DNAP) writes time-varying signals into DNA in the form of nucleotide misincorporation patterns. Here, we define a theoretical framework quantifying the expected capabilities of molecular ticker tapes as a function of experimental parameters. We present a decoding algorithm for estimating time-dependent input signals, and DNAP kinetic parameters, directly from misincorporation rates as determined by sequencing. We explore the requirements for accurate signal decoding, particularly the constraints on (1) the polymerase biochemical parameters, and (2) the amplitude, temporal resolution, and duration of the time-varying input signals. Our results suggest that molecular recording devices with kinetic properties similar to natural polymerases could be used to perform experiments in which neural activity is compared across several experimental conditions, and that devices engineered by combining favorable biochemical properties from multiple known polymerases could potentially measure faster phenomena such as slow synchronization of neuronal oscillations. Sophisticated engineering of DNAPs is likely required to achieve molecular recording of neuronal activity with single-spike temporal resolution over experimentally relevant timescales.

Recording of physiological signals from inaccessible microenvironments is often hampered by the macroscopic sizes of current recording devices. A signal-recording device constructed on a molecular scale could advance biology by enabling the simultaneous recording from millions or billions of cells. We recently proposed a molecular device for recording time-varying ion concentration signals: DNA polymerases (DNAPs) copy known template DNA strands with an error rate dependent on the local ion concentration. The resulting DNA polymers could then be sequenced, and with the help of statistical techniques, used to estimate the time-varying ion concentration signal experienced by the polymerase. We develop a statistical framework to treat this inverse problem and describe a technique to decode the ion concentration signals from DNA sequencing data. We also provide a novel method for estimating properties of DNAP dynamics, such as polymerization rate and pause frequency, directly from sequencing data. We use this framework to explore potential application scenarios for molecular recording devices, achievable via molecular engineering within the biochemical parameter ranges of known polymerases. We find that accurate recording of neural firing rate responses across several experimental conditions would likely be feasible using molecular recording devices with kinetic properties similar to those of known polymerases.

First passage times in homogeneous nucleation and self-assembly

Motivated by nucleation and molecular aggregation in physical, chemical, and biological settings, we present a thorough analysis of the general problem of stochastic self-assembly of a fixed number of identical particles in a finite volume. We derive the backward Kolmogorov equation (BKE) for the cluster probability distribution. From the BKE, we study the distribution of times it takes for a single maximal cluster to be completed, starting from any initial particle configuration. In the limits of slow and fast self-assembly, we develop analytical approaches to calculate the mean cluster formation time and to estimate the first assembly time distribution. We find, both analytically and numerically, that faster detachment can lead to a shorter mean time to first completion of a maximum-sized cluster. This unexpected effect arises from a redistribution of trajectory weights such that upon increasing the detachment rate, paths that take a shorter time to complete a cluster become more likely.

Sequence-dependent base pair stepping dynamics in XPD helicase unwinding

Helicases couple the chemical energy of ATP hydrolysis to directional translocation along nucleic acids and transient duplex separation. Understanding helicase mechanism requires that the basic physicochemical process of base pair separation be understood. This necessitates monitoring helicase activity directly, at high spatio-temporal resolution. Using optical tweezers with single base pair (bp) resolution, we analyzed DNA unwinding by XPD helicase, a Superfamily 2 (SF2) DNA helicase involved in DNA repair and transcription initiation. We show that monomeric XPD unwinds duplex DNA in 1-bp steps, yet exhibits frequent backsteps and undergoes conformational transitions manifested in 5-bp backward and forward steps. Quantifying the sequence dependence of XPD stepping dynamics with near base pair resolution, we provide the strongest and most direct evidence thus far that forward, single-base pair stepping of a helicase utilizes the spontaneous opening of the duplex. The proposed unwinding mechanism may be a universal feature of DNA helicases that move along DNA phosphodiester backbones. DOI:http://dx.doi.org/10.7554/eLife.00334.001.

Mfd as a central partner of transcription coupled repair

Transcription-coupled repair (TCR) is one of the key of the nucleotide excision repair (NER) pathways required to preserve genome integrity. Although understanding TCR is still a major challenge, recent single-molecule experiments have brought new insights into the initial steps of TCR leading to new perspectives.

Error correction during DNA replication

DNA polymerase (DNAP) is a dual-purpose enzyme that plays two opposite roles in two different situations during DNA replication. It plays its a normal role as a polymerase catalyzing the elongation of a new DNA molecule by adding a monomer. However, it can switch to the role of an exonuclease and shorten the same DNA by cleavage of the last incorporated monomer from the nascent DNA. Just as misincorporated nucleotides can escape exonuclease causing a replication error, the correct nucleotide may get sacrificed unnecessarily by erroneous cleavage. The interplay of polymerase and exonuclease activities of a DNAP is explored here by developing a minimal stochastic kinetic model of DNA replication. Exact analytical expressions are derived for a few key statistical distributions; these characterize the temporal patterns in the mechanical stepping and the chemical (cleavage) reaction. The Michaelis-Menten-like analytical expression derived for the average rates of these two processes not only demonstrate the effects of their coupling, but are also utilized to measure the extent of replication error and erroneous cleavage.

Single-molecule tracking of collagenase on native type I collagen fibrils reveals degradation mechanism

BACKGROUND

Collagen, the most abundant human protein, is the principal component of the extracellular matrix and plays important roles in maintaining tissue and organ integrity. Highly resistant to proteolysis, fibrillar collagen is degraded by specific matrix metalloproteases (MMPs). Degradation of fibrillar collagen underlies processes including tissue remodeling, wound healing, and cancer metastasis. However, the mechanism of native collagen fibril degradation remains poorly understood.

RESULTS

Here we present the results of high-resolution tracking of individual MMPs degrading type I collagen fibrils. MMP1 exhibits cleavage-dependent biased and hindered diffusion but spends 90% ± 3% of the time in one of at least two distinct pause states. One class of exponentially distributed pauses (class I pauses) occurs randomly along the fibril, whereas a second class of pauses (class II pauses) exhibits multistep escape kinetics and occurs periodically at intervals of 1.3 ± 0.2 μm and 1.5 ± 0.2 μm along the fibril. After these class II pauses, MMP1 moved faster and farther in one direction along the fibril, indicative of biased motion associated with cleavage. Simulations indicate that 5% ± 2% of the class II pauses result in the initiation of processive collagen degradation, which continues for bursts of 15 ± 4 consecutive cleavage events.

CONCLUSIONS

These findings provide a mechanistic paradigm for type I collagen degradation by MMP1 and establish a general approach to investigate MMP-fibrillar collagen interactions. More generally, this work demonstrates the fundamental role of enzyme-substrate interactions including binding and motion in determining the activity of an enzyme on an extended substrate.

A universal pathway for kinesin stepping

Kinesin-1 is an ATP-driven, processive motor that transports cargo along microtubules in a tightly regulated stepping cycle. Efficient gating mechanisms ensure that the sequence of kinetic events proceeds in proper order, generating a large number of successive reaction cycles. To study gating, we created two mutant constructs with extended neck-linkers and measured their properties using single-molecule optical trapping and ensemble fluorescence techniques. Due to a reduction in the inter-head tension, the constructs access an otherwise rarely populated conformational state where both motor heads remain bound to the microtubule. ATP-dependent, processive backstepping and futile hydrolysis were observed under moderate hindering loads. Based on measurements, we formulated a comprehensive model for kinesin motion that incorporates reaction pathways for both forward and backward stepping. In addition to inter-head tension, we find that neck-linker orientation is also responsible for ensuring gating in kinesin.

Distribution of dwell times of a ribosome: effects of infidelity, kinetic proofreading and ribosome crowding

Ribosome is a molecular machine that polymerizes a protein where the sequence of the amino acid residues, the monomers of the protein, is dictated by the sequence of codons (triplets of nucleotides) on a messenger RNA (mRNA) that serves as the template. The ribosome is a molecular motor that utilizes the template mRNA strand also as the track. Thus, in each step the ribosome moves forward by one codon and, simultaneously, elongates the protein by one amino acid. We present a theoretical model that captures most of the main steps in the mechanochemical cycle of a ribosome. The stochastic movement of the ribosome consists of an alternating sequence of pause and translocation; the sum of the durations of a pause and the following translocation is the time of dwell of the ribosome at the corresponding codon. We derive the analytical expression for the distribution of the dwell times of a ribosome in our model. Wherever experimental data are available, our theoretical predictions are consistent with those results. We suggest appropriate experiments to test the new predictions of our model, particularly the effects of the quality control mechanism of the ribosome and that of their crowding on the mRNA track.

Mechanistic constraints from the substrate concentration dependence of enzymatic fluctuations

The time it takes an enzyme to complete its reaction is a stochastic quantity governed by thermal fluctuations. With the advent of high-resolution methods of single-molecule manipulation and detection, it is now possible to observe directly this natural variation in the enzymatic cycle completion time and extract kinetic information from the statistics of its fluctuations. To this end, the inverse of the squared coefficient of variation, which we term n(min), is a useful measure of fluctuations because it places a strict lower limit on the number of kinetic states in the enzymatic mechanism. Here we show that there is a single general expression for the substrate dependence of n(min) for a wide range of kinetic models. This expression is governed by three kinetic parameters, which we term N(L), N(S), and alpha. These parameters have simple geometric interpretations and provide clear constraints on possible kinetic mechanisms. As a demonstration of this analysis, we fit the fluctuations in the dwell times of the packaging motor of the bacteriophage varphi29, revealing additional features of the nucleotide loading process in this motor. Because a diverse set of kinetic models display the same substrate dependence for their fluctuations, the expression for this general dependence may prove of use in the characterization and study of the dynamics of a wide range of enzymes.

Revealing the base pair stepping dynamics of nucleic acid motor proteins with optical traps

Nearly all aspects of nucleic acid metabolism involve motor proteins. This diverse group of enzymes, which includes DNA and RNA polymerases, the ribosome, helicases, and other translocases, converts chemical energy in the form of bond hydrolysis into concerted motion along nucleic acid filaments. The direct observation of this motion at its fundamental distance scale of one base pair has required the development of new ultrasensitive techniques. Recent advances in optical traps have now made these length scales, once the exclusive realm of crystallographic techniques, accessible. Several new studies using optical traps have revealed for the first time how motor proteins translocate along their substrates in a stepwise fashion. Though these techniques have only begun to be applied to biological problems, the unprecedented access into nucleic acid motor protein movement has already provided important insights into their mechanism. In this perspective, we review these advances and offer our view on the future of this exciting development.

Simulation and analysis of single-ribosome translation

In the cell, proteins are synthesized by ribosomes in a multi-step process called translation. The ribosome translocates along the messenger RNA to read the codons that encode the amino acid sequence of a protein. Elongation factors, including EF-G and EF-Tu, are used to catalyze the process. Recently, we have shown that translation can be followed at the single-molecule level using optical tweezers; this technique allows us to study the kinetics of translation by measuring the lifetime the ribosome spends at each codon. Here, we analyze the data from single-molecule experiments and fit the data with simple kinetic models. We also simulate the translation kinetics based on a multi-step mechanism from ensemble kinetic measurements. The mean lifetimes from the simulation were consistent with our experimental single-molecule measurements. We found that the calculated lifetime distributions were fit in general by equations with up to five rate-determining steps. Two rate-determining steps were only obtained at low concentrations of elongation factors. These analyses can be used to design new single-molecule experiments to better understand the kinetics and mechanism of translation.

Stochastic kinetics of ribosomes: single motor properties and collective behavior

Syntheses of protein molecules in a cell are carried out by ribosomes. A ribosome can be regarded as a molecular motor which utilizes the input chemical energy to move on a messenger RNA (mRNA) track that also serves as a template for the polymerization of the corresponding protein. The forward movement, however, is characterized by an alternating sequence of translocation and pause. Using a quantitative model, which captures the mechanochemical cycle of an individual ribosome, we derive an exact analytical expression for the distribution of its dwell times at the successive positions on the mRNA track. Inverse of the average dwell time satisfies a "Michaelis-Menten-type" equation and is consistent with the general formula for the average velocity of a molecular motor with an unbranched mechanochemical cycle. Extending this formula appropriately, we also derive the exact force-velocity relation for a ribosome. Often many ribosomes simultaneously move on the same mRNA track, while each synthesizes a copy of the same protein. We extend the model of a single ribosome by incorporating steric exclusion of different individuals on the same track. We draw the phase diagram of this model of ribosome traffic in three-dimensional spaces spanned by experimentally controllable parameters. We suggest new experimental tests of our theoretical predictions.