Mechanistic constraints from the substrate concentration dependence of enzymatic fluctuations

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.

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.

SlyD Accelerates trans-to-cis Prolyl Isomerization in a Mechanosignaling Protein under Load

Prolyl isomerization is recognized as one of the key regulatory mechanisms, which plays a crucial role in cell signaling, ion channel gating, phage virus infection, and molecular timing. This isomerization is usually slow but often accelerated by an enzyme, called peptidyl-prolyl isomerase (PPIase). In the current project, we investigate using single-molecule force spectroscopy (SMFS) the impact of a bacterial PPIase, SlyD, on the cis-trans isomerization of the proline 2225 (P2225) in an isolated 20th domain of a cytoskeletal mechanosensing protein filamin-A (FlnA20). To explore the FlnA20-PPIase interaction, we have used multiple SMFS modes, like constant velocity, constant distance, and jumping trap experiments. In our previous study, we reported the unique nature of the P2225, which is conserved in all naturally occurring filamins and can slowly (minutes) interconvert between cis-trans isomers, in absence of any PPIase. Our current results show a staggering 25-fold acceleration of the trans-to-cis isomerization rate in the presence of saturating SlyD concentration (7.25 μM) compared to the unenzymatic condition. A SlyD concentration-dependent depletion of the trans isomeric lifetime was also observed. Additionally, we observed that SlyD stabilizes the cis-isomer in the native state of FlnA20 by ∼2 k_BT. This is the first single-molecule observation of the cis-trans isomerization catalysis by a PPIase in a mechanosensing protein.

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.

Motion of kinesin in a viscoelastic medium

Kinesin is a molecular motor that transports cargo along microtubules. The results of many in vitro experiments on kinesin-1 are described by kinetic models in which one transition corresponds to the forward motion and subsequent binding of the tethered motor head. We argue that in a viscoelastic medium like the cytosol of a cell this step is not Markov and has to be described by a nonexponential waiting time distribution. We introduce a semi-Markov kinetic model for kinesin that takes this effect into account. We calculate, for arbitrary waiting time distributions, the moment generating function of the number of steps made, and determine from this the average velocity and the diffusion constant of the motor. We illustrate our results for the case of a waiting time distribution that is Weibull. We find that for realistic parameter values, viscoelasticity decreases the velocity and the diffusion constant, but increases the randomness (or Fano factor).

Single-molecule theory of enzymatic inhibition

The classical theory of enzymatic inhibition takes a deterministic, bulk based approach to quantitatively describe how inhibitors affect the progression of enzymatic reactions. Catalysis at the single-enzyme level is, however, inherently stochastic which could lead to strong deviations from classical predictions. To explore this, we take the single-enzyme perspective and rebuild the theory of enzymatic inhibition from the bottom up. We find that accounting for multi-conformational enzyme structure and intrinsic randomness should strongly change our view on the uncompetitive and mixed modes of inhibition. There, stochastic fluctuations at the single-enzyme level could make inhibitors act as activators; and we state—in terms of experimentally measurable quantities—a mathematical condition for the emergence of this surprising phenomenon. Our findings could explain why certain molecules that inhibit enzymatic activity when substrate concentrations are high, elicit a non-monotonic dose response when substrate concentrations are low.

Single molecule approaches demonstrated that enzymatic catalysis is stochastic which could lead to deviations from classical predictions. Here authors rebuild the theory of enzymatic inhibition to show that stochastic fluctuations on the single enzyme level could make inhibitors act as activators.

Active translocation of a semiflexible polymer assisted by an ATP-based molecular motor

In this work we study the assisted translocation of a polymer across a membrane nanopore, inside which a molecular motor exerts a force fuelled by the hydrolysis of ATP molecules. In our model the motor switches to its active state for a fixed amount of time, while it waits for an ATP molecule which triggers the motor, during an exponentially distributed time lapse. The polymer is modelled as a beads-springs chain with both excluded volume and bending contributions, and moves in a stochastic three dimensional environment modelled with a Langevin dynamics at a fixed temperature. The resulting dynamics shows a Michaelis-Menten translocation velocity that depends on the chain flexibility. The scaling behavior of the mean translocation time with the polymer length for different bending values is also investigated.

Variance-corrected Michaelis-Menten equation predicts transient rates of single-enzyme reactions and response times in bacterial gene-regulation

Many chemical reactions in biological cells occur at very low concentrations of constituent molecules. Thus, transcriptional gene-regulation is often controlled by poorly expressed transcription-factors, such as E.coli lac repressor with few tens of copies. Here we study the effects of inherent concentration fluctuations of substrate-molecules on the seminal Michaelis-Menten scheme of biochemical reactions. We present a universal correction to the Michaelis-Menten equation for the reaction-rates. The relevance and validity of this correction for enzymatic reactions and intracellular gene-regulation is demonstrated. Our analytical theory and simulation results confirm that the proposed variance-corrected Michaelis-Menten equation predicts the rate of reactions with remarkable accuracy even in the presence of large non-equilibrium concentration fluctuations. The major advantage of our approach is that it involves only the mean and variance of the substrate-molecule concentration. Our theory is therefore accessible to experiments and not specific to the exact source of the concentration fluctuations.

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.

Stepping and crowding of molecular motors: statistical kinetics from an exclusion process perspective

Motor enzymes are remarkable molecular machines that use the energy derived from the hydrolysis of a nucleoside triphosphate to generate mechanical movement, achieved through different steps that constitute their kinetic cycle. These macromolecules, nowadays investigated with advanced experimental techniques to unveil their molecular mechanisms and the properties of their kinetic cycles, are implicated in many biological processes, ranging from biopolymerization (e.g., RNA polymerases and ribosomes) to intracellular transport (motor proteins such as kinesins or dyneins). Although the kinetics of individual motors is well studied on both theoretical and experimental grounds, the repercussions of their stepping cycle on the collective dynamics still remains unclear. Advances in this direction will improve our comprehension of transport process in the natural intracellular medium, where processive motor enzymes might operate in crowded conditions. In this work, we therefore extend contemporary statistical kinetic analysis to study collective transport phenomena of motors in terms of lattice gas models belonging to the exclusion process class. Via numerical simulations, we show how to interpret and use the randomness calculated from single particle trajectories in crowded conditions. Importantly, we also show that time fluctuations and non-Poissonian behavior are intrinsically related to spatial correlations and the emergence of large, but finite, clusters of comoving motors. The properties unveiled by our analysis have important biological implications on the collective transport characteristics of processive motor enzymes in crowded conditions.

Thermodynamic uncertainty relation for biomolecular processes

Biomolecular systems like molecular motors or pumps, transcription and translation machinery, and other enzymatic reactions, can be described as Markov processes on a suitable network. We show quite generally that, in a steady state, the dispersion of observables, like the number of consumed or produced molecules or the number of steps of a motor, is constrained by the thermodynamic cost of generating it. An uncertainty ε requires at least a cost of 2k(B)T/ε2 independent of the time required to generate the output.

Ribosome excursions during mRNA translocation mediate broad branching of frameshift pathways

Programmed ribosomal frameshifting produces alternative proteins from a single transcript. -1 frameshifting occurs on Escherichia coli's dnaX mRNA containing a slippery sequence AAAAAAG and peripheral mRNA structural barriers. Here, we reveal hidden aspects of the frameshifting process, including its exact location on the mRNA and its timing within the translation cycle. Mass spectrometry of translated products shows that ribosomes enter the -1 frame from not one specific codon but various codons along the slippery sequence and slip by not just -1 but also -4 or +2 nucleotides. Single-ribosome translation trajectories detect distinctive codon-scale fluctuations in ribosome-mRNA displacement across the slippery sequence, representing multiple ribosomal translocation attempts during frameshifting. Flanking mRNA structural barriers mechanically stimulate the ribosome to undergo back-and-forth translocation excursions, broadly exploring reading frames. Both experiments reveal aborted translation around mutant slippery sequences, indicating that subsequent fidelity checks on newly adopted codon position base pairings lead to either resumed translation or early termination.

Analytic approaches to stochastic gene expression in multicellular systems

Deterministic thermodynamic models of the complex systems, which control gene expression in metazoa, are helping researchers identify fundamental themes in the regulation of transcription. However, quantitative single cell studies are increasingly identifying regulatory mechanisms that control variability in expression. Such behaviors cannot be captured by deterministic models and are poorly suited to contemporary stochastic approaches that rely on continuum approximations, such as Langevin methods. Fortunately, theoretical advances in the modeling of transcription have assembled some general results that can be readily applied to systems being explored only through a deterministic approach. Here, I review some of the recent experimental evidence for the importance of genetically regulating stochastic effects during embryonic development and discuss key results from Markov theory that can be used to model this regulation. I then discuss several pairs of regulatory mechanisms recently investigated through a Markov approach. In each case, a deterministic treatment predicts no difference between the mechanisms, but the statistical treatment reveals the potential for substantially different distributions of transcriptional activity. In this light, features of gene regulation that seemed needlessly complex evolutionary baggage may be appreciated for their key contributions to reliability and precision of gene expression.

A viral packaging motor varies its DNA rotation and step size to preserve subunit coordination as the capsid fills

Multimeric, ring-shaped molecular motors rely on the coordinated action of their subunits to perform crucial biological functions. During these tasks, motors often change their operation in response to regulatory signals. Here, we investigate a viral packaging machine as it fills the capsid with DNA and encounters increasing internal pressure. We find that the motor rotates the DNA during packaging and that the rotation per base pair increases with filling. This change accompanies a reduction in the motor's step size. We propose that these adjustments preserve motor coordination by allowing one subunit to make periodic, specific, and regulatory contacts with the DNA. At high filling, we also observe the downregulation of the ATP-binding rate and the emergence of long-lived pauses, suggesting a throttling-down mechanism employed by the motor near the completion of packaging. This study illustrates how a biological motor adjusts its operation in response to changing conditions, while remaining highly coordinated.

Role of substrate unbinding in Michaelis-Menten enzymatic reactions

The Michaelis-Menten equation provides a hundred-year-old prediction by which any increase in the rate of substrate unbinding will decrease the rate of enzymatic turnover. Surprisingly, this prediction was never tested experimentally nor was it scrutinized using modern theoretical tools. Here we show that unbinding may also speed up enzymatic turnover--turning a spotlight to the fact that its actual role in enzymatic catalysis remains to be determined experimentally. Analytically constructing the unbinding phase space, we identify four distinct categories of unbinding: inhibitory, excitatory, superexcitatory, and restorative. A transition in which the effect of unbinding changes from inhibitory to excitatory as substrate concentrations increase, and an overlooked tradeoff between the speed and efficiency of enzymatic reactions, are naturally unveiled as a result. The theory presented herein motivates, and allows the interpretation of, groundbreaking experiments in which existing single-molecule manipulation techniques will be adapted for the purpose of measuring enzymatic turnover under a controlled variation of unbinding rates. As we hereby show, these experiments will not only shed first light on the role of unbinding but will also allow one to determine the time distribution required for the completion of the catalytic step in isolation from the rest of the enzymatic turnover cycle.

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.

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.

High degree of coordination and division of labor among subunits in a homomeric ring ATPase

Ring NTPases of the ASCE superfamily perform a variety of cellular functions. An important question about the operation of these molecular machines is how the ring subunits coordinate their chemical and mechanical transitions. Here, we present a comprehensive mechanochemical characterization of a homomeric ring ATPase-the bacteriophage φ29 packaging motor-a homopentamer that translocates double-stranded DNA in cycles composed of alternating dwells and bursts. We use high-resolution optical tweezers to determine the effect of nucleotide analogs on the cycle. We find that ATP hydrolysis occurs sequentially during the burst and that ADP release is interlaced with ATP binding during the dwell, revealing a high degree of coordination among ring subunits. Moreover, we show that the motor displays an unexpected division of labor: although all subunits of the homopentamer bind and hydrolyze ATP during each cycle, only four participate in translocation, whereas the remaining subunit plays an ATP-dependent regulatory role.

Single-molecule studies of viral DNA packaging

Many double-stranded DNA bacteriophages and viruses use specialized ATP-driven molecular machines to package their genomes into tightly confined procapsid shells. Over the last decade, single-molecule approaches - and in particular, optical tweezers - have made key contributions to our understanding of this remarkable process. In this chapter, we review these advances and the insights they have provided on the packaging mechanisms of three bacteriophages: φ 29, λ, and T4.