Molecular motors: thermodynamics and the random walk

The biochemical cycle of a molecular motor provides the essential link between its thermodynamics and kinetics. The thermodynamics of the cycle determine the motor's ability to perform mechanical work, whilst the kinetics of the cycle govern its stochastic behaviour. We concentrate here on tightly coupled, processive molecular motors, such as kinesin and myosin V, which hydrolyse one molecule of ATP per forward step. Thermodynamics require that, when such a motor pulls against a constant load f, the ratio of the forward and backward products of the rate constants for its cycle is exp [-(DeltaG + u(0)f)/kT], where -DeltaG is the free energy available from ATP hydrolysis and u(0) is the motor's step size. A hypothetical one-state motor can therefore act as a chemically driven ratchet executing a biased random walk. Treating this random walk as a diffusion problem, we calculate the forward velocity v and the diffusion coefficient D and we find that its randomness parameter r is determined solely by thermodynamics. However, real molecular motors pass through several states at each attachment site. They satisfy a modified diffusion equation that follows directly from the rate equations for the biochemical cycle and their effective diffusion coefficient is reduced to D-v(2)tau, where tau is the time-constant for the motor to reach the steady state. Hence, the randomness of multistate motors is reduced compared with the one-state case and can be used for determining tau. Our analysis therefore demonstrates the intimate relationship between the biochemical cycle, the force-velocity relation and the random motion of molecular motors.     

Distributions of activation energy barriers that produce stretched exponential probability distributions for the time spent in each state of the two state reaction A⇌B

Many biochemical reactions consist of the spontaneous fluctuation between two states: A⇌B. For example these two states could be a ligand bound to an enzyme and the ligand and the enzyme separated from each other. A typical case would be the unbinding of CO from myoglobin (Mb), namely, MbCO⇌Mb+CO. Another example is the fluctuation in the ion channel protein in the cell membrane between conformations that are closed to the passage of ions and those that are open to the passage of ions, namely, closed⇌open. Such chemical reactions can be described as two energy levels corresponding to the two states, separated by a distribution of activation energy barriers. Since a kinetic rate can be associated with each energy barrier, this is also equivalent to a distribution of kinetic rate constants. We derive the distribution of the kinetic rates that produces the stretched exponential probability distribution, exp(−at ^b ) where 0<b≤1, which has been observed for such reactions. We also derive the form of the cumulative probability distribution when the pathways connecting the states have minimum or maximum rate constants.     

Dwell time symmetry in random walks and molecular motors

The statistics of steps and dwell times in reversible molecular motors differ from those of cycle completion in enzyme kinetics. The reason is that a step is only one of several transitions in the mechanochemical cycle. As a result, theoretical results for cycle completion in enzyme kinetics do not apply to stepping data. To allow correct parameter estimation, and to guide data analysis and experiment design, a theoretical treatment is needed that takes this observation into account. In this article, we model the distribution of dwell times and number of forward and backward steps using first passage processes, based on the assumption that forward and backward steps correspond to different directions of the same transition. We extend recent results for systems with a single cycle and consider the full dwell time distributions as well as models with multiple pathways, detectable substeps, and detachments. Our main results are a symmetry relation for the dwell time distributions in reversible motors, and a relation between certain relative step frequencies and the free energy per cycle. We demonstrate our results by analyzing recent stepping data for a bacterial flagellar motor, and discuss the implications for the efficiency and reversibility of the force-generating subunits.     

Relationship between extreme pathways and structurally minimal pathways

The determination of reaction pathways is one of the most important functions that should be performed in exploring the kinetics of catalyzed chemical reactions or biochemical reactions, the latter being generally catalyzed by enzymes. It is proven that the terms, “type-I extreme pathway” and “structurally minimal pathway”, both introduced to characterize the kinetics of a catalyzed reaction are equivalent. These two terms are based on two distinct methodologies, one mainly rooted in convex analysis and the other in graph theory. The equivalence promises further even more effective methods for reaction-pathway identification by synergistic integration of existing ones.     

Tools and strategies for constructing cell-free enzyme pathways

Single enzyme systems or engineered microbial hosts have been used for decades but the notion of assembling multiple enzymes into cell-free synthetic pathways is a relatively new development. The extensive possibilities that stem from this synthetic concept makes it a fast growing and potentially high impact field for biomanufacturing fine and platform chemicals, pharmaceuticals and biofuels. However, the translation of individual single enzymatic reactions into cell-free multi-enzyme pathways is not trivial. In reality, the kinetics of an enzyme pathway can be very inadequate and the production of multiple enzymes can impose a great burden on the economics of the process. We examine here strategies for designing synthetic pathways and draw attention to the requirements of substrates, enzymes and cofactor regeneration systems for improving the effectiveness and sustainability of cell-free biocatalysis. In addition, we comment on methods for the immobilisation of members of a multi-enzyme pathway to enhance the viability of the system. Finally, we focus on the recent development of integrative tools such as in silico pathway modelling and high throughput flux analysis with the aim of reinforcing their indispensable role in the future of cell-free biocatalytic pathways for biomanufacturing.     

Detection of elementary flux modes in biochemical networks: a promising tool for pathway analysis and metabolic engineering

Rational metabolic engineering requires powerful theoretical methods such as pathway analysis, in which the topology of metabolic networks is considered. All metabolic capabilities in steady states are composed of elementary flux modes, which are minimal sets of enzymes that can each generate valid steady states. The modes of the fructose-2,6-bisphosphate cycle, the combined tricarboxylic-acid-glyoxylate-shunt system and tryptophan synthesis are used here for illustration. This approach can be used for many biotechnological applications such as increasing the yield of a product, channelling a product into desired pathways and in functional reconstruction from genomic data.     

Metabolites: a helping hand for pathway evolution?

The evolution of enzymes and pathways is under debate. Recent studies show that recruitment of single enzymes from different pathways could be the driving force for pathway evolution. Other mechanisms of evolution, such as pathway duplication, enzyme specialization, de novo invention of pathways or retro-evolution of pathways, appear to be less abundant. Twenty percent of enzyme superfamilies are quite variable, not only in changing reaction chemistry or metabolite type but in changing both at the same time. These variable superfamilies account for nearly half of all known reactions. The most frequently occurring metabolites provide a helping hand for such changes because they can be accommodated by many enzyme superfamilies. Thus, a picture is emerging in which new pathways are evolving from central metabolites by preference, thereby keeping the overall topology of the metabolic network.     

Mechanochemistry of t7 DNA helicase

The bacteriophage T7 helicase is a ring-shaped hexameric motor protein that unwinds double-stranded DNA during DNA replication and recombination. To accomplish this it couples energy from the nucleotide hydrolysis cycle to translocate along one of the DNA strands. Here, we combine computational biology with new biochemical measurements to infer the following properties of the T7 helicase: (1) all hexameric subunits are catalytic; (2) the mechanical movement along the DNA strand is driven by the binding transition of nucleotide into the catalytic site; (3) hydrolysis is coordinated between adjacent subunits that bind DNA; (4) the hydrolysis step changes the affinity of a subunit for DNA allowing passage of DNA from one subunit to the next. We construct a numerical optimization scheme to analyze transient and steady-state biochemical measurements to determine the rate constants for the hydrolysis cycle and determine the flux distribution through the reaction network. We find that, under physiological and experimental conditions, there is no dominant pathway; rather there is a distribution of pathways that varies with the ambient conditions. Our analysis methods provide a systematic procedure to study kinetic pathways of multi-subunit, multi-state cooperative enzymes.     

An Algebraic Approach to Signaling Cascades with n Layers

Posttranslational modification of proteins is key in transmission of signals in cells. Many signaling pathways contain several layers of modification cycles that mediate and change the signal through the pathway. Here, we study a simple signaling cascade consisting of n layers of modification cycles such that the modified protein of one layer acts as modifier in the next layer. Assuming mass-action kinetics and taking the formation of intermediate complexes into account, we show that the steady states are solutions to a polynomial in one variable and in fact that there is exactly one steady state for any given total amounts of substrates and enzymes.     

Alamethicin channel kinetics: Studies using fluctuation analysis and multifractal fluctuation analysis

The kinetics of the alamethicine channel incorporated into BLM was studied using detrended fluctuation analysis and multifractal detrended fluctuation analysis. Detrended fluctuation analysis has shown that both the event series formed with dwell times on fixed channel conductivity levels and the event series formed with opened states on all channel conductivity levels are random processes. However, multifractal detrended fluctuation analysis indicated that only dwell times series on fixed channel conductivity levels are random, but the event series formed with opened states on all channel conductivity levels are the correlated multifractal processes.     

Free energy transduction in a chemical motor model

Motor enzymes catalyse chemical reactions, like the hydrolysis of ATP, and in the process they also perform work. Recent studies indicate that motor enzymes perform work with specific biochemical steps in their catalysed reactions, challenging the classical view that work can only be performed within a biochemical state. To address these studies an alternative class of models, often referred to as chemical motor models, has emerged in which motors perform work with biochemical transitions. In this paper, I develop a novel, self-consistent framework for chemical motor models, which accommodates multiple pathways for free energy transfer, predicts rich behaviors from the simplest multi-motor systems, and provides important new insights into muscle and motor function.     

Modeling of uncertainties in biochemical reactions

Mathematical modeling is an indispensable tool for research and development in biotechnology and bioengineering. The formulation of kinetic models of biochemical networks depends on knowledge of the kinetic properties of the enzymes of the individual reactions. However, kinetic data acquired from experimental observations bring along uncertainties due to various experimental conditions and measurement methods. In this contribution, we propose a novel way to model the uncertainty in the enzyme kinetics and to predict quantitatively the responses of metabolic reactions to the changes in enzyme activities under uncertainty. The proposed methodology accounts explicitly for mechanistic properties of enzymes and physico‐chemical and thermodynamic constraints, and is based on formalism from systems theory and metabolic control analysis. We achieve this by observing that kinetic responses of metabolic reactions depend: (i) on the distribution of the enzymes among their free form and all reactive states; (ii) on the equilibrium displacements of the overall reaction and that of the individual enzymatic steps; and (iii) on the net fluxes through the enzyme. Relying on this observation, we develop a novel, efficient Monte Carlo sampling procedure to generate all states within a metabolic reaction that satisfy imposed constrains. Thus, we derive the statistics of the expected responses of the metabolic reactions to changes in enzyme levels and activities, in the levels of metabolites, and in the values of the kinetic parameters. We present aspects of the proposed framework through an example of the fundamental three‐step reversible enzymatic reaction mechanism. We demonstrate that the equilibrium displacements of the individual enzymatic steps have an important influence on kinetic responses of the enzyme. Furthermore, we derive the conditions that must be satisfied by a reversible three‐step enzymatic reaction operating far away from the equilibrium in order to respond to changes in metabolite levels according to the irreversible Michelis–Menten kinetics. The efficient sampling procedure allows easy, scalable, implementation of this methodology to modeling of large‐scale biochemical networks. Biotechnol. Bioeng. 2011;108: 413–423. © 2010 Wiley Periodicals, Inc.     

Brownian motion, fluctuation and life

The measurements of dynamic behaviors of biomolecules in relation to their functions have been allowed using single molecule measurements. Thermal Brownian motion causes random step motion of motor proteins and structural fluctuation of protein molecules between multiple states. In hierarchic structure of life, the fluctuation is modulated. Random fluctuation is biased to directional motion and reactions as a result of interaction of proteins. The fluctuation of kinetic state of signaling proteins results in polarization and localization of cells. A recognition process in brain is also explained by the equation analogous to biochemical reaction at the molecular level. Thus dynamic processes originated from thermal motion may play an important role in activation processes in life.     

Using gene knockouts to investigate plant metabolism

Arabidopsis functional genomics resources now make the isolation of knockout mutants in any gene of choice both realistic and increasingly straightforward. Coupled with the completion of the genome sequence, this reverse genetics approach provides a platform facilitating dramatic progress in our understanding of fundamental aspects of plant metabolism. Recent experience shows that knockouts of genes encoding enzymes of primary metabolism can produce mutants with clear and sometimes unexpected phenotypes. They can provide new information about old pathways. Specific functions for individual members of multigene families can be revealed. Knockouts of enzymes of undefined function can lead to the discovery of those functions, and the analysis of enzymes which have previously never been studied at the biochemical level offers the potential to reveal new pathways of plant metabolism. Furthermore, the mutants isolated provide the starting point for genetic modification experiments to determine exactly how metabolism fuels growth and development, so providing a rational basis for the future modification of plant productivity.     

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.     

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.     

Revisiting a ‘simple’ fungal metabolic pathway reveals redundancy,complexity and diversity

Next to d -glucose, the pentoses l -arabinose and d -xylose are the main monosaccharide components of plant cell wall polysaccharides and are therefore of major importance in biotechnological applications that use plant biomass as a substrate. Pentose catabolism is one of the best-studied pathways of primary metabolism of Aspergillus niger, and an initial outline of this pathway with individual enzymes covering each step of the pathway has been previously established. However, although growth on l -arabinose and/or d -xylose of most pentose catabolic pathway (PCP) single deletion mutants of A. niger has been shown to be negatively affected, it was not abolished, suggesting the involvement of additional enzymes. Detailed analysis of the single deletion mutants of the known A. niger PCP genes led to the identification of additional genes involved in the pathway. These results reveal a high level of complexity and redundancy in this pathway, emphasizing the need for a comprehensive understanding of metabolic pathways before entering metabolic engineering of such pathways for the generation of more efficient fungal cell factories.     

Probing the kinetics of single molecule protein folding

We propose an approach to integrate the theory, simulations, and experiments in protein-folding kinetics. This is realized by measuring the mean and high-order moments of the first-passage time and its associated distribution. The full kinetics is revealed in the current theoretical framework through these measurements. In the experiments, information about the statistical properties of first-passage times can be obtained from the kinetic folding trajectories of single molecule experiments (for example, fluorescence). Theoretical/simulation and experimental approaches can be directly related. We study in particular the temperature-varying kinetics to probe the underlying structure of the folding energy landscape. At high temperatures, exponential kinetics is observed; there are multiple parallel kinetic paths leading to the native state. At intermediate temperatures, nonexponential kinetics appears, revealing the nature of the distribution of local traps on the landscape and, as a result, discrete kinetic paths emerge. At very low temperatures, exponential kinetics is again observed; the dynamics on the underlying landscape is dominated by a single barrier. The ratio between first-passage-time moments is proposed to be a good variable to quantitatively probe these kinetic changes. The temperature-dependent kinetics is consistent with the strange kinetics found in folding dynamics experiments. The potential applications of the current results to single-molecule protein folding are discussed.