A simple theory of motor protein kinetics and energetics. II

A three-state stochastic model of motor protein [Qian, Biophys. Chem. 67 (1997) pp. 263-267] is further developed to illustrate the relationship between the external load on an individual motor protein in aqueous solution with various ATP concentrations and its steady-state velocity. A wide variety of dynamic motor behavior are obtained from this simple model. For the particular case of free-load translocation being the most unfavorable step within the hydrolysis cycle, the load-velocity curve is quasi-linear, V/Vmax = (cF/Fmax-c)/(1-c), in contrast to the hyperbolic relationship proposed by A.V. Hill for macroscopic muscle. Significant deviation from the linearity is expected when the velocity is less than 10% of its maximal (free-load) value--a situation under which the processivity of motor diminishes and experimental observations are less certain. We then investigate the dependence of load-velocity curve on ATP (ADP) concentration. It is shown that the free load Vmax exhibits a Michaelis-Menten like behavior, and the isometric Fmax increases linearly with ln([ATP]/[ADP]). However, the quasi-linear region is independent of the ATP concentration, yielding an apparently ATP-independent maximal force below the true isometric force. Finally, the heat production as a function of ATP concentration and external load are calculated. In simple terms and solved with elementary algebra, the present model provides an integrated picture of biochemical kinetics and mechanical energetics of motor proteins.     

Protein folding as a nonlinear nonequilibrium thermodynamic process

Biofurcational theory of protein folding is developed describing the process of formation of protein native structure as a sequence of non-equilibrium irreversible fluctuations, specific for a particular protein. The model gives explanation to all characteristic features of the folding process: stochastic mechanism, short time and precision of proteins self-assembling. A constructive role of entropy in the formation of a highly ordered structure out of disorder is discussed. A numerical method of a priori calculations of polypeptide structures basing on principles of bifractional folding model is presented.     

Mechanical Model of Nuclei Ordering in Drosophila Embryos Reveals Dilution of Stochastic Forces

During the initial development of syncytial embryos, nuclei go through cycles of nuclear division and spatial rearrangement. The arising spatial pattern of nuclei is important for subsequent cellularization and morphing of the embryo. Although nuclei are contained within a common cytoplasm, cytoskeletal proteins are nonuniformly packaged into regions around every nucleus. In fact, cytoskeletal elements like microtubules and their associated motor proteins exert stochastic forces between nuclei, actively driving their rearrangement. Yet, it is unknown how the stochastic forces are balanced to maintain nuclear order in light of increased nuclear density upon every round of divisions. Here, we investigate the nuclear arrangements in Drosophila melanogaster over the course of several nuclear divisions starting from interphase 11. We develop a theoretical model in which we distinguish long-ranged passive forces due to the nuclei as inclusions in the elastic matrix, namely the cytoplasm, and active, stochastic forces arising from the cytoskeletal dynamics mediated by motor proteins. We perform computer simulations and quantify the observed degree of orientational and spatial order of nuclei. Solely doubling the nuclear density upon nuclear division, the model predicts a decrease in nuclear order. Comparing results to experimental recordings of tracked nuclei, we make contradictory observations, finding an increase in nuclear order upon nuclear divisions. Our analysis of model parameters resulting from this comparison suggests that overall motor protein density as well as relative active-force amplitude has to decrease by a factor of about two upon nuclear division to match experimental observations. We therefore expect a dilution of cytoskeletal motors during the rapid nuclear division to account for the increase in nuclear order during syncytial embryo development. Experimental measurements of kinesin-5 cluster lifetimes support this theoretical finding.     

Spatial modeling of vesicle transport and the cytoskeleton: the challenge of hitting the right road

The membrane trafficking machinery provides a transport and sorting system for many cellular proteins. We propose a mechanistic agent-based computer simulation to integrate and test the hypothesis of vesicle transport embedded into a detailed model cell. The method tracks both the number and location of the vesicles. Thus both the stochastic properties due to the low numbers and the spatial aspects are preserved. The underlying molecular interactions that control the vesicle actions are included in a multi-scale manner based on the model of Heinrich and Rapoport (2005). By adding motor proteins we can improve the recycling process of SNAREs and model cell polarization. Our model also predicts that coat molecules should have a high turnover at the compartment membranes, while the turnover of motor proteins has to be slow. The modular structure of the underlying model keeps it tractable despite the overall complexity of the vesicle system. We apply our model to receptor-mediated endocytosis and show how a polarized cytoskeleton structure leads to polarized distributions in the plasma membrane both of SNAREs and the Ste2p receptor in yeast. In addition, we can couple signal transduction and membrane trafficking steps in one simulation, which enables analyzing the effect of receptor-mediated endocytosis on signaling.     

Absorbent Markov chains as a model for the study of the evolution of proteins

The formalism of absorbent Markov chains, previously developed by Kemeny & Snell (1960) is used as a model for the study of the evolution of proteins. Within the limits of statistical analysis used, the amino acid substitution frequencies of McLachlan (1972) are explained by the numerical values derived from the model used. In addition, the amino acid composition of proteins is partially explained and the relative mutability of amino acids receives a new interpretation in the light of the above mentioned stochastic model. The results show that some basic aspect of protein evolution can be predicted by a stochastic model and therefore a significant component of protein evolution is driven by a random element.     

Active Degradation Explains the Distribution of Nuclear Proteins during Cellular Senescence

The amount of cellular proteins is a crucial parameter that is known to vary between cells as a function of the replicative passages, and can be important during physiological aging. The process of protein degradation is known to be performed by a series of enzymatic reactions, ranging from an initial step of protein ubiquitination to their final fragmentation by the proteasome. In this paper we propose a stochastic dynamical model of nuclear proteins concentration resulting from a balance between a constant production of proteins and their degradation by a cooperative enzymatic reaction. The predictions of this model are compared with experimental data obtained by fluorescence measurements of the amount of nuclear proteins in murine tail fibroblast (MTF) undergoing cellular senescence. Our model provides a three-parameter stationary distribution that is in good agreement with the experimental data even during the transition to the senescent state, where the nuclear protein concentration changes abruptly. The estimation of three parameters (cooperativity, saturation threshold, and maximal velocity of the reaction), and their evolution during replicative passages shows that only the maximal velocity varies significantly. Based on our modeling we speculate the reduction of functionality of the protein degradation mechanism as a possible competitive inhibition of the proteasome.     

Protein localization with flexible DNA or RNA

Localization of activity is ubiquitous in life, and also within sub-cellular compartments. Localization provides potential advantages as different proteins involved in the same cellular process may supplement each other on a fast timescale. It might also prevent proteins from being active in other regions of the cell. However localization is at odds with the spreading of unbound molecules by diffusion. We model the cost and gain for specific enzyme activity using localization strategies based on binding to sites of intermediate specificity. While such bindings in themselves decrease the activity of the protein on its target site, they may increase protein activity if stochastic motion allows the acting protein to touch both the intermediate binding site and the specific site simultaneously. We discuss this strategy in view of recent suggestions on long non-coding RNA as a facilitator of localized activity of chromatin modifiers.     

Regular gaits and optimal velocities for motor proteins

It has been observed in numerical experiments that adding a cargo to a motor protein can regularize its gait. Here we explain these results via asymptotic analysis on a general stochastic motor protein model. This analysis permits a computation of various observables (e.g., the mean velocity) of the motor protein and shows that the presence of the cargo also makes the velocity of the motor nonmonotone in certain control parameters (e.g., ATP concentration). As an example, we consider the case of a single myosin-V protein transporting a cargo and show that, at realistic concentrations of ATP, myosin-V operates in the regime which maximizes motor velocity. Our analysis also suggests an experimental regimen which can test the efficacy of any specific motor protein model to a greater degree than was heretofore possible.     

Estimating the number of motor units using random sums with independently thinned terms

The problem of estimating the numbers of motor units N in a muscle is embedded in a general stochastic model using the notion of thinning from point process theory. In the paper a new moment type estimator for the numbers of motor units in a muscle is denned, which is derived using random sums with independently thinned terms. Asymptotic normality of the estimator is shown and its practical value is demonstrated with bootstrap and approximative confidence intervals for a data set from a 31-year-old healthy right-handed, female volunteer. Moreover simulation results are presented and Monte-Carlo based quantiles, means, and variances are calculated for N in{300,600,1000}.     

Meiotic chromosome distribution in Drosophila oocytes: Roles of two kinesin-related proteins

Summary The recent finding that two proteins required for proper chromosome distribution in Drosophila oocytes are related to the microtubule motor protein, kinesin, provides new insights into the forces involved in meiotic chromosome movement. ncd is a spindle motor in meiosis but may perform a different role in the early mitotic divisions of the embryo. nod, until recently, has been thought to be a component of the distributive process of chromosome segregation. The finding that nod is a kinesin protein provides an alternative explanation of the effect of mutants on nonexchange chromosomes and suggests that nonexchange chromosomes segregate with exchange chromosomes in a single process, rather than via a two-step distributive system.     

Mitotic motors in Saccharomyces cerevisiae

The budding yeast Saccharomyces cerevisiae provides a unique opportunity for study of the microtubule-based motor proteins that participate in mitotic spindle function. The genome of Saccharomyces encodes a relatively small and genetically tractable set of microtubule-based motor proteins. The single cytoplasmic dynein and five of the six kinesin-related proteins encoded have been implicated in mitotic spindle function. Each motor protein is unique in amino acid sequence. On account of functional overlap, no single motor is uniquely required for cell viability, however. The ability to create and analyze multiple mutants has allowed experimental dissection of the roles performed by each mitotic motor. Some of the motors operate within the nucleus to assemble and elongate the bipolar spindle (kinesin-related Cin8p, Kip1p, Kip3p and Kar3p). Others operate on the cytoplasmic microtubules to effect spindle and nuclear positioning within the cell (dynein and kinesin-related Kip2p, Kip3p and Kar3p). The six motors apparently contribute three fundamental activities to spindle function: motility, microtubule cross-linking and regulation of microtubule dynamics.     

The coordination of protein motors and the kinetic behavior of microtubule--a computational study

Utilizing the mechanical energy converted from chemical energy through hydrolysis of ATP, motor proteins drive cytoskeleton filaments to move in various biological systems. Recent technological advance has shown the potential of the motor proteins for powering future nano-bio-mechanical systems. In order to effectively use motor proteins as a biological motor, the interaction between the protein motors and bio-filaments needs to be well clarified, since such interaction is largely influenced by many factors, such as the coordination among the motors, their dynamic behavior, physical properties of microtubules, and the viscosity of solution involved, etc. In this study, a two-dimensional model was proposed to simulate the motion of a microtubule driven by protein motors based on a dissipative particle dynamics (DPD) method with attempt to correlate the microtubule's kinetic behavior to the coordination among protein motors.     

On the lack of specificity of proteins and its consequences for a theory of biological organization

It is now widely recognized that gene expression and cellular processes include a probabilistic component. However, this does not essentially modify the theory of genetic programming. This stochastic aspect, which is called noise, is usually conceived as a margin of fluctuation in the way the genetic program functions and the latter remains understood as a specific mechanism guided by genetic information. In contrast, recent data show that proteins do not possess a high level of specificity. They can interact with numerous molecular partners. As a consequence molecular interactions are not simply “noisy”. Because they are subject to large combinatorial interaction possibilities, they are also intrinsically stochastic and must be sorted out by the cell structure. This contradicts the genetic programming theory which is based on the idea that protein interactions are directed by their stereospecificity and genetic information. Taking into account the lack of protein specificity leads to a new theory. Natural selection acts not only in evolution but also in ontogenesis by sorting stochastic molecular interactions. In this frame, the making up of an organism, instead of being a simple bottom-top process in which information flows from genes to phenotypes, is both a bottom-top and top-bottom process. Genes provide proteins, but their stochastic interactions are sorted by selective constraints arising from the cell and multi-cellular structures, which are themselves subject to the action of natural selection.     

Genetic distance and electrophoretic identity of proteins between taxa

Summary The relationship between amino acid substitution and charge change of proteins in the evolutionary process is studied by using a stochastic model. A mathematical formula is developed for the electrophoretic identity of proteins between two different taxa for a given number of average codon differences per protein locus. Using this formula, a reference figure is constructed for estimating the average number of codon differences per locus between taxa.     

A kinetic model for the impact of packaging signal mimics on genome encapsulation

Inspired by recent experiments on the spontaneous assembly of virus-like particles from a solution containing a synthetic coat protein and double-stranded DNA, we put forward a kinetic model that has as main ingredients a stochastic nucleation and a deterministic growth process. The efficiency and rate of DNA packaging strongly increase after tiling the DNA with CRISPR-Cas proteins at predesignated locations, mimicking assembly signals in viruses. Our model shows that treating these proteins as nucleation-inducing diffusion barriers is sufficient to explain the experimentally observed increase in encapsulation efficiency, but only if the nucleation rate is sufficiently high. We find an optimum in the encapsulation kinetics for conditions where the number of packaging signal mimics is equal to the number of nucleation events that can occur during the time required to fully encapsulate the DNA template, presuming that the nucleation events can only take place adjacent to a packaging signal. Our theory is in satisfactory agreement with the available experimental data.     

Probing force-induced unfolding intermediates of a single staphylococcal nuclease molecule and the effect of ligand binding

Single-molecule manipulation techniques have given experimental access to unfolding intermediates of proteins that are inaccessible in conventional experiments. A detailed characterization of the intermediates is a challenging problem that provides new possibilities for directly probing the energy landscape of proteins. We investigated single-molecule mechanical unfolding of a small globular protein, staphylococcal nuclease (SNase), using atomic force microscopy. The unfolding trajectories of the protein displayed sub-molecular and stochastic behavior with typical lengths corresponding to the size of the unfolded substructures. Our results support the view that the single protein unfolds along multiple pathways as suggested in recent theoretical studies. Moreover, we found the drastic change, caused by the ligand and inhibitor bindings, in the mechanical unfolding dynamics.     

Reconstitution of the bacterial chemotaxis signal transduction system from purified components.

In bacterial chemotaxis, transmembrane receptor proteins detect attractants and repellents in the medium and send intracellular signals that control motility. The cytoplasmic proteins that transduce information from the receptors to the flagellar motor have previously been purified and many of their enzymatic activities have been identified. Here we report the reconstitution of the complete signal transduction system from purified components. The protein kinase, CheA, plays a central role in both the initial excitation response to stimuli as well as subsequent events associated with adaptation. This kinase provides phosphoryl groups to two acceptor proteins, CheY, which interacts with the flagellar motor, and CheB, which demethylates the receptors. The purified aspartate receptor, Tar, reconstituted into phospholipid vesicles, acts in conjunction with an auxiliary protein, CheW, to stimulate the rate of kinase autophosphorylation greater than 10-fold. This stimulation is inhibited by aspartate. The activity of the kinase is increased by increased levels of receptor methylation. This effect provides a mechanism that explains how changes in receptor methylation mediate adaptive responses to attractant and repellant stimuli.