The starting point of many fundamental biological processes is associated with protein molecules finding and recognizing specific sites on DNA. However, despite a large number of experimental and theoretical studies on protein search for targets on DNA, many molecular aspects of underlying mechanisms are still not well understood. Experiments show that proteins bound to DNA can switch between slow recognition and fast search conformations. However, from a theoretical point of view, such conformational transitions should slow down the protein search for specific sites on DNA, in contrast to available experimental observations. In addition, experiments indicate that the nucleotide composition near the target site is more symmetrically homogeneous, leading to stronger effective interactions between proteins and DNA at these locations. However, as has been shown theoretically, this should also make the search less efficient, which is not observed. We propose a possible resolution of these problems by suggesting that conformational transitions occur only within a segment around the target where stronger interactions between proteins and DNA are observed. Two theoretical methods, based on continuum and discrete-state stochastic calculations, are developed, allowing us to obtain a comprehensive dynamic description for the protein search process in this system. The existence of an optimal length of the conformational transition zone with the shortest mean search time is predicted.

Enzymes are biological catalysts that play a fundamental role in all living systems by supporting the selectivity and the speed for almost all cellular processes. While the general principles of enzyme functioning are known, the specific details of how they work at the microscopic level are not always available. Simple Michaelis–Menten kinetics assumes that the enzyme–substrate complex has only one conformation that decays as a single exponential. As a consequence, the enzymatic velocity decreases as the dissociation (off) rate constant of the complex increases. Recently, Reuveni et al. [ Proc. Natl. Acad. Sci. USA 2014, 111, 4391−4396] showed that it is possible for the enzymatic velocity to increase when the off rate becomes higher, if the enzyme–substrate complex has many conformations which dissociate with the same off rate constant. This was done using formal mathematical arguments, without specifying the nature of the dynamics of the enzyme–substrate complex. In order to provide a physical basis for this unexpected result, we derive an analytical expression for the enzymatic velocity assuming that the enzyme–substrate complex has multiple states and its conformational dynamics is described by rate equations with arbitrary rate constants. By applying our formalism to a complex with two conformations, we show that the unexpected off rate dependence of the velocity can be readily understood: If one of the conformations is unproductive, the system can escape from this “trap” by dissociating, thereby giving the enzyme another chance to form the productive enzyme–substrate complex. We also demonstrate that the nonmonotonic off rate dependence of the enzymatic velocity is possible not only when all off rate constants are identical, but even when they are different. We show that for typical experimentally determined rate constants, the nonmonotonic off rate dependence can occur for micromolar substrate concentrations. Finally, we discuss the relation of this work to the problem of optimizing the flux through singly occupied membrane channels and transporters.

The ability to precisely edit and modify a genome opens endless opportunities to investigate fundamental properties of living systems as well as to advance various medical techniques and bioengineering applications. This possibility is now close to reality due to a recent discovery of the adaptive bacterial immune system, which is based on clustered regularly interspaced short palindromic repeats (CRISPR)-associated proteins (Cas) that utilize RNA to find and cut the double-stranded DNA molecules at specific locations. Here we develop a quantitative theoretical approach to analyze the mechanism of target search on DNA by CRISPR RNA-guided Cas9 proteins, which is followed by a selective cleavage of nucleic acids. It is based on a discrete-state stochastic model that takes into account the most relevant physical-chemical processes in the system. Using a method of first-passage processes, a full dynamic description of the target search is presented. It is found that the location of specific sites on DNA by CRISPR Cas9 proteins is governed by binding first to protospacer adjacent motif sequences on DNA, which is followed by reversible transitions into DNA interrogation states. In addition, the search dynamics is strongly influenced by the off-target cutting. Our theoretical calculations allow us to explain the experimental observations and to give experimentally testable predictions. Thus, the presented theoretical model clarifies some molecular aspects of the genome interrogation by CRISPR RNA-guided Cas9 proteins.

Genetic stability is a key factor in maintaining, survival, and reproduction of biological cells. It relies on many processes, but one of the most important is a homologous recombination, in which the repair of breaks in double-stranded DNA molecules is taking place with a help of several specific proteins. In bacteria, this task is accomplished by RecA proteins that are active as nucleoprotein filaments formed on single-stranded segments of DNA. A critical step in the homologous recombination is a search for a corresponding homologous region on DNA, which is called a homology search. Recent single-molecule experiments clarified some aspects of this process, but its molecular mechanisms remain not well understood. We developed a quantitative theoretical approach to analyze the homology search. It is based on a discrete-state stochastic model that takes into account the most relevant physical-chemical processes in the system. Using a method of first-passage processes, a full dynamic description of the homology search is presented. It is found that the search dynamics depends on the degree of extension of DNA molecules and on the size of RecA nucleoprotein filaments, in agreement with experimental single-molecule measurements of DNA pairing by RecA proteins. Our theoretical calculations, supported by extensive Monte Carlo computer simulations, provide a molecular description of the mechanisms of the homology search.

Major cellular processes are supported by various biomolecular motors that usually operate together as teams. We present an overview of the collective dynamics of processive cytokeletal motor proteins based on recent experimental and theoretical investigations. Experimental studies show that multiple motors function with different degrees of cooperativity, ranging from negative to positive. This effect depends on the mechanical properties of individual motors, the geometry of their connections, and the surrounding cellular environment. Theoretical models based on stochastic approaches underline the importance of intermolecular interactions, the properties of single motors, and couplings with cellular medium in predicting the collective dynamics. We discuss several features that specify the cooperativity in motor proteins. Based on this approach a general picture of collective dynamics of motor proteins is formulated, and the future directions and challenges are discussed.

Biological signaling is a crucial natural process that governs the formation of all multicellular organisms. It relies on efficient and fast transfer of information between different cells and tissues. It has been presumed for a long time that these long-distance communications in most systems can take place only indirectly via the diffusion of signaling molecules, also known as morphogens, through the extracellular fluid; however, recent experiments indicate that there is also an alternative direct delivery mechanism. It utilizes dynamic tubular cellular extensions, called cytonemes, that directly connect cells, supporting the flux of morphogens to specific locations. We present a first quantitative analysis of the cytoneme-mediated mechanism of biological signaling. Dynamics of the formation of signaling molecule profiles, which are also known as morphogen gradients, is discussed. It is found that the direct-delivery mechanism is more robust with respect to fluctuations in comparison with the passive diffusion mechanism. In addition, we show that the direct transport of morphogens through cytonemes simultaneously delivers the information to all cells, which is also different from the diffusional indirect delivery; however, it requires energy dissipation and it might be less efficient at large distances due to intermolecular interactions of signaling molecules.

Successful biological development via spatial and temporal regulations of cell differentiation relies on the action of multiple signaling molecules that are known as morphogens. It is now well established that biological signaling molecules create nonuniform concentration profiles, called morphogen gradients, that activate different genes, leading to patterning in the developing organisms. The current view of the formation of morphogen gradients is that it is a result of complex reaction–diffusion processes that include production, diffusion, and degradation of signaling molecules. Recent studies also suggest that the degradation of morphogens is a critically important step in the whole process. We develop a theoretical model that allows us to investigate the role of a spatially varying degradation in the formation of morphogen gradients. Our analysis shows that the spatial inhomogeneities in degradation might strongly influence the dynamics of formation of signaling profiles. Physical–chemical mechanisms of the underlying processes are discussed.

Proteins searching and recognizing specific sites on DNA is required for initiating all major biological processes. While the details of the protein search for targets on DNA in purified in vitro systems are reasonably well understood, the situation in real cells is much less clear. The presence of other types of molecules on DNA should prevent reaching the targets, but experiments show that, surprisingly, the molecular crowding on DNA influences the search dynamics much less than expected. We develop a theoretical method that allowed us to clarify the mechanisms of the protein search on DNA in the presence of crowding. It is found that the dimensionality of the search trajectories specifies whether the crowding will affect the target finding. For 3D search pathways it is minimal, while the strongest effect is for 1D search pathways when the crowding particle can block the search. In addition, for 1D search we determined that the critical parameter is a mobility of crowding agents: highly mobile molecules do not affect the search dynamics, while the slow particles can significantly slow down the process. Physical-chemical explanations of the observed phenomena are presented. Our theoretical predictions thus explain the experimental observations, and they are also supported by extensive numerical simulations.

ERK2 are protein kinases that during the enzymatic catalysis, in contrast to traditional enzymes, utilize additional interactions with substrates outside of the active sites. It is widely believed that these docking interactions outside of the enzymatic pockets enhance the specificity of these proteins. However, the molecular mechanisms of how the docking interactions affect the catalysis remain not well understood. Here, we develop a simple theoretical approach to analyze the enzymatic catalysis in ERK2 proteins. Our method is based on first-passage process analysis, and it provides explicit expressions for all dynamic properties of the system. It is found that there are specific binding energies for substrates in docking and catalytic domains that lead to maximal enzymatic reaction rates. Thus, we propose that the role of the docking interactions is not only to increase the enzymatic specificity but also to optimize the dynamics of the catalytic process. Our theoretical results are utilized to describe experimental observations on ERK2 enzymatic activities.

Protein search for specific sequences on DNA marks the beginning of major biological processes. Experiments indicate that proteins find and recognize their targets quickly and efficiently. Because of the large number of experimental and theoretical investigations, there is a reasonable understanding of the protein search processes in purified in vitro systems. However, the situation is much more complex in live cells where multiple biochemical and biophysical processes can interfere with the protein search dynamics. In this study, we develop a theoretical method that explores the effect of crowding on DNA chains during the protein search. More specifically, the role of static and dynamic obstacles is investigated. The method employs a discrete-state stochastic framework that accounts for most relevant physical and chemical processes in the system. Our approach also provides an analytical description for all dynamic properties. It is found that the presence of the obstacles can significantly modify the protein search dynamics. This effect depends on the size of the obstacles, on the spatial positions of the target and the obstacles, on the nature of the search regime, and on the dynamic nature of the obstacles. It is argued that the crowding on DNA can accelerate or slow down the protein search dynamics depending on these factors. A comparison with existing experimental and theoretical results is presented. Theoretical results are discussed using simple physical-chemical arguments, and they are also tested with extensive Monte Carlo computer simulations.

Unimolecular submersible nanomachines (USNs) bearing light-driven motors and fluorophores are synthesized. NMR experiments demonstrate that the rotation of the motor is not quenched by the fluorophore and that the motor behaves in the same manner as the corresponding motor without attached fluorophores. No photo or thermal decomposition is observed. Through careful design of control molecules with no motor and with a slow motor, we found using single molecule fluorescence correlation spectroscopy that only the molecules with fast rotating speed (MHz range) show an enhancement in diffusion by 26% when the motor is fully activated by UV light. This suggests that the USN molecules give ∼9 nm steps upon each motor actuation. A non-unidirectional rotating motor also results in a smaller, 10%, increase in diffusion. This study gives new insight into the light actuation of motorized molecules in solution.

Microtubules and actin filaments are biopolymer molecules that are major components of cytoskeleton networks in biological cells. They play important roles in supporting fundamental cellular processes such as cell division, signaling, locomotion, and intracellular transport. In cells, cytoskeleton proteins function under nonequilibrium conditions that are powered by hydrolysis of adenosine triphosphate (ATP) or guanosine triphosphate (GTP) molecules attached to them. Although these biopolymers are critically important for all cellular processes, the mechanisms that govern their complex dynamics and force generation remain not well explained. One of the most difficult fundamental issues is to understand how different components of cytoskeleton proteins interact together. We develop an approximate theoretical approach for analyzing complex processes in cytoskeleton proteins that takes into account the multifilament structure, lateral interactions between parallel protofilaments, and the most relevant biochemical transitions during the biopolymer growth. It allows us to fully evaluate collective dynamic properties of cytoskeleton filaments as well as the effect of external forces on them. It is found that for the case of strong lateral interactions the stall force of the multifilament protein is a linear function of the number of protofilaments. However, for weak lateral interactions, deviations from the linearity are observed. We also show that stall forces, mean velocities, and dispersions are increasing functions of the lateral interactions. Physical–chemical explanations of these phenomena are presented. Our theoretical predictions are supported by extensive Monte Carlo computer simulations.

Protein search for specific binding sites on DNA is a fundamental biological phenomenon associated with the beginning of most major biological processes. It is frequently found that proteins find and recognize their specific targets quickly and efficiently despite the complex nature of protein–DNA interactions in living cells. Although significant experimental and theoretical efforts were made in recent years, the mechanisms of these processes remain not well-clarified. We present a theoretical study of the protein target search dynamics in the presence of semispecific binding sites which are viewed as traps. Our theoretical approach employs a discrete-state stochastic method that accounts for the most important physical and chemical processes in the system. It also leads to a full analytical description for all dynamic properties of the protein search. It is found that the presence of traps can significantly modify the protein search dynamics. This effect depends on the spatial positions of the targets and traps, on distances between them, on the average sliding length of the protein along the DNA, and on the total length of DNA. Theoretical predictions are discussed using simple physical–chemical arguments, and they are also validated with extensive Monte Carlo computer simulations.

A simple three-state model for the dynamics of the singlet fission (SF) process is developed. The model facilitates the analysis of the relative significance of different factors, such as electronic energies, couplings, and the entropic contributions. The entropic contributions to the rates are important; they drive the SF process in endoergic cases (such as tetracene). The anticipated magnitude of entropic contributions is illustrated by simple calculations. By considering a series of three acenes (tetracene, pentacene, and hexacene), we explained the experimentally observed 3 orders of magnitude difference in the rate of SF in tetracene and pentacene and predicted that the rate in hexacene will be slightly faster than in pentacene. This trend is driven by the increased thermodynamic drive for SF (Gibbs free energy difference of the initial excitonic state and two separated triplets). The model also explains experimentally observed fast SF in 5,12-diphenyltetracene. Consistently with the experimental observations, the model predicts weak temperature dependence of the multiexciton formation rate in tetracene as well as a reduced rate of this step in solutions and in isolated dimers.

Cytoskeleton proteins are filament structures that support a large number of important biological processes. These dynamic biopolymers exist in nonequilibrium conditions stimulated by hydrolysis chemical reactions in their monomers. Current theoretical methods provide a comprehensive picture of biochemical and biophysical processes in cytoskeleton proteins. However, the description is only qualitative under biologically relevant conditions because utilized theoretical mean-field models neglect correlations. We develop a new theoretical method to describe dynamic processes in cytoskeleton proteins that takes into account spatial correlations in the chemical composition of these biopolymers. Our approach is based on analysis of probabilities of different clusters of subunits. It allows us to obtain exact analytical expressions for a variety of dynamic properties of cytoskeleton filaments. By comparing theoretical predictions with Monte Carlo computer simulations, it is shown that our method provides a fully quantitative description of complex dynamic phenomena in cytoskeleton proteins under all conditions.

Most chemical and biological processes can be viewed as reaction networks in which different pathways often compete kinetically for transformation of substrates into products. An enzymatic process is an example of such phenomena when biological catalysts create new routes for chemical reactions to proceed. It is typically assumed that the general process of product formation is governed by the pathway with the fastest kinetics at all time scales. In contrast to the expectation, here we show theoretically that at time scales sufficiently short, reactions are predominantly determined by the shortest pathway (in the number of intermediate states), regardless of the average turnover time associated with each pathway. This universal phenomenon is demonstrated by an explicit calculation for a system with two competing reversible (or irreversible) pathways. The time scales that characterize this regime and its relevance for single-molecule experimental studies are also discussed.

Microtubules are biopolymers consisting of tubulin dimer subunits. As a major component of cytoskeleton they are essential for supporting most important cellular processes such as cell division, signaling, intracellular transport and cell locomotion. The hydrolysis of guanosine triphosphate (GTP) molecules attached to each tubulin subunit supports the nonequilibrium nature of microtubule dynamics. One of the most spectacular properties of microtubules is their dynamic instability when their growth from continuous attachment of tubulin dimers stochastically alternates with periods of shrinking. Despite the critical importance of this process to all cellular activities, its mechanism remains not fully understood. We investigated theoretically microtubule dynamics at all times by analyzing explicitly temporal evolution of various length clusters of unhydrolyzed subunits. It is found that the dynamic behavior of microtubules depends strongly on initial conditions. Our theoretical findings provide a microscopic explanation for recent experiments which found that the frequency of catastrophes increases with the lifetime of microtubules. It is argued that most growing microtubule configurations cannot transit in one step into a shrinking state, leading to a complex overall temporal behavior. Theoretical calculations combined with Monte Carlo computer simulations are also directly compared with experimental observations, and good agreement is found.

Protein search for targets on DNA starts all major biological processes. Although significant experimental and theoretical efforts have been devoted to investigation of these phenomena, mechanisms of protein–DNA interactions during the search remain not fully understood. One of the most surprising observations is known as a speed-selectivity paradox. It suggests that experimentally observed fast findings of targets require smooth protein–DNA binding potentials, while the stability of the specific protein–DNA complex imposes a large energy gap which should significantly slow down the protein molecule. We developed a discrete-state stochastic approach that allowed us to investigate explicitly target search phenomena and to analyze the speed-selectivity paradox. A general dynamic phase diagram for different search regimes is constructed. The effect of the target position on search dynamics is investigated. Using experimentally observed parameters, it is found that slow protein diffusion on DNA does not lead to an increase in the search times. Thus, our theory resolves the speed-selectivity paradox by arguing that it does not exist. It is just an artifact of using approximate continuum theoretical models for analyzing protein search in the region of the parameter space beyond the range of validity of these models. In addition, the presented method, for the first time, provides an explanation for fast target search at the level of single protein molecules. Our theoretical predictions agree with all available experimental observations, and extensive Monte Carlo computer simulations are performed to support analytical calculations.

Intracellular transport is a fundamental biological process during which cellular materials are driven by enzymatic molecules called motor proteins. Recent optical trapping experiments and theoretical analysis have uncovered many features of cargo transport by multiple kinesin motor protein molecules under applied loads. These studies suggest that kinesins cooperate negatively under typical transport conditions, although some productive cooperation could be achieved under higher applied loads. However, the microscopic origins of this complex behavior are still not well understood. Using a discrete-state stochastic approach we analyze factors that affect the cooperativity among kinesin motors during cargo transport. Kinesin cooperation is shown to be largely unaffected by the structural and mechanical parameters of a multiple motor complex connected to a cargo, but much more sensitive to biochemical parameters affecting motor–filament affinities. While such behavior suggests the net negative cooperative responses of kinesins will persist across a relatively wide range of cargo types, it is also shown that the rates with which cargo velocities relax in time upon force perturbations are influenced by structural factors that affect the free energies of and load distributions within a multiple kinesin complex. The implications of these later results on transport phenomena where loads change temporally, as in the case of bidirectional transport, are discussed.

Microtubules are cytoskeleton multifilament proteins that support many fundamental biological processes such as cell division, cellular transport, and motility. They can be viewed as dynamic polymers that function in nonequilibrium conditions stimulated by hydrolysis of GTP (guanosine triphosphate) molecules bound to their monomers. We present a theoretical description of microtubule dynamics based on discrete-state stochastic models that explicitly takes into account all relevant biochemical transitions. In contrast to previous theoretical analysis, a more realistic physical–chemical description of GTP hydrolysis is presented, in which the hydrolysis rate at a given monomer depends on the chemical composition of the neighboring monomers. This dependence naturally leads to a cooperativity in the hydrolysis. It is found that this cooperativity significantly influences all dynamic properties of microtubules. It is suggested that the dynamic instability in cytoskeleton proteins might be observed only for weak cooperativity, while the strong cooperativity in hydrolysis suppresses the dynamic instability. The presented microscopic analysis is compared with existing phenomenological descriptions of hydrolysis processes. Our analytical calculations, supported by computer Monte Carlo simulations, are also compared with available experimental observations.

Intracellular transport is supported by enzymes called motor proteins that are often coupled to the same cargo and function collectively. Recent experiments and theoretical advances have been able to explain certain behaviors of multiple motor systems by elucidating how unequal load sharing between coupled motors changes how they bind, step, and detach. However, nonmechanical interactions are typically overlooked despite several studies suggesting that microtubule-bound kinesins interact locally via short-range nonmechanical potentials. This work develops a new stochastic model to explore how these types of interactions influence multiple kinesin functions in addition to mechanical coupling. Nonmechanical interactions are assumed to affect kinesin mechanochemistry only when the motors are separated by less than three microtubule lattice sites, and it is shown that relatively weak interaction energies (∼2 kBT) can have an appreciable influence over collective motor velocities and detachment rates. In agreement with optical trapping experiments on structurally defined kinesin complexes, the model predicts that these effects primarily occur when cargos are transported against loads exceeding single-kinesin stalling forces. Overall, these results highlight the interdependent nature of factors influencing collective motor functions, namely, that the way the bound configuration of a multiple motor system evolves under load determines how local nonmechanical interactions influence motor cooperation.

Understanding microscopic mechanisms of motion of artificial molecular machines is fundamentally important for scientific and technological progress. It is known that electric field might strongly influence structures and dynamic properties of molecules at the nanoscale level. Specifically, it is possible to induce conformational changes and the directional motion in many surface-bound molecules by electric field in scanning tunneling microscopy (STM) experiments. Utilizing a recently developed theoretical method to describe charge transfer phenomena for fullerenes near metal surfaces, in this work we theoretically investigated dynamics of fullerene-based nanocars in the presence of external electric field. Our approach is based on classical rigid-body molecular dynamics simulations that allow us to fully analyze dynamics of nanocars on gold surfaces. Theoretical calculations predict that it is possible to drive nonpolar nanocars unidirectionally with the help of external electric field. It is shown also that charge transfer effects play a critical role in driving nanocars and for understanding mechanisms of the directionality of the observed motion. Our theoretical predictions explain experimental observations on moving nanocars along metal surfaces.

Rotating surface-mounted molecules have attracted the attention of many research groups as a way to develop new nanoscale devices and materials. However, mechanisms of motion of these rotors at the single-molecule level are still not well-understood. Theoretical and experimental studies on thioether molecular rotors on gold surfaces suggest that the size of the molecules, their flexibility, and steric repulsions with the surface are important for dynamics of the system. A complex combination of these factors leads to the observation that the rotation speeds have not been hindered by increasing the length of the alkyl chains. However, experiments on diferrocene derivatives indicated a significant increase in the rotational barriers for longer molecules. We present here a comprehensive theoretical study that combines molecular dynamics simulations and simple models to investigate what factors influence single-molecule rotations on the surfaces. Our results suggest that rotational dynamics is determined by the size and by the symmetry of the molecules and surfaces and by interactions with surfaces. Our theoretical predictions are in excellent agreement with current experimental observations.

Studies of motion of carborane wheel-like molecules on glassy surfaces provide an important contribution to the practical goal of designing molecular nanoscale transporters called nanocars. In these vehicles, carborane wheels are chemically coupled to chassis allowing these devices to participate in translations over the surface. As a preliminary step toward modeling dynamics of these species, we have investigated interactions of the p-carborane molecule with the hydrated (1010) surface of α-quartz. Quantum calculations with the CP2K package (cp2k.berlioz.de) have been performed for a series of model systems in order to explicitly estimate interaction energies between the carborane wheel and the surface. It was found that 8 kJ/mol for absorption of carborane on the surface provides a reasonable estimate for this interaction. Correspondingly, the four-wheeled vehicle would require at least 32 kJ/mol to activate its diffusion on the glassy surface. The latter value is consistent with experimental estimates of 42 kJ/mol for the activation energy as follows from single-molecule measurements of temperature dependence of the diffusion coefficient. It is argued that the translation of the four-wheeled carborane nanocars might be consistent with both rolling and hopping mechanisms of motion.

In recent years molecular rotors have attracted the attention of many research groups for possible applications as new nanoscale devices and materials with controlled chemical, physical, and mechanical properties. One of the most unique systems with molecular rotations is amphidynamic molecular crystals, also known as crystalline molecular gyroscopes. This system can be viewed as a solid-state assembly of molecules that cannot move translationally but show internal rotations. Recent experiments on amphidynamic crystals indicate importance of rotational symmetry for describing their dynamics. However, mechanisms and rotational dynamic properties of molecular gyroscopes are still not well understood. We present here a theoretical investigation of amphidynamic crystals by utilizing extensive rigid-body molecular dynamics simulations and simple phenomenological arguments. Theoretical analysis suggests that intramolecular interactions within stator and rotator segments of molecular rotors as well as their flexibility strongly affect their crystal packing, energies and rotational behavior. Our quantitative predictions for dynamic properties agree well with available experimental results.

Concentration profiles of signaling molecules, known as morphogen gradients, determine polarity and spatial patterning in the development of all multicellular organisms. A widely used approach to explain the establishment of morphogen concentration gradients assumes that signaling molecules are produced locally, then spread via a free diffusion along the line of developing cells and degraded uniformly. However, recent experiments have produced controversial observations concerning the feasibility of this theoretical description. Some experimentally measured dispersions for morphogens cannot support fast formation of stationary concentration profiles. In addition, the latest theoretical analyses of times to establish the morphogen gradient yield a surprising linear scaling as a function of length from the source that is not expected for the unbiased diffusion process. We propose here a theoretical approach that provides a possible physical–chemical mechanism to explain these observations. It is argued that relaxation times to establish morphogen gradients are mostly determined by first arrival times, and the degradation plays a critical role in this mechanism by effectively accelerating diffusion of signaling molecules via removal of slow moving particles. This coupling between diffusion and degradation is analogous to the action of the effective field that drives particles away from the local source.Keywords: bicoid molecules; morphogen gradient; synthesis−diffusion−degradation (SDD) model;

Methodical problems of coarse-grained-type molecular dynamics, namely, rigid-body molecular dynamics (RB MD), are studied by investigating the dynamics of nanosized molecular vehicles called nanocars that move on gold and silver surfaces. Specifically, we analyzed the role of thermostats and the effects of temperature, couplings, and correlations between rigid fragments of the nanocar molecule in extensive RB MD simulations. It is found that the use of the Nosé−Poincaré thermostat does not introduce systematic errors, but the time trajectories might be required to be limited to not accumulate large numerical integration errors. Correlations in the motion of different fragments of the molecules are also analyzed. Our theoretical computations also point to the importance of temperature, interfragment interactions, and interactions with surfaces and to the nature of the surface for understanding mechanisms of motion of single-molecule transporters.

A starting point of many biological processes is protein binding to specific regions on DNA. Although typical concentrations of DNA-binding proteins are low, and target sites are typically buried among huge number of non-specific sites, the search process is frequently achieved at a remarkably fast rate. For some proteins it has been confirmed that association rates might be even larger than the maximal allowed three-dimensional diffusion rates. The current theoretical view of this phenomenon is based on the idea of lowering dimensionality, i.e., the overall search process is viewed as a combination of uncorrelated three-dimensional excursions in the solution and one-dimensional hoppings on DNA. However, some predictions of this theoretical picture contradict recent single-molecule measurements of protein diffusion processes. An alternative theoretical approach points out the importance of correlations during the search process that appear due to non-specific interactions between protein and DNA molecules. To test different theoretical ideas we performed extensive lattice Monte Carlo computer simulations of the facilitated diffusion. Our results revealed that correlations are important, and the acceleration in the search could only be achieved at some intermediate non-specific binding energies and protein concentrations. Physico-chemical aspects and the origins of these correlations are discussed.

Recent single-molecule experiments indicated that thioethers (dialkyl sulfides) on gold surfaces act as thermally- or mechanically activated molecular rotors, although the mechanisms for these phenomena are not yet clearly understood. Here we present theoretical and experimental investigations of the rotational dynamics of these thioether molecules. Single-molecule studies utilizing low-temperature scanning tunneling microscopy allowed us to determine rotational rates and activation energies for the rotation of symmetric dialkyl sulfides. It was found that the rotational energy barriers increased as a function of alkyl chain length but then quickly saturated. Molecular dynamics simulations have also been performed in order to understand the molecular rotations of thioethers, and our theoretical calculations agree well with experimental observations. It is argued that the observed rotational dynamics of dialkyl sulfides are determined by the effective interactions with the surface and the flexibility of the alkyl chains. These results suggest possible ways to control and utilize thioether rotors at the single-molecule level.

We developed molecular models describing the thermally initiated motion of nanocars, nanosized vehicles composed of two to four spherical fullerene wheels chemically coupled to a planar chassis, on a metal surface. The simulations were aimed at reproducing qualitative features of the experimentally observed migration of nanocars over gold crystals as determined by scanning tunneling microscopy. Coarse-grained-type molecular dynamics simulations were carried out for the species “Trimer” and “Nanotruck”, the simplified versions of the experimentally studied nanomachines. Toward this goal, we developed a version of the rigid body molecular dynamics based on the symplectic quaternion scheme in conjunction with the Nose−Poincare thermostat approach. Interactions between rigid fragments were described by using the corrected CHARMM force field parameters, while several empirical models were introduced for interactions of nanocars with gold crystals. With the single adjusted potential parameter, the computed trajectories are consistent with the qualitative features of the thermally activated migration of the nanocars: the primary pivoting motion of Trimer and the two-dimensional combination of translations and pivoting of Nanotruck. This work presents a first attempt at a theoretical analysis of nanocarsʼ dynamics on a surface by providing a computationally minimalist approach.

Motor proteins are active biological molecules that perform their functions by converting chemical energy into mechanical work. They move unidirectionally along rigid protein filaments or DNA and RNA molecules in discrete steps by hydrolyzing ATP (adenosine triphsophate) or related energy-rich compounds. Recent single-molecule experiments have shown that motor proteins experience significant spatial fluctuations during its motion, leading to broad step-size distributions. The effect of these spatial fluctuations is analyzed explicitly by considering discrete-state stochastic models that allow us to compute exactly all dynamic properties. It is shown that for symmetric spatial fluctuations there is no change in mean velocities for weak external forces, while dispersions and stall forces are strongly affected at all conditions. These results are illustrated by several simple examples. Our method is also applied to analyze the effect of step-size fluctuations on dynamics of myosin V motor proteins. It is argued that spatial fluctuations might be used to control and regulate the dynamics of motor proteins.

Uncovering mechanisms that control the dynamics of microtubules is fundamental for our understanding of multiple cellular processes such as chromosome separation and cell motility. Building on previous theoretical work on the dynamic instability of microtubules, we propose here a stochastic model that includes all relevant biochemical processes that affect the dynamics of microtubule plus-end, namely, the binding of GTP-bound monomers, unbinding of GTP- and GDP-bound monomers, and hydrolysis of GTP monomers. The inclusion of dissociation processes, present in our approach but absent from many previous studies, is essential to guarantee the thermodynamic consistency of the model. Our theoretical method allows us to compute all dynamic properties of microtubules explicitly. Using experimentally determined rates, it is found that the cap size is ∼3.6 layers, an estimate that is compatible with several experimental observations. In the end, our model provides a comprehensive description of the dynamic instability of microtubules that includes not only the statistics of catastrophes but also the statistics of rescues.

A starting point of many biological processes is protein binding to specific regions on DNA. Although typical concentrations of DNA-binding proteins are low, and target sites are typically buried among huge number of non-specific sites, the search process is frequently achieved at a remarkably fast rate. For some proteins it has been confirmed that association rates might be even larger than the maximal allowed three-dimensional diffusion rates. The current theoretical view of this phenomenon is based on the idea of lowering dimensionality, i.e., the overall search process is viewed as a combination of uncorrelated three-dimensional excursions in the solution and one-dimensional hoppings on DNA. However, some predictions of this theoretical picture contradict recent single-molecule measurements of protein diffusion processes. An alternative theoretical approach points out the importance of correlations during the search process that appear due to non-specific interactions between protein and DNA molecules. To test different theoretical ideas we performed extensive lattice Monte Carlo computer simulations of the facilitated diffusion. Our results revealed that correlations are important, and the acceleration in the search could only be achieved at some intermediate non-specific binding energies and protein concentrations. Physico-chemical aspects and the origins of these correlations are discussed.