Diffusible crosslinkers generate directed forces in microtubule networks.

Cytoskeletal remodeling is essential to eukaryotic cell division and morphogenesis. The mechanical forces driving the restructuring are attributed to the action of molecular motors and the dynamics of cytoskeletal filaments, which both consume chemical energy. By contrast, non-enzymatic filament crosslinkers are regarded as mere friction-generating entities. Here, we experimentally demonstrate that diffusible microtubule crosslinkers of the Ase1/PRC1/Map65 family generate directed microtubule sliding when confined between partially overlapping microtubules. The Ase1-generated forces, directly measured by optical tweezers to be in the piconewton-range, were sufficient to antagonize motor-protein driven microtubule sliding. Force generation is quantitatively explained by the entropic expansion of confined Ase1 molecules diffusing within the microtubule overlaps. The thermal motion of crosslinkers is thus harnessed to generate mechanical work analogous to compressed gas propelling a piston in a cylinder. As confinement of diffusible proteins is ubiquitous in cells, the associated entropic forces are likely of importance for cellular mechanics beyond cytoskeletal networks.

Trade-offs between cost and information in cellular prediction.

Living cells can leverage correlations in environmental fluctuations to predict the future environment and mount a response ahead of time. To this end, cells need to encode the past signal into the output of the intracellular network from which the future input is predicted. Yet, storing information is costly while not all features of the past signal are equally informative on the future input signal. Here, we show for two classes of input signals that cellular networks can reach the fundamental bound on the predictive information as set by the information extracted from the past signal: Push-pull networks can reach this information bound for Markovian signals, while networks that take a temporal derivative can reach the bound for predicting the future derivative of non-Markovian signals. However, the bits of past information that are most informative about the future signal are also prohibitively costly. As a result, the optimal system that maximizes the predictive information for a given resource cost is, in general, not at the information bound. Applying our theory to the chemotaxis network of Escherichia coli reveals that its adaptive kernel is optimal for predicting future concentration changes over a broad range of background concentrations, and that the system has been tailored to predicting these changes in shallow gradients.

Cross-linkers at growing microtubule ends generate forces that drive actin transport.

SignificanceComplex cellular processes such as cell migration require coordinated remodeling of both the actin and the microtubule cytoskeleton. The two networks for instance exert forces on each other via active motor proteins. Here we show that, surprisingly, coupling via passive cross-linkers can also result in force generation. We specifically study the transport of actin filaments by growing microtubule ends. We show by cell-free reconstitution experiments, computer simulations, and theoretical modeling that this transport is driven by the affinity of the cross-linker for the chemically distinct microtubule tip region. Our work predicts that growing microtubules could potentially rapidly relocate newly nucleated actin filaments to the leading edge of the cell and thus boost migration.

Constriction of Actin Rings by Passive Crosslinkers.

In many organisms, cell division is driven by the constriction of a cytokinetic ring, which consists of actin filaments and crosslinking proteins. While it has long been believed that the constriction is driven by motor proteins, it has recently been discovered that passive crosslinkers that do not turn over fuel are able to generate enough force to constrict actin filament rings. To study the ring constriction dynamics, we develop a model that includes the driving force of crosslinker condensation and the opposing forces of friction and filament bending. We analyze the constriction force as a function of ring topology and crosslinker concentration, and predict forces that are sufficient to constrict an unadorned plasma membrane. Our model also predicts that actin-filament sliding arises from an interplay between filament rotation and crosslinker hopping, producing frictional forces that are low compared with those of crosslinker-mediated microtubule sliding.

Nonequilibrium correlations in minimal dynamical models of polymer copying.

Living systems produce "persistent" copies of information-carrying polymers, in which template and copy sequences remain correlated after physically decoupling. We identify a general measure of the thermodynamic efficiency with which these nonequilibrium states are created and analyze the accuracy and efficiency of a family of dynamical models that produce persistent copies. For the weakest chemical driving, when polymer growth occurs in equilibrium, both the copy accuracy and, more surprisingly, the efficiency vanish. At higher driving strengths, accuracy and efficiency both increase, with efficiency showing one or more peaks at moderate driving. Correlations generated within the copy sequence, as well as between template and copy, store additional free energy in the copied polymer and limit the single-site accuracy for a given chemical work input. Our results provide insight into the design of natural self-replicating systems and can aid the design of synthetic replicators.

Shape Transformations of Vesicles Induced by Swim Pressure.

While the behavior of vesicles in thermodynamic equilibrium has been studied extensively, how active forces control vesicle shape transformations is not understood. Here, we combine theory and simulations to study the shape behavior of vesicles containing active Brownian particles. We show that the combination of active forces, dimensionality, and membrane bending free energy creates a plethora of novel phase transitions. At low swim pressure, the vesicle exhibits a discontinuous transition from a spherical to a prolate shape, which has no counterpart in two dimensions. At high swim pressure it exhibits stochastic spatiotemporal oscillations. Our work helps researchers to understand and control the shape dynamics of membranes in active-matter systems.

Changes in microtubule overlap length regulate kinesin-14-driven microtubule sliding.

Microtubule-crosslinking motor proteins, which slide antiparallel microtubules, are required for the remodeling of microtubule networks. Hitherto, all microtubule-crosslinking motors have been shown to slide microtubules at a constant velocity until no overlap remains between them, leading to the breakdown of the initial microtubule geometry. Here, we show in vitro that the sliding velocity of microtubules, driven by human kinesin-14 HSET, decreases when microtubules start to slide apart, resulting in the maintenance of finite-length microtubule overlaps. We quantitatively explain this feedback using the local interaction kinetics of HSET with overlapping microtubules that cause retention of HSET in shortening overlaps. Consequently, the increased HSET density in the overlaps leads to a density-dependent decrease in sliding velocity and the generation of an entropic force that antagonizes the force exerted by the motors. Our results demonstrate that a spatial arrangement of microtubules can regulate the collective action of molecular motors through the local alteration of their individual interaction kinetics.

Frustrated binding of biopolymer crosslinkers.

Transiently crosslinked actin filament networks allow cells to combine elastic rigidity with the ability to deform viscoelastically. Theoretical models of semiflexible polymer networks predict that the crosslinker unbinding rate governs the timescale beyond which viscoelastic flow occurs. However a direct comparison between network and crosslinker dynamics is lacking. Here we measure the network's stress relaxation timescale using rheology and the lifetime of bound crosslinkers using fluorescence recovery after photobleaching (FRAP). Intriguingly, we observe that the crosslinker unbinding rate measured by FRAP is more than an order of magnitude slower than the rate measured by rheology. We rationalize this difference with a three-state model where crosslinkers are bound to either 0, 1 or 2 filaments, which allows us to extract crosslinker transition rates that are otherwise difficult to access. We find that the unbinding rate of singly bound crosslinkers is nearly two orders of magnitude slower than for doubly bound ones. We attribute the increased unbinding rate of doubly bound crosslinkers to the high stiffness of biopolymers, which frustrates crosslinker binding.

Kinetics and equilibrium constants of oligonucleotides at low concentrations. Hybridization and melting study.

Although DNA hybridization/melting is one of the most important biochemical reactions, the non-trivial kinetics of the process is not yet fully understood. In this work, we use Förster resonance energy transfer (FRET) to investigate the influence of temperature, ionic strength, and oligonucleotide length on the kinetic and equilibrium constants of DNA oligonucleotide binding and dissociation. We show that at low reagent concentrations and ionic strength, the time needed to establish equilibrium between single and double strand forms may be of the order of days, even for simple oligonucleotides of a length of 20 base pairs or less. We also identify and discuss the possible artifacts related to fluorescence-based experiments conducted in extremely dilute solutions. The results should prove useful for the judicious design of technologies based on DNA-matching, including sensors, DNA multiplication, sequencing, and gene manipulation.

eGFRD in all dimensions.

Biochemical reactions often occur at low copy numbers but at once in crowded and diverse environments. Space and stochasticity therefore play an essential role in biochemical networks. Spatial-stochastic simulations have become a prominent tool for understanding how stochasticity at the microscopic level influences the macroscopic behavior of such systems. While particle-based models guarantee the level of detail necessary to accurately describe the microscopic dynamics at very low copy numbers, the algorithms used to simulate them typically imply trade-offs between computational efficiency and biochemical accuracy. eGFRD (enhanced Green's Function Reaction Dynamics) is an exact algorithm that evades such trade-offs by partitioning the N-particle system into M ≤ N analytically tractable one- and two-particle systems; the analytical solutions (Green's functions) then are used to implement an event-driven particle-based scheme that allows particles to make large jumps in time and space while retaining access to their state variables at arbitrary simulation times. Here we present "eGFRD2," a new eGFRD version that implements the principle of eGFRD in all dimensions, thus enabling efficient particle-based simulation of biochemical reaction-diffusion processes in the 3D cytoplasm, on 2D planes representing membranes, and on 1D elongated cylinders representative of, e.g., cytoskeletal tracks or DNA; in 1D, it also incorporates convective motion used to model active transport. We find that, for low particle densities, eGFRD2 is up to 6 orders of magnitude faster than conventional Brownian dynamics. We exemplify the capabilities of eGFRD2 by simulating an idealized model of Pom1 gradient formation, which involves 3D diffusion, active transport on microtubules, and autophosphorylation on the membrane, confirming recent experimental and theoretical results on this system to hold under genuinely stochastic conditions.

Discrete gene replication events drive coupling between the cell cycle and circadian clocks.

Many organisms possess both a cell cycle to control DNA replication and a circadian clock to anticipate changes between day and night. In some cases, these two rhythmic systems are known to be coupled by specific, cross-regulatory interactions. Here, we use mathematical modeling to show that, additionally, the cell cycle generically influences circadian clocks in a nonspecific fashion: The regular, discrete jumps in gene-copy number arising from DNA replication during the cell cycle cause a periodic driving of the circadian clock, which can dramatically alter its behavior and impair its function. A clock built on negative transcriptional feedback either phase-locks to the cell cycle, so that the clock period tracks the cell division time, or exhibits erratic behavior. We argue that the cyanobacterium Synechococcus elongatus has evolved two features that protect its clock from such disturbances, both of which are needed to fully insulate it from the cell cycle and give it its observed robustness: a phosphorylation-based protein modification oscillator, together with its accompanying push-pull read-out circuit that responds primarily to the ratios of different phosphoform concentrations, makes the clock less susceptible to perturbations in protein synthesis; the presence of multiple, asynchronously replicating copies of the same chromosome diminishes the effect of replicating any single copy of a gene.

Chimera proteins with affinity for membranes and microtubule tips polarize in the membrane of fission yeast cells.

Cell polarity refers to a functional spatial organization of proteins that is crucial for the control of essential cellular processes such as growth and division. To establish polarity, cells rely on elaborate regulation networks that control the distribution of proteins at the cell membrane. In fission yeast cells, a microtubule-dependent network has been identified that polarizes the distribution of signaling proteins that restricts growth to cell ends and targets the cytokinetic machinery to the middle of the cell. Although many molecular components have been shown to play a role in this network, it remains unknown which molecular functionalities are minimally required to establish a polarized protein distribution in this system. Here we show that a membrane-binding protein fragment, which distributes homogeneously in wild-type fission yeast cells, can be made to concentrate at cell ends by attaching it to a cytoplasmic microtubule end-binding protein. This concentration results in a polarized pattern of chimera proteins with a spatial extension that is very reminiscent of natural polarity patterns in fission yeast. However, chimera levels fluctuate in response to microtubule dynamics, and disruption of microtubules leads to disappearance of the pattern. Numerical simulations confirm that the combined functionality of membrane anchoring and microtubule tip affinity is in principle sufficient to create polarized patterns. Our chimera protein may thus represent a simple molecular functionality that is able to polarize the membrane, onto which additional layers of molecular complexity may be built to provide the temporal robustness that is typical of natural polarity patterns.

Robustness of Clocks to Input Noise.

To estimate the time, many organisms, ranging from cyanobacteria to animals, employ a circadian clock which is based on a limit-cycle oscillator that can tick autonomously with a nearly 24 h period. Yet, a limit-cycle oscillator is not essential for knowing the time, as exemplified by bacteria that possess an "hourglass": a system that when forced by an oscillatory light input exhibits robust oscillations from which the organism can infer the time, but that in the absence of driving relaxes to a stable fixed point. Here, using models of the Kai system of cyanobacteria, we compare a limit-cycle oscillator with two hourglass models, one that without driving relaxes exponentially and one that does so in an oscillatory fashion. In the limit of low input noise, all three systems are equally informative on time, yet in the regime of high input-noise the limit-cycle oscillator is far superior. The same behavior is found in the Stuart-Landau model, indicating that our result is universal.

A thermodynamically consistent model of the post-translational Kai circadian clock.

The principal pacemaker of the circadian clock of the cyanobacterium S. elongatus is a protein phosphorylation cycle consisting of three proteins, KaiA, KaiB and KaiC. KaiC forms a homohexamer, with each monomer consisting of two domains, CI and CII. Both domains can bind and hydrolyze ATP, but only the CII domain can be phosphorylated, at two residues, in a well-defined sequence. While this system has been studied extensively, how the clock is driven thermodynamically has remained elusive. Inspired by recent experimental observations and building on ideas from previous mathematical models, we present a new, thermodynamically consistent, statistical-mechanical model of the clock. At its heart are two main ideas: i) ATP hydrolysis in the CI domain provides the thermodynamic driving force for the clock, switching KaiC between an active conformational state in which its phosphorylation level tends to rise and an inactive one in which it tends to fall; ii) phosphorylation of the CII domain provides the timer for the hydrolysis in the CI domain. The model also naturally explains how KaiA, by acting as a nucleotide exchange factor, can stimulate phosphorylation of KaiC, and how the differential affinity of KaiA for the different KaiC phosphoforms generates the characteristic temporal order of KaiC phosphorylation. As the phosphorylation level in the CII domain rises, the release of ADP from CI slows down, making the inactive conformational state of KaiC more stable. In the inactive state, KaiC binds KaiB, which not only stabilizes this state further, but also leads to the sequestration of KaiA, and hence to KaiC dephosphorylation. Using a dedicated kinetic Monte Carlo algorithm, which makes it possible to efficiently simulate this system consisting of more than a billion reactions, we show that the model can describe a wealth of experimental data.

Rate constants for proteins binding to substrates with multiple binding sites using a generalized forward flux sampling expression.

To predict the response of a biochemical system, knowledge of the intrinsic and effective rate constants of proteins is crucial. The experimentally accessible effective rate constant for association can be decomposed in a diffusion-limited rate at which proteins come into contact and an intrinsic association rate at which the proteins in contact truly bind. Reversely, when dissociating, bound proteins first separate into a contact pair with an intrinsic dissociation rate, before moving away by diffusion. While microscopic expressions exist that enable the calculation of the intrinsic and effective rate constants by conducting a single rare event simulation of the protein dissociation reaction, these expressions are only valid when the substrate has just one binding site. If the substrate has multiple binding sites, a bound enzyme can, besides dissociating into the bulk, also hop to another binding site. Calculating transition rate constants between multiple states with forward flux sampling requires a generalized rate expression. We present this expression here and use it to derive explicit expressions for all intrinsic and effective rate constants involving binding to multiple states, including rebinding. We illustrate our approach by computing the intrinsic and effective association, dissociation, and hopping rate constants for a system in which a patchy particle model enzyme binds to a substrate with two binding sites. We find that these rate constants increase as a function of the rotational diffusion constant of the particles. The hopping rate constant decreases as a function of the distance between the binding sites. Finally, we find that blocking one of the binding sites enhances both association and dissociation rate constants. Our approach and results are important for understanding and modeling association reactions in enzyme-substrate systems and other patchy particle systems and open the way for large multiscale simulations of such systems.

Period Robustness and Entrainability of the Kai System to Changing Nucleotide Concentrations.

Circadian clocks must be able to entrain to time-varying signals to keep their oscillations in phase with the day-night rhythm. On the other hand, they must also exhibit input compensation: their period must remain approximately one day in different constant environments. The posttranslational oscillator of the Kai system can be entrained by transient or oscillatory changes in the ATP fraction, yet is insensitive to constant changes in this fraction. We study in three different models of this system how these two seemingly conflicting criteria are met. We find that one of these (our recently published Paijmans model) exhibits the best tradeoff between input compensation and entrainability: on the footing of equal phase-response curves, it exhibits the strongest input compensation. Performing stochastic simulations at the level of individual hexamers allows us to identify a new, to our knowledge, mechanism, which is employed by the Paijmans model to achieve input compensation: at lower ATP fraction, the individual hexamers make a shorter cycle in the phosphorylation state space, which compensates for the slower pace at which they traverse the cycle.

Publisher's Note: Biochemical Machines for the Interconversion of Mutual Information and Work [Phys. Rev. Lett. 118, 028101 (2017)].

This corrects the article DOI: 10.1103/PhysRevLett.118.028101.

Biochemical Machines for the Interconversion of Mutual Information and Work.

We propose a physically realizable information-driven device consisting of an enzyme in a chemical bath, interacting with pairs of molecules prepared in correlated states. These correlations persist without direct interaction and thus store free energy equal to the mutual information. The enzyme can harness this free energy, and that stored in the individual molecular states, to do chemical work. Alternatively, the enzyme can use the chemical driving to create mutual information. A modified system can function without external intervention, approaching biological systems more closely.

The magnitude of the intrinsic rate constant: How deep can association reactions be in the diffusion limited regime?

Intrinsic and effective rate constants have an important role in the theory of diffusion-limited reactions. In a previous paper, we provide detailed microscopic expressions for these intrinsic rates [A. Vijaykumar, P. G. Bolhuis, and P. R. ten Wolde, Faraday Discuss. 195, 421 (2016)], which are usually considered as abstract quantities and assumed to be implicitly known. Using these microscopic expressions, we investigate how the rate of association depends on the strength and the range of the isotropic potential and the strength of the non-specific attraction in case of the anisotropic potential. In addition, we determine the location of the interface where these expressions become valid for anisotropic potentials. In particular, by investigating the particles' orientational distributions, we verify whether the interface at which these distributions become isotropic agrees with the interface predicted by the effective association rate constant. Finally, we discuss how large the intrinsic association rate can become, and what are the consequences for the existence of the diffusion limited regime.

Multiscale simulations of anisotropic particles combining molecular dynamics and Green's function reaction dynamics.

The modeling of complex reaction-diffusion processes in, for instance, cellular biochemical networks or self-assembling soft matter can be tremendously sped up by employing a multiscale algorithm which combines the mesoscopic Green's Function Reaction Dynamics (GFRD) method with explicit stochastic Brownian, Langevin, or deterministic molecular dynamics to treat reactants at the microscopic scale [A. Vijaykumar, P. G. Bolhuis, and P. R. ten Wolde, J. Chem. Phys. 143, 214102 (2015)]. Here we extend this multiscale MD-GFRD approach to include the orientational dynamics that is crucial to describe the anisotropic interactions often prevalent in biomolecular systems. We present the novel algorithm focusing on Brownian dynamics only, although the methodology is generic. We illustrate the novel algorithm using a simple patchy particle model. After validation of the algorithm, we discuss its performance. The rotational Brownian dynamics MD-GFRD multiscale method will open up the possibility for large scale simulations of protein signalling networks.