A coarse-grained model for force-induced protein deformation and kinetics

Solving the inverse problem of time independent Fokker-Planck equation with a self supervised neural network method

The Fokker–Planck equation (FPE) has been used in many important applications to study stochastic processes with the evolution of the probability density function (pdf). Previous studies on FPE mainly focus on solving the forward problem which is to predict the time-evolution of the pdf from the underlying FPE terms. However, in many applications the FPE terms are usually unknown and roughly estimated, and solving the forward problem becomes more challenging. In this work, we take a different approach of starting with the observed pdfs to recover the FPE terms using a self-supervised machine learning method. This approach, known as the inverse problem, has the advantage of requiring minimal assumptions on the FPE terms and allows data-driven scientific discovery of unknown FPE mechanisms. Specifically, we propose an FPE-based neural network (FPE-NN) which directly incorporates the FPE terms as neural network weights. By training the network on observed pdfs, we recover the FPE terms. Additionally, to account for noise in real-world observations, FPE-NN is able to denoise the observed pdfs by training the pdfs alongside the network weights. Our experimental results on various forms of FPE show that FPE-NN can accurately recover FPE terms and denoising the pdf plays an essential role.

Solving Fokker-Planck equation using deep learning

The probability density function of stochastic differential equations is governed by the Fokker-Planck (FP) equation. A novel machine learning method is developed to solve the general FP equations based on deep neural networks. The proposed algorithm does not require any interpolation and coordinate transformation, which is different from the traditional numerical methods. The main novelty of this paper is that penalty factors are introduced to overcome the local optimization for the deep learning approach, and the corresponding setting rules are given. Meanwhile, we consider a normalization condition as a supervision condition to effectively avoid that the trial solution is zero. Several numerical examples are presented to illustrate performances of the proposed algorithm, including one-, two-, and three-dimensional systems. All the results suggest that the deep learning is quite feasible and effective to calculate the FP equation. Furthermore, influences of the number of hidden layers, the penalty factors, and the optimization algorithm are discussed in detail. These results indicate that the performances of the machine learning technique can be improved through constructing the neural networks appropriately.

Inward open characterization of EmrD transporter with molecular dynamics simulation

EmrD is a member of the multidrug resistance exporter family. Up to now, little is known about the structural dynamics that underline the function of the EmrD protein in inward-facing open state and how the EmrD transits from an occluded state to an inward open state. For the first time the article applied the AT simulation to investigate the membrane transporter protein EmrD, and described the dynamic features of the whole protein, the domain, the helices, and the amino acid residues during an inward-open process from its occluded state. The gradual inward-open process is different from the current model of rigid-body domain motion in alternating-access mechanism. Simulation results show that the EmrD inward-open conformational fluctuation propagates from a C-terminal domain to an N-terminal domain via the linker region during the transition from its occluded state. The conformational fluctuation of the C-terminal domain is larger than that of the N-terminal domain. In addition, it is observed that the helices exposed to the surrounding membrane show a higher level of flexibility than the other regions, and the protonated E227 plays a key role in the transition from the occluded to the open state.

Cell-biomaterial mechanical interaction in the framework of tissue engineering: insights, computational modeling and perspectives

Tissue engineering is an emerging field of research which combines the use of cell-seeded biomaterials both in vitro and/or in vivo with the aim of promoting new tissue formation or regeneration. In this context, how cells colonize and interact with the biomaterial is critical in order to get a functional tissue engineering product. Cell-biomaterial interaction is referred to here as the phenomenon involved in adherent cells attachment to the biomaterial surface, and their related cell functions such as growth, differentiation, migration or apoptosis. This process is inherently complex in nature involving many physico-chemical events which take place at different scales ranging from molecular to cell body (organelle) levels. Moreover, it has been demonstrated that the mechanical environment at the cell-biomaterial location may play an important role in the subsequent cell function, which remains to be elucidated. In this paper, the state-of-the-art research in the physics and mechanics of cell-biomaterial interaction is reviewed with an emphasis on focal adhesions. The paper is focused on the different models developed at different scales available to simulate certain features of cell-biomaterial interaction. A proper understanding of cell-biomaterial interaction, as well as the development of predictive models in this sense, may add some light in tissue engineering and regenerative medicine fields.

Energy landscape, structure and rate effects on strength properties of alpha-helical proteins

The strength of protein domains is crucial to identify the mechanical role of protein domains in biological processes such as mechanotransduction, tissue mechanics and tissue remodeling. Whereas the concept of strength has been widely investigated for engineered materials, the strength of fundamental protein material building blocks and how it depends on structural parameters such as the chemical bonding, the protein filament length and the timescale of observation or deformation velocity remains poorly understood. Here we report a systematic analysis of the influence of key parameters that define the energy landscape of the strength properties of alpha-helical protein domains, including energy barriers, unfolding and refolding distances, the locations of folded and unfolded states, as well as variations of the length and pulling velocity of alpha-helical protein filaments. The analysis is facilitated by the development of a double-well mesoscale potential formulation, utilized here to carry out a systematic numerical analysis of the behavior of alpha-helices. We compare the results against widely used protein strength models based on the Bell model, one of the simplest models used to characterize the strength of protein filaments. We find that, whereas Bell-type models are a reasonable approximation to describe the rupture of alpha-helical protein domains for a certain range of pulling speeds and values of energy barriers, the model ceases to hold for very large energy barriers and for very small pulling speeds, in agreement with earlier findings. We conclude with an application of our mesoscale model to investigate the effect of the length of alpha-helices on their mechanical strength. We find a weakening effect as the length of alpha-helical proteins increases, followed by an asymptotic regime in which the strength remains constant. We compare strand lengths found in biological proteins with the scaling law of strength versus alpha-helix filament length. The mesoscale model reported here is generally applicable to other protein filaments that feature a serial array of domains that unfold under applied strain, where a similar length-dependent strength could be observed.

Mechanics of forced unfolding of proteins

We describe and solve a two-state kinetic model for the forced unfolding of proteins. The protein oligomer is modeled as a heterogeneous, freely jointed chain with two possible values of Kuhn length and contour length representing its folded and unfolded configurations. We obtain analytical solutions for the force-extension response of the protein oligomer for different types of loading conditions. We fit the analytical solutions for constant-velocity pulling to the force-extension data for ubiquitin and fibrinogen and obtain model parameters, such as Kuhn lengths and kinetic coefficients, for both proteins. We then predict their response under a linearly increasing force and find that our solutions for ubiquitin are consistent with a different set of experiments. Our calculations suggest that the refolding rate of proteins at low forces is several orders larger than the unfolding rate, and neglecting it can lead to lower predictions for the unfolding force, especially at high stretching velocities. By accounting for the refolding of proteins we obtain a critical force below which equilibrium is biased in favor of the folded state. Our calculations also suggest new methods to determine the distance of the transition state from the energy wells representing the folded and unfolded states of a protein.

Analyzing single-bond experiments: influence of the shape of the energy landscape and universal law between the width, depth, and force spectrum of the bond

Experimentalists who measure the rupture force of a single molecular bond usually pull on that bond at a constant speed, keeping the loading rate r=df/dt constant. The challenge is to extract the energy landscape of the interaction between the two molecules involved from the experimental rupture force distribution under several loading rates. This analysis requires the use of a model for the shape of this energy landscape. Several barriers can compose the landscape, though molecular bonds with a single barrier are often observed. The Bell model is commonly used for the analysis of rupture force measurements with bonds displaying a single barrier. It provides an analytical expression of the most likely rupture force which makes it very simple to use. However, in principle, it can only be applied to landscapes with extrema whose positions do not vary under force. Here, we evaluate the general relevance of the Bell model by comparing it with another analytical model for which the landscape is harmonic in the vicinity of its extrema. Similar shapes of force distributions are obtained with both models, making it difficult to confirm the validity of the Bell model for a given set of experimental data. Nevertheless, we show that the analysis of rupture force experiments on such harmonic landscapes with the Bell model provides excellent results in most cases. However, numerical computation of the distributions of the rupture forces on piecewise-linear energy landscapes indicates that the blind use of any model such as the Bell model may be risky, since there often exist several landscapes compatible with a given set of experimental data. Finally, we derive a universal relation between the range and energy of the bond and the force spectrum. This relation does not depend on the shape of the energy landscape and can thus be used to characterize unambiguously any one-barrier landscape from experiments. All the results are illustrated with the streptavidin-biotin bond.

Extending Bell's model: how force transducer stiffness alters measured unbinding forces and kinetics of molecular complexes

Forced unbinding of complementary macromolecules such as ligand-receptor complexes can reveal energetic and kinetic details governing physiological processes ranging from cellular adhesion to drug metabolism. Although molecular-level experiments have enabled sampling of individual ligand-receptor complex dissociation events, disparities in measured unbinding force F(R) among these methods lead to marked variation in inferred binding energetics and kinetics at equilibrium. These discrepancies are documented for even the ubiquitous ligand-receptor pair, biotin-streptavidin. We investigated these disparities and examined atomic-level unbinding trajectories via steered molecular dynamics simulations, as well as via molecular force spectroscopy experiments on biotin-streptavidin. In addition to the well-known loading rate dependence of F(R) predicted by Bell's model, we find that experimentally accessible parameters such as the effective stiffness of the force transducer k can significantly perturb the energy landscape and the apparent unbinding force of the complex for sufficiently stiff force transducers. Additionally, at least 20% variation in unbinding force can be attributed to minute differences in initial atomic positions among energetically and structurally comparable complexes. For force transducers typical of molecular force spectroscopy experiments and atomistic simulations, this energy barrier perturbation results in extrapolated energetic and kinetic parameters of the complex that depend strongly on k. We present a model that explicitly includes the effect of k on apparent unbinding force of the ligand-receptor complex, and demonstrate that this correction enables prediction of unbinding distances and dissociation rates that are decoupled from the stiffness of actual or simulated molecular linkers.

A structural, kinetic model of soft tissue thermomechanics

A structure-based kinetic model was developed to predict the thermomechanical response of collagenous soft tissues. The collagen fibril was represented as an ensemble of molecular arrays with cross-links connecting the collagen molecules within the same array. A two-state kinetic model for protein folding was employed to represent the native and the denatured states of the collagen molecule. The Monte Carlo method was used to determine the state of the collagen molecule when subjected to thermal and mechanical loads. The model predictions were compared to existing experimental data for New Zealand white rabbit patellar tendons. The model predictions for one-dimensional tissue shrinkage and the corresponding mechanical property degradation agreed well with the experimental data, showing that the gross tissue behavior is dictated by molecular-level phenomena.

Force-induced activation of talin and its possible role in focal adhesion mechanotransduction

It is now well established that cells can sense mechanical force, but the mechanisms by which force is transduced into a biochemical signal remain poorly understood. One example is the recruitment of vinculin to reinforce initial contacts between a cell and the extracellular matrix (ECM) due to tensile force. Talin, an essential linking protein in an initial contact, contains at least one vinculin-binding site (VBS) that is cryptic and inactive in the native state. The N-terminal five-helix bundle of talin rod is a stable structure with a known cryptic VBS1. The perturbation of this stable structure through elevated temperature or destabilizing mutation activates vinculin binding. Although the disruption of this subdomain by transmitted mechanical force is a potential cue for the force-induced focal adhesion strengthening, the molecular basis for this mechanism remains elusive. Here, molecular dynamics (MD) is employed to demonstrate a force-induced conformational change that exposes the cryptic vinculin-binding residues of VBS1 to solvent under applied force along a realistic pulling direction. VBS1 undergoes a rotation of 62.0 +/- 9.5 degrees relative to its native state as its vinculin-binding residues are released from the tight hydrophobic core. Charged and polar residues on the VBS1 surface are the site of force transmission that strongly interact with an adjacent alpha-helix, and in effect, apply torque to the VBS1 to cause its rotation. Activation was observed with mean force of 13.2 +/-8.0 pN during constant velocity simulation and with steady force greater than 18.0 pN.