Yaroslav Kholodov

Sop-GPU: accelerating biomolecular simulations in the centisecond timescale using graphics processors.

Theoretical exploration of fundamental biological processes involving the forced unraveling of multimeric proteins, the sliding motion in protein fibers and the mechanical deformation of biomolecular assemblies under physiological force loads is challenging even for distributed computing systems. Using a Cα‐based coarse‐grained self organized polymer (SOP) model, we implemented the Langevin simulations of proteins on graphics processing units (SOP‐GPU program). We assessed the computational performance of an end‐to‐end application of the program, where all the steps of the algorithm are running on a GPU, by profiling the simulation time and memory usage for a number of test systems. The ∼90‐fold computational speedup on a GPU, compared with an optimized central processing unit program, enabled us to follow the dynamics in the centisecond timescale, and to obtain the force‐extension profiles using experimental pulling speeds (vf = 1–10 μm/s) employed in atomic force microscopy and in optical tweezers‐based dynamic force spectroscopy. We found that the mechanical molecular response critically depends on the conditions of force application and that the kinetics and pathways for unfolding change drastically even upon a modest 10‐fold increase in vf. This implies that, to resolve accurately the free energy landscape and to relate the results of single‐molecule experiments in vitro and in silico, molecular simulations should be carried out under the experimentally relevant force loads. This can be accomplished in reasonable wall‐clock time for biomolecules of size as large as 105 residues using the SOP‐GPU package. Proteins 2010;

Tubulin bond energies and microtubule biomechanics determined from nanoindentation in silico.

Microtubules, the primary components of the chromosome segregation machinery, are stabilized by longitudinal and lateral noncovalent bonds between the tubulin subunits. However, the thermodynamics of these bonds and the microtubule physicochemical properties are poorly understood. Here, we explore the biomechanics of microtubule polymers using multiscale computational modeling and nanoindentations in silico of a contiguous microtubule fragment. A close match between the simulated and experimental force–deformation spectra enabled us to correlate the microtubule biomechanics with dynamic structural transitions at the nanoscale. Our mechanical testing revealed that the compressed MT behaves as a system of rigid elements interconnected through a network of lateral and longitudinal elastic bonds. The initial regime of continuous elastic deformation of the microtubule is followed by the transition regime, during which the microtubule lattice undergoes discrete structural changes, which include first the reversible dissociation of lateral bonds followed by irreversible dissociation of the longitudinal bonds. We have determined the free energies of dissociation of the lateral (6.9 ± 0.4 kcal/mol) and longitudinal (14.9 ± 1.5 kcal/mol) tubulin–tubulin bonds. These values in conjunction with the large flexural rigidity of tubulin protofilaments obtained (18,000–26,000 pN·nm2) support the idea that the disassembling microtubule is capable of generating a large mechanical force to move chromosomes during cell division. Our computational modeling offers a comprehensive quantitative platform to link molecular tubulin characteristics with the physiological behavior of microtubules. The developed in silico nanoindentation method provides a powerful tool for the exploration of biomechanical properties of other cytoskeletal and multiprotein assemblies.

Generation of random numbers on graphics processors: forced indentation in silico of the bacteriophage HK97.

The use of graphics processing units (GPUs) in simulation applications offers a significant speed gain as compared to computations on central processing units (CPUs). Many simulation methods require a large number of independent random variables generated at each step. We present two approaches for implementation of random number generators (RNGs) on a GPU. In the one-RNG-per-thread approach, one RNG produces a stream of random numbers in each thread of execution, whereas the one-RNG-for-all-threads method builds on the ability of different threads to communicate, thus, sharing random seeds across an entire GPU device. We used these approaches to implement Ran2, Hybrid Taus, and Lagged Fibonacci algorithms on a GPU. We profiled the performance of these generators in terms of the computational time, memory usage, and the speedup factor (CPU time/GPU time). These generators have been incorporated into the program for Langevin simulations of biomolecules fully implemented on the GPU. The ∼250-fold computational speedup on the GPU allowed us to carry out single-molecule dynamic force measurements in silico to explore the mechanical properties of the bacteriophage HK97 in the experimental subsecond time scale. We found that the nanomechanical response of HK97 depends on the conditions of force application, including the rate of change and geometry of the mechanical perturbation. Hence, using the GPU-based implementation of RNGs, presented here, in conjunction with Langevin simulations, makes it possible to directly compare the results of dynamic force measurements in vitro and in silico.

Fluctuating Nonlinear Spring Model of Mechanical Deformation of Biological Particles

The mechanical properties of virus capsids correlate with local conformational dynamics in the capsid structure. They also reflect the required stability needed to withstand high internal pressures generated upon genome loading and contribute to the success of important events in viral infectivity, such as capsid maturation, genome uncoating and receptor binding. The mechanical properties of biological nanoparticles are often determined from monitoring their dynamic deformations in Atomic Force Microscopy nanoindentation experiments; but a comprehensive theory describing the full range of observed deformation behaviors has not previously been described. We present a new theory for modeling dynamic deformations of biological nanoparticles, which considers the non-linear Hertzian deformation, resulting from an indenter-particle physical contact, and the bending of curved elements (beams) modeling the particle structure. The beams’ deformation beyond the critical point triggers a dynamic transition of the particle to the collapsed state. This extreme event is accompanied by a catastrophic force drop as observed in the experimental or simulated force (F)-deformation (X) spectra. The theory interprets fine features of the spectra, including the nonlinear components of the FX-curves, in terms of the Young’s moduli for Hertzian and bending deformations, and the structural damage dependent beams’ survival probability, in terms of the maximum strength and the cooperativity parameter. The theory is exemplified by successfully describing the deformation dynamics of natural nanoparticles through comparing theoretical curves with experimental force-deformation spectra for several virus particles. This approach provides a comprehensive description of the dynamic structural transitions in biological and artificial nanoparticles, which is essential for their optimal use in nanotechnology and nanomedicine applications.

Protein-protein docking by fast generalized Fourier transforms on 5D rotational manifolds.

Significance Expressing the interaction energy as sum of correlation functions, fast Fourier transform (FFT) based methods speed the calculation, enabling the sampling of billions of putative protein–protein complex conformations. However, such acceleration is currently achieved only on a 3D subspace of the full 6D rotational/translational space, and the remaining dimensions must be sampled using conventional slow calculations. Here we present an algorithm that employs FFT-based sampling on the 5D rotational space, and only the 1D translations are sampled conventionally. The accuracy of the results is the same as those of earlier methods, but the calculation is an order of magnitude faster. Also, it is inexpensive computationally to add more correlation function terms to the scoring function compared with classical approaches. Energy evaluation using fast Fourier transforms (FFTs) enables sampling billions of putative complex structures and hence revolutionized rigid protein–protein docking. However, in current methods, efficient acceleration is achieved only in either the translational or the rotational subspace. Developing an efficient and accurate docking method that expands FFT-based sampling to five rotational coordinates is an extensively studied but still unsolved problem. The algorithm presented here retains the accuracy of earlier methods but yields at least 10-fold speedup. The improvement is due to two innovations. First, the search space is treated as the product manifold SO(3)×(SO(3)∖S1), where SO(3) is the rotation group representing the space of the rotating ligand, and (SO(3)∖S1) is the space spanned by the two Euler angles that define the orientation of the vector from the center of the fixed receptor toward the center of the ligand. This representation enables the use of efficient FFT methods developed for SO(3). Second, we select the centers of highly populated clusters of docked structures, rather than the lowest energy conformations, as predictions of the complex, and hence there is no need for very high accuracy in energy evaluation. Therefore, it is sufficient to use a limited number of spherical basis functions in the Fourier space, which increases the efficiency of sampling while retaining the accuracy of docking results. A major advantage of the method is that, in contrast to classical approaches, increasing the number of correlation function terms is computationally inexpensive, which enables using complex energy functions for scoring.

SOP‐GPU: influence of solvent‐induced hydrodynamic interactions on dynamic structural transitions in protein assemblies

Hydrodynamic interactions (HI) are incorporated into Langevin dynamics of the Cα‐based protein model using the Truncated Expansion approximation (TEA) to the Rotne–Prager–Yamakawa diffusion tensor. Computational performance of the obtained GPU realization demonstrates the models capability for describing protein systems of varying complexity (102–105 residues), including biological particles (filaments, virus shells). Comparison of numerical accuracy of the TEA versus exact description of HI reveals similar results for the kinetics and thermodynamics of protein unfolding. The HI speed up and couple biomolecular transitions through cross‐communication among protein domains, which result in more collective displacements of structure elements governed by more deterministic (less variable) dynamics. The force‐extension/deformation spectra from nanomanipulations in silico exhibit sharper force signals that match well the experimental profiles. Hence, biomolecular simulations without HI overestimate the role of tension/stress fluctuations. Our findings establish the importance of incorporating implicit water‐mediated many‐body effects into theoretical modeling of dynamic processes involving biomolecules.

Computational Study of the Vibrating Disturbances to the Lung Function

Frequently during its lifetime a human organism is subjected to the acoustical and similar to them vibrating impacts. Under the certain conditions such influence may cause physiological changes in the organs functioning. Thus the study of the oscillatory mechanical impacts to the organism is very important task of the numerical physiology. It allows to investigate the endurance limits of the organism and to develop protective measures in order to extend them. The noise nuisances affects to the most parts of the organism disrupting their functions. The vibrating disturbances caused to the lung function as one of the most sensitive to the acoustical impacts is considered in this work. The model proposed to describe the air motion in trachea-bronchial tree is based on the one dimensional no-linear theory including mass and momentum conservation for the air flow in flexible tubes similar to the model of blood flow in large vessels [1]. It combined with the single-component model of alveole [1], [2]. Two types of vibrating impacts were simulated that affect the thorax and the nasopharynx. The conducted simulations allowed us to detect two resonance frequencies that lay in the ranges from 3 Hz to 8 Hz and from 40 to 70 Hz when the thorax was affected (fig.1). For the nasopharynx disturbances no resonance states were found. Open image in new window Fig. 1. Dependencies of the integral volume and pressure of the lungs from oscillatory impacts.

Global Dynamical Model of the Cardiovascular System

Blood system functions are very diverse and important for most processes in human organism. One of its primary functions is matter transport among different parts of the organism including tissue supplying with oxygen, carbon dioxide excretion, drug propagation etc. Forecasting of these processes under normal conditions and in the presence of different pathologies like atherosclerosis, loss of blood, anatomical abnormalities, pathological changing in chemical transformations and others is significant issue for many physiologists. In this connection should be pointed out that global processes are of special interest as they include feedbacks and interdependences among different regions of the organism. At the modern level of computer engineering the most adequate physical model for the dynamical description of cardiovascular system is the model of non-stationary flow of incompressible fluid through the system of elastic tubes. Mechanics of such flow is described by nonlinear set of hyperbolic equations including mass and momentum conservation joined with equation of state that determines elastic properties of the tube [1]. As we interested in global processes the models of the four vascular trees (arterial and venous parts of systemic and pulmonary circulation) must be closed with heart and peripheral circulation models. Heart operation is described by the model of fluid flow averaged by volume through the system of extensible chambers that results in the set of stiff ordinary differential equations [1]. When combined these models allow us to consider functional changes and responses as during one cardiac cycle and at a longer periods upon 10 minutes that Open image in new window Fig. 1. Pressure wave propagation through the large pulmonary arteries during one cardiac cycle. Grayscale designates divergence from the minimum pressure in each vessel.

Computational Models on Graphs for Nonlinear Hyperbolic and Parabolic System of Equations

For each graph edge with length X k we consider 1D nonlinear hyperbolic system of equations \( \overrightarrow \nu _t + \overrightarrow F _{x_k } \left( {\overrightarrow \nu } \right) = \overrightarrow f ,\overrightarrow \nu = \left\{ {\nu _1 , \ldots ,\nu _1 } \right\},t \geqslant 0,0 \leqslant x_k \leqslant X_k ,k = 1, \ldots ,K \) (1) with initial conditions \( \overrightarrow \nu \left( {0,x_k } \right) = \overrightarrow \nu ^0 \left( {x_k } \right),k = 1, \ldots ,K \) and the next boundary conditions: for graph enters \( \left( {l^0 = 1, \ldots L^0 ,x_{k_ * } = 0} \right)\varphi _{li}^0 \left( {t,\overrightarrow \nu \left( {t,0} \right)} \right) = 0,i = 1, \ldots r_k^0 \leqslant I \) (2), for graph exits \( \left( {l^ * = 1, \ldots L^ * ,x_k = X_k } \right)\varphi _{li} \left( {t,\overrightarrow \nu \left( {t,X_k } \right)} \right) = 0,i = 1, \ldots ,r_k^ * \leqslant I \) (3) and for graph branchpoints \( l^ * = 1, \ldots L\psi _{lm} \left( {t,w_l ,\overrightarrow \nu _{l1} , \ldots \overrightarrow \nu _{lM_1 } } \right) = 0m = 1, \ldots M_l \) (4). Here K is the number of graph edges, LO - enters, LO - exits, L - branchpoints, M l - incoming and outgoing graph edges for the lth branchpoint, \( \overrightarrow \nu _{l1} , \ldots \overrightarrow \nu _{lM_l } \) - required vectors in the ends of edges adjoining to branchpoin l, W l - required vector for the branchpoint l. The matrix \( {{\partial \overrightarrow F } \mathord{\left/ {\vphantom {{\partial \overrightarrow F } {\partial \overrightarrow \nu }}} \right. \kern-\nulldelimiterspace} {\partial \overrightarrow \nu }} = A = \left\{ {a_{ij} } \right\}i,j = 1, \ldots ,I \) is Jacobi matrix and we can apply the identity \( A = \Omega ^{ - 1} \Lambda \Omega \), where \( \Lambda = \left\{ {\lambda _i } \right\} \) is the diagonal matrix of the matrix A eigenvalues, Ώ is the nonsingular matrix whose rows are linearly independent left-hand eigenvectors of the matrix A \( \left( {Det\Omega \ne 0} \right) \) and Ώ −1 is the matrix inverse to Ώ.

52 Tubulin bond energies and microtubule biomechanics determined from nanoindentation in silico

Microtubules, the primary components of the chromosome segregation machinery, are stabilized by longitudinal and lateral noncovalent bonds between the tubulin subunits. However, the thermodynamics of these bonds and the microtubule physicochemical properties are poorly understood. Here, we explore the biomechanics of microtubule polymers using multiscale computational modeling and nanoindentations in silico of a contiguous microtubule fragment. A close match between the simulated and experimental force–deformation spectra enabled us to correlate the microtubule biomechanics with dynamic structural transitions at the nanoscale. Our mechanical testing revealed that the compressed MT behaves as a system of rigid elements interconnected through a network of lateral and longitudinal elastic bonds. The initial regime of continuous elastic deformation of the microtubule is followed by the transition regime, during which the microtubule lattice undergoes discrete structural changes, which include first the revers...