macromolecular
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We focus on understanding the stability and the dynamics of mechanical structures of biological interests: from equilibrium statistical mechanics of internal motions of biomolecules to far-from-equilibrium assembly/disassembly of macromolecules and cells.
macromolecular
configurationsModern biology identifies many diseases as malfunctions at the molecular level. As a consequence, understanding the mechanisms of these macromolecules function is of enormous importance in the discovery of new therapies and diagnostics for diseases. From the physical prospect, the functions of molecules relate to the changing of states/configurations of those molecules.
To study the biological functions at the most fundamental level, we are curious about how stable each state is and how fast the molecules make transitions between these configurations. Indeed as complex as these molecules are, there are multiple stable configurations and the dynamics have a variety of flavors, from simple dynamics with Poisson statistics of torsional angles of peptides, complex fractal gating motions of enzymes, to the versatile and complicated dynamics on folding landscapes. Theoretical modeling and computational methods have extremely important applications and have been widely adopted as useful complements to experimental methods. Often experiments observe indirectly what is going on and could have certain limitations. They are also very labor intensive at times. Computational methods may provide a more holistic result which is crucial for the full understanding of biomolecular function and further bioengineering innovation.
- folding & unfolding
At the largest scale of molecule's internal motion, the folding/unfolding is intriguing. From a physical viewpoint, the process of the assembly of these long biopolymers chain into almost unique structures can be view as a disorder to order transformation. The ordered state is favored by the contact energy formed between the monomers while the disordered state ensemble is naturally preferred by the shear large number of unfolded states. Often there is free energy barrier between these two states that symbols neither energy nor entropy is being favored enough for the molecule to stay at that conformation. Theoretical methods can quantitatively calculate the relative stability of the free energy basins, the locations and heights of barriers and how they response to various external perturbations and mutations.
- enzymatic motions
Using AChE (acetylcholinesterase) as an example, it is an enzyme having a critical biological function, to break down one type of neurotransmitter and thus to terminate signaling in many synapses, including the neuromuscular junction. Thus it is the focus of variety of studies, from neurotoxins to Alzheimer' disease. The great speed of the enzyme is essential for rapid modulation of synaptic activity. However, there is a puzzle about how this enzyme can function effectively. The experimental result reveals an inactive conformation of this enzyme. The configuration of the enzyme from the crystal structure does not allow the neurotransmitter to get to the correct position, thus the neurotransmitter cannot be broken down at all. How can AChE catalyze at high speed? Theoretical studies lead to the analyses of active conformations based on inactive conformations of the enzyme.
- cis-trans isomerization
For small scale motion, torsional angle dynamics of biopolymers stands out as the most biologically important. One tough torsional angle to investigate is the omega angle of protein's backbone (CA-C-N'-CA'). The majority of times it has the trans conformation, while for residue proline and a few other situations it has a considerable probability of being cis. The barriers between these two conformations are very high [~ 15-20 kcal/mol] and lead to very slow dynamics [seconds to minutes]. We can use advanced computational methods to obtain both equilibrium and dynamic properties for various scenarios.
supramolecular assembly & disassembly, cellular tectonicsIt is import to look into the stability and dynamics of these larger cellular structures. These objects, with a size-scale quite beyond nanometers, are much more complicated in terms of these properties. Often the assembly/disassembly is no longer an equilibrium spontaneous process involving the objects themselves. Rather helper enzymes and energy sources are required to facilitate one state's transformation into another.
- nonequilibrium fibers: microtubule & f-actin
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- cell wall: cellulose, hemicellulose, & lignin
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flow of cellular informationCell as a complicated machine is constantly computing both external signals and intrinsic dynamics through a network of signaling biomolecules changing their states: from simple changes of chemistry, conformations, or positions of these players in the cell such as binding, induced conformational changes, posttranslational modification of proteins to more dramatic means, such as creating or deconstructing the players. Even though at this large scale the concepts of equilibrium statistical physics and classic Hamiltonian are of little help owing to the extreme nonequilibrium and heterogeneity nature of the system, some of the nonequilibrium physics concepts (and some more yet to be developed) are invaluable to exam the stability of different states of the cell and the dynamics of transformation from one state to another.
- posttranslational mod: phosphorylation, glycosylation
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Cellulosic biomass has the potential to serve as a major renewable energy source. However, its strong recalcitrance to degradation hampers its large-scale use in biofuel production. To overcome this problem, a detailed understanding of the origins of the recalcitrance is required. One main biophysical phenomenon leading to the recalcitrance is the high structural ordering of natural cellulose fibrils, that arises largely from an extensive hydrogen-bond network between and within cellulose polymers. Here, we present a lattice-based model of cellulose Ia, one of the two major natural forms, at the resolution of explicit hydrogen bonds. The partition function and thermodynamic properties are evaluated using the transfer matrix method. Two competing hydrogen-bond patterns are found. This plasticity of the hydrogen-bond network leads to an entropic contribution stabilizing the crystalline fibril at intermediate temperatures. At these temperatures, an enhanced probability of bonding between the individual cellulose chains gives rise to increased resistance of the entire cellulose fibril to degradation, before the final disassembly temperature is reached. The results are consistent with the available crystallographic and IR spectroscopic experiments on the thermostability of cellulose Ia.
Lectins are a class of proteins known for their novel binding to saccharides. Understanding this sugar recognition process can be crucial in creating structure-based designs of proteins with various biological roles. We focus on the sugar binding of a particular lectin, ricin, which has two beta-trefoil carbohydrate-binding domains (CRDs) found in several plant protein toxins. The binding ability of possible sites of ricin-like CRD has been puzzling. The apo and various (multiple) ligand-bound forms of the sugar-binding domains of ricin were studied by molecular dynamics simulations. By evaluating structural stability, hydrogen bond dynamics, flexibility, and binding energy, we obtained a detailed picture of the sugar recognition of the ricin-like CRD. Unlike what was previously believed, we found that the binding abilities of the two known sites are not independent of each other. The binding ability of one site is positively affected by the other site. While the mean positions of different binding scenarios are not altered significantly, the flexibility of the binding pockets visibly decreases upon multiple ligand binding. This change in flexibility seems to be the origin of the binding cooperativity. All the hydrogen bonds that are strong in the monoligand state are also strong in the double-ligand complex, although the stability is much higher in the latter form due to cooperativity. These strong hydrogen bonds in a monoligand state are deemed to be the essential hydrogen bonds. Furthermore, by examining the structural correlation matrix, the two domains are structurally one entity. Galactose hydroxyl groups, OH4 and OH3, are the most critical parts in both site 1-alpha and site 2-gamma recognition.
The interplay between cytoskeletal architecture and the nonlinearity of the interactions due to bucklable filaments plays a key role in modulating the cell's mechanical stability and affecting its structural rearrangements. We study a model of cytoskeletal structure treating it as an amorphous network of hard centers rigidly cross-linked by nonlinear elastic strings, neglecting the effects of motorization. Using simulations along with a self-consistent phonon method, we show that this minimal model exhibits diverse thermodynamically stable mechanical phases that depend on excluded volume, cross-link concentration, filament length, and stiffness. Within the framework set by the free energy functional formulation and making use of the random first order transition theory of structural glasses, we further estimate the characteristic densities for a kinetic glass transition to occur in this model system. Network connectivity strongly modulates the transition boundaries between various equilibrium phases, as well as the kinetic glass transition density.
Water plays a very important role in the dynamics and function of proteins. Apart from protein-protein and protein-water interactions, protein motions are accompanied by the formation and breakage of hydrogen-bonding network of the surrounding water molecules. This ordering and reordering of water also adds to the underlying roughness of the energy landscape of proteins and thereby alters their dynamics. Here, we extract the contribution of water to the ruggedness (in terms of an energy scale e) of the energy landscape from molecular dynamics simulations of a peptide substrate analogue of prolyl cis-trans isomerases. In order to do so, we develop and implement a model based on the position space analog of the Ornstein-Uhlenbeck process and Zwanzig's theory of diffusion on a rough potential. This allows us to also probe an important property of the widely used atomistic simulation water models that directly affects the dynamics of biomolecular systems and highlights the importance of the choice of the water model in studying protein dynamics. We show that water contributes an additional roughness to the energy landscape. At lower temperatures this roughness, which becomes comparable to k_B T, can considerably slow down protein dynamics. These results also have much broader implications for the function of some classes of enzymes, since the landscape topology of their substrates may change upon moving from an aqueous environment into the binding site.
Structures, dynamics, and stabilities of different sized cellulosic oligomers need to be considered when designing enzymatic cocktails for the conversion of biomass to biofuels since they can be both productive substrates and inhibitors of the overall process. In the present work, the conformational variability, hydrogen bonding and mechanical properties of short, soluble cellulose chains are investigated as a function of chain length. Cellulose oligomers consisting 2, 4, and 6 beta-D glucose units are examined in explicit solvent using the replica exchange molecular dynamics (REMD) which provides a rigorous evaluation of the relative stabilities of different conformations and their temperature dependence. This application of REMD to oligosaccharides in solution also allows evaluation of the quality of the force-field and its suitability for sampling carbohydrates efficiently. Simulation results are analyzed in synergy with polymer theory and compared to known measurements of oligomers and crystals. As chain length is increased, the conformations of the oligomers become more rigid and likely to form intra-chain hydrogen bonds, like those found in crystals. Several other conformations and hydrogen bonding patterns distinguish these short cellulose chains from those in cellulose crystals. These studies have also addressed the key role played by solvent on shifting the conformational preferences of the oligosaccharides with respect to vacuum and crystals. Correlation between pyranose ring flipping and the conformation of the 1,4-glycosidic bond was observed.
A critical roadblock to the production of biofuels from lignocellulosic biomass is the efficient degradation of crystalline microfibrils of cellulose to glucose. A microscopic understanding of how different physical conditions affect the overall stability of the crystalline structure of microfibrils could facilitate the design of more effective protocols for their degradation. One of the essential physical interactions that stabilizes microfibrils is a network of hydrogen (H) bonds: both intra-chain H-bonds between neighboring monomers of a single cellulose polymer chain and inter-chain H-bonds between adjacent chains. We construct a statistical mechanical model of cellulose assembly at the resolution of explicit hydrogen bond networks. Using the transfer matrix method, the partition function and the subsequent statistical properties are evaluated. With the help of this lattice-based model, we capture the plasticity of the H-bond network in cellulose due to frustration and redundancy in the placement of H-bonds. This plasticity is responsible for the stability of cellulose over a wide range of temperatures. Stable intra-chain and inter-chain H-bonds are identified as a function of temperature that could possibly be manipulated towards rational destruction of crystalline cellulose.
Human immunodeficiency virus type 1 (HIV-1) group M is responsible for the current AIDS pandemic and exhibits exceedingly high levels of viral genetic diversity around the world, necessitating categorization of viruses into distinct lineages, or subtypes. These subtypes can differ by around 35% in the envelope (Env) glycoproteins of the virus, which are displayed on the surface of the virion and are targets for both neutralizing antibody and cell-mediated immune responses. This diversity reflects the remarkable ability of the virus to adapt to selective pressures, the bulk of which is applied by the host immune response, and represents a serious obstacle for developing an effective vaccine with broad coverage. Thus, it is important to understand the underlying biological consequences of intersubtype diversity. Recent studies have revealed that some of the HIV-1 subtypes exhibit phenotypic differences stemming from subtle changes in Env structure, particularly within the highly immunogenic V3 domain, which participates directly in viral entry. This review will therefore explore current research that describes subtype differences in Env at the genetic and phenotypic level, focusing in particular on V3, and highlighting recent discoveries about the unique features of subtype C Env, which is the most globally prevalent subtype.
Various advanced simulation techniques, which are used to sample the statistical ensemble of systems with complex Hamiltonians, such as those displayed in condensed matters and biomolecular systems, rely heavily on successfully reweighting the sampled configurations. The sampled points of a system from an elevated thermal environment or on a modified Hamiltonian are reused with different statistical weights to evaluate its properties at the initial desired temperature or of the original Hamiltonian. Often, the decrease of accuracy induced by this procedure is ignored and the final results can be far from what is expected. We have addressed the reasons behind such a phenomenon and have provided a quantitative method to estimate the number of sampled points required in the crucial step of reweighting of these advanced simulation methods. We also provided examples from temperature histogram reweighting and accelerated molecular dynamics reweighting to illustrate this idea, which can be generalized to the dynamic reweighting as well. The study shows that this analysis may provide /a priori/ guidance for the strategy of setting up the parameters of advanced simulations before a lengthy one is carried out. The method can therefore provide insights for optimizing the parameters for high accuracy simulations with finite amount of computational resources.
Characterizing the phase diagram for proteins is important both for laboratory studies and for the development of structure prediction algorithms. Using a variational scheme, we calculated the generic features of the protein thermostability over a large range of temperatures for a set of more than 50 different proteins using a model based on native structure alone. Focusing on a specific system, protein G, we further examined, using a more realistic model that includes the nonnative interaction, the thermostability of both the native state and a collection of trap structures. By surveying the native structures for many proteins and by paying closer attention to the various trap structures of protein G, we obtained an overall understanding of the folding dynamics far from the conditions usually focused on; namely, those near the folding temperature alone. Two characteristic temperatures (shown to scale with folding temperature in general) signal drastic changes in the folding mechanism. The variational calculations suggest that most proteins would, indeed, fold in a barrierless manner below a critical temperature analogous to a spinodal in crystallization. For fixed interaction strengths, this temperature, however, seems to be generally very low, ~50% of the equilibrium folding temperature. Likewise, native proteins, in general, would unfold in a completely barrierless way at a temperature 25% above folding temperature according to these variational calculations. We also studied the distribution of free energy profiles for escape from a set of trap structures generated by simulations
We investigate how post-translational phosphorylation modifies the global conformation of a protein by changing its free energy landscape using two test proteins, cystatin and NtrC. We first examine the changes in a free energy landscape caused by phosphorylation using a model containing information about both structural forms. For cystatin the free energy cost is fairly large indicating a low probability of sampling the phosphorylated conformation in a perfectly funneled landscape. The predicted barrier for NtrC conformational transition is several times larger than the barrier for cystatin, indicating that the switch protein NtrC most probably follows a partial unfolding mechanism to move from one basin to the other. Principal component analysis and linear response theory show how the naturally occurring conformational changes in unmodified proteins are captured and stabilized by the change of interaction potential. We also develop a partially guided structure prediction Hamiltonian which is capable of predicting the global structure of a phosphorylated protein using only knowledge of the structure of the unphosphorylated protein or vice versa. This algorithm makes use of a generic transferable long-range residue contact potential along with details of structure short range in sequence. By comparing the results obtained with this guided transferable potential to those from the native-only, perfectly funneled Hamiltonians, we show that the transferable Hamiltonian correctly captures the nature of the global conformational changes induced by phosphorylation and can sample substantially correct structures for the modified protein with high probability.
The dynamics and diversity of proliferating cellular populations are governed by the interplay between the growth and death rates among the various phenotypes within a colony. In addition, epigenetic multistability can cause cells to spontaneously switch from one phenotype to another. By examining a generalized form of the relative variance of populations and classifying it into intra-colony and cross-colony contributions, we study the origins and consequences of cellular population variability. We find that the variability can be highly dependent on the initial conditions and the constraints placed on the population by the growth environment. We construct a two-phenotype model system and examine, analytically and numerically, its time-dependent variability in both unbounded and population limited growth environments. We find that in unbounded growth environments the overall variability is strictly governed by the initial conditions. In contrast, when the overall population is limited by the environment, the system eventually relaxes to a unique fixed point regardless of the initial conditions. However, the transient decay to the fixed point is highly dependent on initial conditions and the time scale over which the variability decays can be very long, depending on the intrinsic time scales of the system. These results provide insights into the origins of population variability and suggest mechanisms in which variability can arise in commonly used experimental approaches.
Phosphorylation of proteins by kinases is the most commonly studied class of posttranslational modification, yet its structural consequences are not well understood. The human SR (serine-arginine) protein ASF/SF2 relies on the processive phosphorylation of the serine residues of eight consecutive arginine-serine (RS) dipeptide repeats at the C terminus by SRPK1 before it can be transported into the nucleus. This SR protein plays critical roles in spliceosome assembly, pre-mRNA splicing, and mRNA export, and the phosphorylation process of the RS repeats has been extensively studied experimentally. However, knowledge of the conformational changes associated with the phosphorylation of this simple sequence and how it triggers the importation of the SR protein is lacking. Here, we have carried out extensive molecular dynamics simulations to show that phosphorylation of the eight RS repeats significantly alters the peptide's conformation and leads to the formation of very stable structures that are likely to be involved in the recognition, binding, and transport of the SR protein. Specifically, we found an unusual symmetry-broken phase of conformations of the repetitive and quasi-symmetric phosphorylated peptide sequence. One of the main characteristics of these conformations is the exposed phosphate groups on the periphery, which possibly could serve as the recognition platform for the transport protein transportin-SR2.
Understanding how the folding of proteins establishes their functional characteristics at the molecular level challenges both theorists and experimentalists. The simplest test beds for confronting this issue are provided by electron transfer proteins. The environment provided by the folded protein to the cofactor tunes the metal's electron transport capabilities as envisioned in the entatic hypothesis. To see how the entatic state is achieved one must study how the folding landscape affects and in turn is affected by the metal. Here, we develop a coarse-grained functional to explicitly model how the coordination of the metal (which results in a so-called entatic or rack-induced state) modifies the folding of the metallated Pseudomonas aeruginosa azurin. Our free-energy functional-based approach directly yields the proper nonlinear extra-thermodynamic free energy relationships for the kinetics of folding the wild type and several point-mutated variants of the metallated protein. The results agree quite well with corresponding laboratory experiments. Moreover, our modified free-energy functional provides a sufficient level of detail to explicitly model how the geometric entatic state of the metal modifies the dynamic folding nucleus of azurin.
It is believed that, much like a cat's cradle, the cytoskeleton can be thought of as a network of strings under tension. We show that both regular and random bond-disordered networks having bonds that buckle upon compression exhibit a variety of phase transitions as a function of temperature and extension. The results of self-consistent phonon calculations for the regular networks agree very well with computer simulations at finite temperature. The analytic theory also yields a rigidity onset (mechanical percolation) and the fraction of extended bonds for random networks. There is very good agreement with the simulations by Delaney et al. (2005 Europhys. Lett. 72 990). The mean field theory reveals a nontranslationally invariant phase with self-generated heterogeneity of tautness, representing 'antiferroelasticity'.
Cellular signal transduction often involves a reaction network of phosphorylation and transport events arranged with a ladder topology. If we keep track of the location of the phosphate groups describing an abstract state space, a simple model of signal transduction involving enzymes can be mapped on to a problem of how multiple biased random walkers compete to reach their target in the nucleus yielding a signal. Here, the first passage time probability and the survival probability for multiple walkers can be used to characterize the response of the network. The statistics of the first passage through the network has an asymmetric distribution with a long tail arising from the hierarchical structure of the network. This distribution implies a significant difference between the mean and the most probable signal transduction time. The response patterns for various external inputs generated by our model agree with recent experiments. In addition, the model predicts that there is an optimal phosphorylation enzyme concentration for rapid signal transduction.
The potential energy surface of a protein is rough. This intrinsic energetic roughness affects diffusion, and hence the kinetics. The dynamics of a system undergoing Brownian motion on this surface in an implicit continuum solvent simulation can be tuned via the frictional drag or collision frequency to be comparable to that of experiments or explicit solvent simulations. We show that the kinetic rate constant for a local rotational isomerization in stochastic simulations with continuum solvent and a collision frequency of 2 ps^-1 is about 104 times faster than that in explicit water and experiments. A further increase in the collision frequency to 60 ps^-1 slows down the dynamics, but does not fully compensate for the lack of explicit water. We also show that the addition of explicit water does not only slow down the dynamics by increasing the frictional drag, but also increases the local energetic roughness of the energy landscape by as much as 1.0 kcal/mol.
We calculated the changes of the free energy profile of the peptidyl-prolyl torsional angle of the dipeptide valine-proline under pulling forces by simulations. Using a dynamic model built on the equilibrium properties of this system and previously studied dynamic properties of cis-trans isomerization of other dipeptides, we calculated the dynamic viscoelasticity of this degree of freedom. The results show significant differences between how thermal and mechanical forces alter the equilibrium and the dynamics of the isomerization transition. The former does not change the barrier heights but changes the prefactor of the kinetics owing to temperature effects, while the latter changes minima and thus the population. The force that is required to "excite" this degree of freedom is small. Compared to other systems, we found that this degree of freedom becomes already quite rigid at several hertz, which is a much lower value due to the high barrier of the cis-trans isomerization. We also found that the tensile elastic modulus of densely packed omega bonds is at the order of GPa, which is comparable to that of polymer materials. These results give mechanical properties of polyproline elasticity of a local nature and provide guidance for future experimental designs.
The internal motions of proteins may serve as a "gate" in some systems, which controls ligand-protein association. This study applies Brownian dynamics simulations in a coarse-grained model to study the gated association rate constants of HIV-1 proteases and drugs. The computed gated association rate constants of three protease mutants, G48V/V82A/I84V/L90M, G48V, and L90M with three drugs, amprenavir, indinavir, and saquinavir, yield good agreements with experiments. The work shows that the flap dynamics leads to "slow gating". The simulations suggest that the flap flexibility and the opening frequency of the wild-type, the G48V and L90M mutants are similar, but the flaps of the variant G48V/V82A/I84V/L90M open less frequently, resulting in a lower gated rate constant. The developed methodology is fast and provides an efficient way to predict the gated association rate constants for various protease mutants and ligands.
Many of the large structures of cells are constructed from fibers. These fibers self-assemble from individual proteins in a far-from-equilibrium fashion. Nonequilibrium self-assembly results in a highly dynamic process at the subcellular level that can be regulated and tuned to carry out many of the biological functions of the cell: growth, division and locomotion. We construct and analyze a nonequilibrium model of the dynamic end of a biological fiber that possesses site-resolved resolution. We solve for the steady states of this nonequilibrium system using a variational method. The results are compared to exact numerical solutions for systems with modest size. Using an effective reaction coordinate, we construct an effective potential from the steady-state distribution. The stochastic transitions of the system can be analyzed in this representation. We then apply this method to model microtubule systems. Predictions for macroscopic catastrophe, rescue and dynamic instability in the steady states are analyzed. We find that the length of the cap of the microtubule is small. The relations between the catastrophe/rescue rate and the growth rate are also discussed.
Pseudomonas aeruginosa azurin is a 128-residue beta-sandwich metalloprotein; in vitro kinetic experiments have shown that it folds in a two-state reaction. Here, we used a variational free energy functional to calculate the characteristics of the transition state ensemble (TSE) for folding of the apo-form of P. aeruginosa azurin and investigate how it responds to thermal and mutational changes. The variational method directly yields predicted chevron plots for wild-type and mutant apo-forms of azurin. In parallel, we performed in vitro kinetic-folding experiments on the same set of azurin variants using chemical perturbation. Like the wild-type protein, all apo-variants fold in apparent two-state reactions both in calculations and in stopped-flow mixing experiments. Comparisons of phi values determined from the experimental and theoretical chevron parameters reveal an excellent agreement for most positions, indicating a polarized, highly structured TSE for folding of P. aeruginosa apo-azurin. We also demonstrate that careful analysis of side-chain interactions is necessary for appropriate theoretical description of core mutants.
The cytoskeleton is not an equilibrium structure. To develop theoretical tools to investigate such nonequilibrium assemblies, we study a statistical physical model of motorized spherical particles. Though simple, it captures some of the key nonequilibrium features of the cytoskeletal networks. Variational solutions of the many-body master equation for a set of motorized particles accounts for their thermally induced Brownian motion as well as for the motorized kicking of the structural elements. These approximations yield stability limits for crystalline phases and for frozen amorphous structures. The methods allow one to compute the effects of nonequilibrium behavior and adhesion (effective cross-linking) on the mechanical stability of localized phases as a function of density, adhesion strength, and temperature. We find that nonequilibrium noise does not necessarily destabilize mechanically organized structures. The nonequilibrium forces strongly modulate the phase behavior and have comparable effect as the adhesion due to cross-linking. Modeling transitions such as these allows the mechanical properties of cytoskeleton to rapidly and adaptively change. The present model provides a statistical mechanical underpinning for a tensegrity picture of the cytoskeleton.
An energy landscape approach predicts the conformational changes of the configurations of the regulatory domain of the protein nuclear factor of activated T cells (NFAT) caused by phosphorylation of specific multiple sites. Structurally local effects and secondary structural changes are modeled using all-atom Brownian dynamics to investigate the changes of the backbone torsional distributions upon phosphorylation. For tertiary and global changes, we employ a coarse-grained model to sample ensembles of conformations both with and without phosphorylation. At the secondary structure level, phosphorylation moderately increases the helical propensity and gives a more rigid local backbone conformation. The tertiary effects of phosphorylation caused by the extensive charge modification are more pronounced and collectively change the conformation of the regulatory domain of NFAT from a flexible globular ensemble to a rather rigid helical bundle, blocking access to the nuclear localization sequence. These studies give computational support to one scenario conjectured from experiments.
We show that our accelerated molecular-dynamics (MD) approach can extend the time scale in all-atom MD simulations of biopolymers. We also show that this technique allows for the kinetic rate information to be recaptured. In deducing the kinetic rates, the relationship between the local energetic roughness of the potential-energy landscape and the effective diffusion coefficient is established. These are demonstrated on a very slow but important biomolecular process: the dynamics of cis/trans-isomerization of Ser-Pro motifs. We do not only recapture the slow kinetic rates, which is difficult in traditional MD, but also obtain the underlying roughness of the energy landscape of proteins at atomistic resolution.
Using a variational free energy functional, we calculate the characteristics of the transition state ensembles (TSE) for the folding of protein U1A and investigate how they respond to thermal and mutational changes. The functional directly yields predicted chevron plots both for the wild-type protein and for various mutants. The detailed variations of the TSE and changes in chevron plots predicted by the theory agree reasonably well with the results of the experiments. We also show how to visualize the folding nuclei using 3D isodensity plots.
The presence of serine/threonine-proline motifs in proteins provides a conformational switching mechanism of the backbone through the cis/trans isomerization of the peptidyl-prolyl (omega) bond. The reversible phosphorylation of the serine/threonine modulates this switching in regulatory proteins to alter signaling and transcription. However, the mechanism is not well understood. This is partly because cis/trans isomerization is a very slow process and, hence, difficult to study. We have used our accelerated molecular dynamics method to study the cis/trans proline isomerization, preferred backbone conformation of a serine-proline motif, and the effects of phosphorylation of the serine residue. We demonstrate that, unlike normal molecular dynamics, the accelerated molecular dynamics allows for the system to escape very easily from the trans isomer to cis isomer, and vice versa. Moreover, for both the unphosphorylated and phosphorylated peptides, the statistical thermodynamic properties are recaptured, and the results are consistent with experimental values. Isomerization of the proline omega bond is shown to be asymmetric and strongly dependent on the phi backbone angle before and after phosphorylation. The rates of escape decrease after phosphorylation. Also, the alpha-helical backbone conformation is more favored after phosphorylation. This accelerated molecular dynamics approach provides a general approach for enhancing the conformational transitions of molecular systems without having prior knowledge of the location of the minima and barriers on the potential-energy landscape.
Protein folding has become one of the best understood biochemical reactions from a kinetic viewpoint. The funneled energy landscape, a consequence of the minimal frustration achieved by evolution in sequences, explains how most proteins fold efficiently and robustly to their functional structure and allows robust prediction of folding kinetics. The folding of Rop (repressor of primer) dimer is exceptional because some of its mutants with a redesigned hydrophobic core both fold and unfold much faster than the WT protein, which seems to conflict with a simple funneled energy landscape for which topology mainly determines the kinetics. We propose that the mystery of Rop folding can be unraveled by assuming a double-funneled energy landscape on which there are two basins that correspond to distinct but related topological structures. Because of the near symmetry of the molecule, mutations can cause a conformational switch to a nearly degenerate yet distinct topology or lead to a mixture of both topologies. The topology predicted to have the lower free-energy barrier height for folding was further found by all-atom modeling to give a better structural fit for those mutants with the extreme folding and unfolding rates. Thus, the non-Hammond effects can be understood within energy-landscape theory if there are in fact two different but nearly degenerate structures for Rop. Mutations in symmetric and regular structures may give rise to frustration and thus result in degeneracy.
Many of the large structures of the cell, such as the cytoskeleton, are assembled and maintained far from equilibrium. We study the stabilities of various structures for a simple model of such a far-from-equilibrium organized assembly in which spherical particles move under the influence of attached motors. From the variational solutions of the many-body master equation for Brownian motion with motorized kicking we obtain a closed equation for the order parameter of localization. Thus, we obtain the transition criterion for localization and stability limits for the crystalline phase and frozen amorphous structures of motorized particles. The theory also allows an estimate of nonequilibrium effective temperatures characterizing the response and fluctuations of motorized assemblies.
This article describes the development and implementation of algorithms to study diffusion in biomolecular systems using continuum mechanics equations. Specifically, finite element methods have been developed to solve the steady-state Smoluchowski equation to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to mouse acetylcholinesterase. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different reaction criteria. The calculated rates were compared with experimental data and show very good agreement when the correct reaction criterion is used. Additionally, these finite element methods require significantly less computational resources than existing particle-based Brownian dynamics methods.
Helix alpha-capping motifs are believed to play an important role in stabilizing -helices and defining helix start and stop signals. We performed microsecond scale Brownian dynamics simulations to study ten XAAD sequences, with X = (A,E,I,L,N,Q,S,T,V,Y), to examine their propensity to form helix capping motifs and correlate these results with those obtained from analyzing a structural database of proteins. For the widely studied capping box motif S**D, where the asterisk can be any amino acid residue, the simulations suggested that one of the two hydrogen bonds proposed earlier as a stabilizing factor might not be as important. On the other hand, side-chain interactions between the capping residue and the third residue downstream on the polypeptide chain might also play a role in stabilizing this motif. These results are consistent with explicit-solvent molecular dynamics simulations of two capping box motifs found in the proteins BPTI and alpha-dendrotoxin. Principal component analysis of the SAAD trajectory showed that the first three principal components, after those corresponding to translational-rotational motion were removed, accounted for more than half of the conformational fluctuations. The first component separated the conformational space into two parts with the all-helical conformation and the capping box motif lying largely in one part. The second component, on the other hand, could be used to describe conformational transitions between the all-helical form and the capping box motif.
We extend the self-consistent pair contact probability method to the evaluation of the partition function for a protein complex at thermodynamic equilibrium. Specifically, we adapt the method for multichain models and introduce a parametrization for amino acid-specific pairwise interactions. This method is similar to the Gaussian network model but allows for the adjusting of the strengths of native state contacts. The method is first validated on a high resolution x-ray crystal structure of bovine Pancreatic Phospholipase A2 by comparing calculated B-factors with reported values. We then examine binding-induced changes in flexibility in protein-protein complexes, comparing computed results with those obtained from x-ray crystal structures and molecular dynamics simulations. In particular, we focus on the mouse acetylcholinesterase:fasciculin II and the human alpha-thrombin:thrombomodulin complexes.
We extend a model of Micheletti et al. [Phys. Rev. Lett. 87, 088102 (2001)] used to study protein conformations to the case in which there is an external force field. Under the self-consistent pair contact probability approximation, this residue-level resolution model can still be solved under pulling forces. We implement the algorithm using heterogeneous parameters and study the force-induced unfolding of a helical segment from the protein transformylase and of the beta-stranded domains from the protein titin. The results are qualitatively consistent with the results from more expensive, atomistic dynamics simulation. Despite the mean-field-like approach, we observed a sharp and cooperative unfolding transition.
Our previous molecular dynamics simulation (10 ns) of mouse acetylcholinesterase (EC 3.1.1.7) revealed complex fluctuation of the enzyme active site gorge. Now we report a 5-ns simulation of acetylcholinesterase complexed with fasciculin 2. Fasciculin 2 binds to the gorge entrance of acetylcholinesterase with excellent complementarity and many polar and hydrophobic interactions. In this simulation of the protein-protein complex, where fasciculin 2 appears to sterically block access of ligands to the gorge, again we observe a two-peaked probability distribution of the gorge width. When fasciculin is present, the gorge width distribution is altered such that the gorge is more likely to be narrow. Moreover, there are large increases in the opening of alternative passages, namely, the side door (near Thr 75) and the back door (near Tyr 449). Finally, the catalytic triad arrangement in the acetylcholinesterase active site is disrupted with fasciculin bound. These data support that, in addition to the steric obstruction seen in the crystal structure, fasciculin may inhibit acetylcholinesterase by combined allosteric and dynamical means. Additional data from these simulations can be found at http://mccammon.ucsd.edu/.
Molecular dynamics simulations are leading to a deeper understanding of the activity of the enzyme acetylcholinesterase. Simulations have shown how breathing motions in the enzyme facilitate the displacement of substrate from the surface of the enzyme to the buried active site. The most recent work points to the complex and spatially extensive nature of such motions and suggests possible modes of regulation of the activity of the enzyme.
A 10-ns trajectory from a molecular dynamics simulation is used to examine the structure and dynamics of water in the active site gorge of acetylcholinesterase to determine what influence water may have on its function. While the confining nature of the deep active site gorge slows down and structures water significantly compared to bulk water, water in the gorge is found to display a number of properties that may aid ligand entry and binding. These properties include fluctuations in the population of gorge waters, moderate disorder and mobility of water in the middle and entrance to the gorge, reduced water hydrogen-bonding ability, and transient cavities in the gorge.
We report the implementation of an all-atom Brownian dynamics simulation model of peptides using the constraint algorithm LINCS. The algorithm has been added as a part of UHBD. It uses adaptive time steps to achieve a balance between computational speed and stability. The algorithm was applied to study the effect of phosphorylation on the conformational preference of the peptide Gly-Ser-Ser-Ser. We find that the middle serine residue experiences considerable conformational change from the C_7eq to the alpha_R structure upon phosphorylation. NMR J3 coupling constants were also computed from the Brownian trajectories using the Karplus equation. The calculated J3 results agree reasonably well with experimental data for phosphorylated peptide but less so for doubly charged phosphorylated one.
A 10-ns molecular dynamics simulation of mouse acetylcholinesterase was analyzed, with special attention paid to the fluctuation in the width of the gorge and opening events of the back door. The trajectory was first verified to ensure its stability. We defined the gorge proper radius as the measure for the extent of gorge opening. We developed an expression of an inter-atom distance representative of the gorge proper radius in terms of projections on the principal components. This revealed the fact that collective motions of many scales contribute to the opening behavior of the gorge. Covariance and correlation results identified the motions of the protein backbone as the gorge opens. In the back-door region, side-chain dihedral angles that define the opening were identified.
The enzyme acetylcholinesterase has an active site that is accessible only by a "gorge" or main channel from the surface, and perhaps by secondary channels such as the "back door." Molecular-dynamics simulations show that these channels are too narrow most of the time to admit substrate or other small molecules. Binding of substrates is therefore "gated" by structural fluctuations of the enzyme. Here, we analyze the fluctuations of these possible channels, as observed in the 10.8-ns trajectory of the simulation. The probability density function of the gorge proper radius (defined in the text) was calculated. A double-peak feature of the function was discovered and therefore two states with a threshold were identified. The relaxation (transition probability) functions of these two states were also calculated. The results revealed a power-law decay trend and an oscillation around it, which show properties of fractal dynamics with a complex exponent. The cross correlation of potential energy versus proper radius was also investigated. We discuss possible physical models behind the fractal protein dynamics; the dynamic hierarchical model for glassy systems is evaluated in detail.
Based on previous molecular dynamics simulation results for acetylcholinesterase dimer, we calculate and analyse the electrostatic field fluctuations around the enzyme. The results show that dynamic features of the electrostatic field favor attraction of the positively-charged substrate. An internet link to an animation of the results is also provided.
Analytical expressions for E1, E2, E3, M1, and M2 transition rates for low-lying negative-parity states in the SU(3) limit of the spdf IBM are given. Applications to some deformed nuclei in the A=150 region and Uranium isotopes have yielded good agreement between calculation and data for E1 transitions. These formulas are useful in studying both positive- and negative-parity states of deformed nuclei.
The SU(3) limit of the isospin invariant IBM-IBM3 is studied. The decomposition rules are given for N <= 9. An analytical formula for the decomposition of the U(6)[N,1] is given. Typical spectrum is discussed. Different forms of the interaction and their relation are obtained. Transition operators are also discussed. PACS numbers: 21.10.Re, 21.60.FW