A generalized Flory-Stockmayer kinetic theory of connectivity percolation and rigidity percolation of cytoskeletal networks.

Actin networks are essential for living cells to move, reproduce, and sense their environments. The dynamic and rheological behavior of actin networks is modulated by actin-binding proteins such as α-actinin, Arp2/3, and myosin. There is experimental evidence that actin-binding proteins modulate the cooperation of myosin motors by connecting the actin network. In this work, we present an analytical mean field model, using the Flory-Stockmayer theory of gelation, to understand how different actin-binding proteins change the connectivity of the actin filaments as the networks are formed. We follow the kinetics of the networks and estimate the concentrations of actin-binding proteins that are needed to reach connectivity percolation as well as to reach rigidity percolation. We find that Arp2/3 increases the actomyosin connectivity in the network in a non-monotonic way. We also describe how changing the connectivity of actomyosin networks modulates the ability of motors to exert forces, leading to three possible phases of the networks with distinctive dynamical characteristics: a sol phase, a gel phase, and an active phase. Thus, changes in the concentration and activity of actin-binding proteins in cells lead to a phase transition of the actin network, allowing the cells to perform active contraction and change their rheological properties.

The role of the Arp2/3 complex in shaping the dynamics and structures of branched actomyosin networks.

Actomyosin networks give cells the ability to move and divide. These networks contract and expand while being driven by active energy-consuming processes such as motor protein walking and actin polymerization. Actin dynamics is also regulated by actin-binding proteins, such as the actin-related protein 2/3 (Arp2/3) complex. This complex generates branched filaments, thereby changing the overall organization of the network. In this work, the spatiotemporal patterns of dynamical actin assembly accompanying the branching-induced reorganization caused by Arp2/3 were studied using a computational model (mechanochemical dynamics of active networks [MEDYAN]); this model simulates actomyosin network dynamics as a result of chemical reactions whose rates are modulated by rapid mechanical equilibration. We show that branched actomyosin networks relax significantly more slowly than do unbranched networks. Also, branched networks undergo rare convulsive movements, "avalanches," that release strain in the network. These avalanches are associated with the more heterogeneous distribution of mechanically linked filaments displayed by branched networks. These far-from-equilibrium events arising from the marginal stability of growing actomyosin networks provide a possible mechanism of the "cytoquakes" recently seen in experiments.

Assemblies of calcium/calmodulin-dependent kinase II with actin and their dynamic regulation by calmodulin in dendritic spines.

Calcium/calmodulin-dependent kinase II (CaMKII) plays a key role in the plasticity of dendritic spines. Calcium signals cause calcium-calmodulin to activate CaMKII, which leads to remodeling of the actin filament (F-actin) network in the spine. We elucidate the mechanism of the remodeling by combining computer simulations with protein array experiments and electron microscopic imaging, to arrive at a structural model for the dodecameric complex of CaMKII with F-actin. The binding interface involves multiple domains of CaMKII. This structure explains the architecture of the micrometer-scale CaMKII/F-actin bundles arising from the multivalence of CaMKII. We also show that the regulatory domain of CaMKII may bind either calmodulin or F-actin, but not both. This frustration, along with the multipartite nature of the binding interface, allows calmodulin transiently to strip CaMKII from actin assemblies so that they can reorganize. This observation therefore provides a simple mechanism by which the structural dynamics of CaMKII establishes the link between calcium signaling and the morphological plasticity of dendritic spines.

Determining the atomic charge of calcium ion requires the information of its coordination geometry in an EF-hand motif.

It is challenging to parameterize the force field for calcium ions (Ca2+) in calcium-binding proteins because of their unique coordination chemistry that involves the surrounding atoms required for stability. In this work, we observed a wide variation in Ca2+ binding loop conformations of the Ca2+-binding protein calmodulin, which adopts the most populated ternary structures determined from the molecular dynamics simulations, followed by ab initio quantum mechanical (QM) calculations on all 12 amino acids in the loop that coordinate Ca2+ in aqueous solution. Ca2+ charges were derived by fitting to the electrostatic potential in the context of a classical or polarizable force field (PFF). We discovered that the atomic radius of Ca2+ in conventional force fields is too large for the QM calculation to capture the variation in the coordination geometry of Ca2+ in its ionic form, leading to unphysical charges. Specifically, we found that the fitted atomic charges of Ca2+ in the context of PFF depend on the coordinating geometry of electronegative atoms from the amino acids in the loop. Although nearby water molecules do not influence the atomic charge of Ca2+, they are crucial for compensating for the coordination of Ca2+ due to the conformational flexibility in the EF-hand loop. Our method advances the development of force fields for metal ions and protein binding sites in dynamic environments.

CTRAMER: An open-source software package for correlating interfacial charge transfer rate constants with donor/acceptor geometries in organic photovoltaic materials.

In this paper, we present CTRAMER (Charge-Transfer RAtes from Molecular dynamics, Electronic structure, and Rate theory)-an open-source software package for calculating interfacial charge-transfer (CT) rate constants in organic photovoltaic (OPV) materials based on ab initio calculations and molecular dynamics simulations. The software is based on identifying representative donor/acceptor geometries within interfacial structures obtained from molecular dynamics simulation of donor/acceptor blends and calculating the corresponding Fermi's golden rule CT rate constants within the framework of the linearized-semiclassical approximation. While the methods used are well established, the integration of these state-of-the-art tools originating from different disciplines to study photoinduced CT processes with explicit treatment of the environment, in our opinion, makes this package unique and innovative. The software also provides tools for investigating other observables of interest. After outlining the features and implementation details, the usage and performance of the software are demonstrated with results from an example OPV system.

Molecular mechanisms of the interhead coordination by interhead tension in cytoplasmic dyneins.

Cytoplasmic dyneins play a major role in retrograde cellular transport by moving vesicles and organelles along microtubule filaments. Dyneins are multidomain motor proteins with two heads that coordinate their motion via their interhead tension. Compared with the leading head, the trailing head has a higher detachment rate from microtubules, facilitating the movement. However, the molecular mechanism of such coordination is unknown. To elucidate this mechanism, we performed molecular dynamics simulations on a cytoplasmic dynein with a structure-based coarse-grained model that probes the effect of the interhead tension on the structure. The tension creates a torque that influences the head rotating about its stalk. The conformation of the stalk switches from the α registry to the ß registry during the rotation, weakening the binding affinity to microtubules. The directions of the tension and the torque of the leading head are opposite to those of the trailing head, breaking the structural symmetry between the heads. The leading head transitions less often to the ß registry than the trailing head. The former thus has a greater binding affinity to the microtubule than the latter. We measured the moment arm of the torque from a dynein structure in the simulations to develop a phenomenological model that captures the influence of the head rotating about its stalk on the differential detachment rates of the two heads. Our study provides a consistent molecular picture for interhead coordination via interhead tension.

Molecular Dynamics Ensemble Refinement of Intrinsically Disordered Peptides According to Deconvoluted Spectra from Circular Dichroism.

We have developed a computational method of atomistically refining the structural ensemble of intrinsically disordered peptides (IDPs) facilitated by experimental measurements using circular dichroism spectroscopy (CD). A major challenge surrounding this approach stems from the deconvolution of experimental CD spectra into secondary structure features of the IDP ensemble. Currently available algorithms for CD deconvolution were designed to analyze the spectra of proteins with stable secondary structures. Herein, our work aims to minimize any bias from the peptide deconvolution analysis by implementing a non-negative linear least-squares fitting algorithm in conjunction with a CD reference data set that contains soluble and denatured proteins (SDP48). The non-negative linear least-squares method yields the best results for deconvolution of proteins with higher disordered content than currently available methods, according to a validation analysis of a set of protein spectra with Protein Data Bank entries. We subsequently used this analysis to deconvolute our experimental CD data to refine our computational model of the peptide secondary structure ensemble produced by all-atom molecular dynamics simulations with implicit solvent. We applied this approach to determine the ensemble structures of a set of short IDPs, that mimic the calmodulin binding domain of calcium/calmodulin-dependent protein kinase II and its 1-amino-acid and 3-amino-acid mutants. Our study offers a, to our knowledge, novel way to solve the ensemble secondary structures of IDPs in solution, which is important to advance the understanding of their roles in regulating signaling pathways through the formation of complexes with multiple partners.

Charge transfer rate constants for the carotenoid-porphyrin-C60 molecular triad dissolved in tetrahydrofuran: The spin-boson model vs the linearized semiclassical approximation.

Charge transfer rate constants were calculated for the carotenoid-porphyrin-C60 (CPC60) molecular triad dissolved in explicit tetrahydrofuran. The calculation was based on mapping the all-atom anharmonic Hamiltonian of this system onto the spin-boson Hamiltonian. The mapping was based on discretizing the spectral density from the time correlation function of the donor-acceptor potential energy gap, as obtained from all-atom molecular dynamics simulations. Different spin-boson Hamiltonians were constructed for each of the possible transitions between the three excited electronic states in two different triad conformations. The rate constants of three possible transitions were calculated via the quantum-mechanically exact Fermi's golden rule (FGR), as well as a progression of more approximate expressions that lead to the classical Marcus expression. The advantage of the spin-boson approach is that once the mapping is established, the quantum-mechanically exact FGR and the hierarchy of approximations are known in closed form. The classical Marcus charge transfer rate constants obtained with the spin-boson Hamiltonians were found to reproduce those obtained from all-atom simulations with the linearized semiclassical approximation, thereby confirming the equivalence of the two approaches for this system. Within the spin-boson Hamiltonian, we also found that the quantum-mechanically exact FGR rate constants were significantly enhanced compared to the classical Marcus theory rate constants for two out of three transitions in one of the two conformations under consideration. The results confirm that mapping to the spin-boson model can yield accurate predictions for charge transfer rate constants in a system as complex as CPC60 dissolved in tetrahydrofuran.

Molecular origin of the weak susceptibility of kinesin velocity to loads and its relation to the collective behavior of kinesins.

Motor proteins are active enzymatic molecules that support important cellular processes by transforming chemical energy into mechanical work. Although the structures and chemomechanical cycles of motor proteins have been extensively investigated, the sensitivity of a motor's velocity in response to a force is not well-understood. For kinesin, velocity is weakly influenced by a small to midrange external force (weak susceptibility) but is steeply reduced by a large force. Here, we utilize a structure-based molecular dynamic simulation to study the molecular origin of the weak susceptibility for a single kinesin. We show that the key step in controlling the velocity of a single kinesin under an external force is the ATP release from the microtubule-bound head. Only under large loading forces can the motor head release ATP at a fast rate, which significantly reduces the velocity of kinesin. It underpins the weak susceptibility that the velocity will not change at small to midrange forces. The molecular origin of this velocity reduction is that the neck linker of a kinesin only detaches from the motor head when pulled by a large force. This prompts the ATP binding site to adopt an open state, favoring ATP release and reducing the velocity. Furthermore, we show that two load-bearing kinesins are incapable of equally sharing the load unless they are very close to each other. As a consequence of the weak susceptibility, the trailing kinesin faces the challenge of catching up to the leading one, which accounts for experimentally observed weak cooperativity of kinesins motors.

Competition of individual domain folding with inter-domain interaction in WW domain engineered repeat proteins.

Engineered repeat proteins have proven to be a fertile ground for studying the competition between folding, misfolding and transient aggregation of tethered protein domains. We examine the interplay between folding and inter-domain interactions of engineered FiP35 WW domain repeat proteins with n = 1 through 5 repeats. We characterize protein expression, thermal and guanidium melts, as well as laser T-jump kinetics. All experimental data is fitted by a global fitting model with two states per domain (U, N), plus a third state M to account for non-native states due to domain interactions present in all but the monomer. A detailed structural model is provided by coarse-grained simulated annealing using the AWSEM Hamiltonian. Tethered FiP35 WW domains with n = 2 and 3 domains are just slightly less stable than the monomer. The n = 4 oligomer is yet less stable, its expression yield is much lower than the monomer's, and depends on the purification tag used. The n = 5 plasmid did not express at all, indicating the sudden onset of aggregation past n = 4. Thus, tethered FiP35 has a critical nucleus size for inter-domain aggregation of n ≈ 4. According to our simulations, misfolded structures become increasingly prevalent as one proceeds from monomer to pentamer, with extended inter-domain beta sheets appearing first, then multi-sheet 'intramolecular amyloid' structures, and finally novel motifs containing alpha helices. We discuss the implications of our results for oligomeric aggregate formation and structure, transient aggregation of proteins whilst folding, as well as for protein evolution that starts with repeat proteins.

Towards developing principles of protein folding and dynamics in the cell.

Proteins must fold and function in the immensely complex environment of a cell, i.e. the cytoplasm-this is far from the ideal test-tube setting of a dilute solution. Here we review the advances in protein folding and dynamics inside the cell. In developing principles of protein behavior in vivo, we also begin to understand the organization and dynamics of the cytoplasm, unifying the single protein scale with the many-protein architectures at the subcellular scale. Our group has significantly contributed to this frontier by characterizing the effect of macromolecular crowding on the distribution of protein conformations. Additionally, we provide a personal perspective on becoming a theoretical biological physicist in the era of interdisciplinary research that has been greatly influenced by Dr Kamal Shukla. We also share our view on the future direction of protein folding inside a cell.

Opposing Intermolecular Tuning of Ca2+ Affinity for Calmodulin by Neurogranin and CaMKII Peptides.

We investigated the impact of bound calmodulin (CaM)-target compound structure on the affinity of calcium (Ca2+) by integrating coarse-grained models and all-atomistic simulations with nonequilibrium physics. We focused on binding between CaM and two specific targets, Ca2+/CaM-dependent protein kinase II (CaMKII) and neurogranin (Ng), as they both regulate CaM-dependent Ca2+ signaling pathways in neurons. It was shown experimentally that Ca2+/CaM (holoCaM) binds to the CaMKII peptide with overwhelmingly higher affinity than Ca2+-free CaM (apoCaM); the binding of CaMKII peptide to CaM in return increases the Ca2+ affinity for CaM. However, this reciprocal relation was not observed in the Ng peptide (Ng13-49), which binds to apoCaM or holoCaM with binding affinities of the same order of magnitude. Unlike the holoCaM-CaMKII peptide, whose structure can be determined by crystallography, the structural description of the apoCaM-Ng13-49 is unknown due to low binding affinity, therefore we computationally generated an ensemble of apoCaM-Ng13-49 structures by matching the changes in the chemical shifts of CaM upon Ng13-49 binding from nuclear magnetic resonance experiments. Next, we computed the changes in Ca2+ affinity for CaM with and without binding targets in atomistic models using Jarzynski's equality. We discovered the molecular underpinnings of lowered affinity of Ca2+ for CaM in the presence of Ng13-49 by showing that the N-terminal acidic region of Ng peptide pries open the ß-sheet structure between the Ca2+ binding loops particularly at C-domain of CaM, enabling Ca2+ release. In contrast, CaMKII peptide increases Ca2+ affinity for the C-domain of CaM by stabilizing the two Ca2+ binding loops. We speculate that the distinctive structural difference in the bound complexes of apoCaM-Ng13-49 and holoCaM-CaMKII delineates the importance of CaM's progressive mechanism of target binding on its Ca2+ binding affinities.

Induced polarization restricts the conformational distribution of a light-harvesting molecular triad in the ground state.

The light-harvesting molecular triad consisting of carotenoid polyene (C), diaryl-porphyrin (P) and pyrrole-fullerene (C60) is a donor-acceptor molecule capable of absorbing incident light in the visible range. Its ability to convert solar energy to electrical excitation and charge separation energy suggests a great potential in real-world applications. The ensemble of its conformations under ambient conditions varies widely according to its electronic state. In previous work, we applied a non-polarizable model to study the conformational distribution of the molecular triad in the ground and charge separated states. However, due to the lack of polarization, which imparts subtle changes in the charge distribution on atoms, molecular simulations fail to produce accurate average dipole moments. We developed the first polarizable model for a molecular triad to investigate the structural and dynamic properties of a molecular triad in the ground state in an explicit organic solvent, tetrahydrofuran (THF). We performed first-principles electronic structure calculations of the individual components in the triad as well as THF and then fit the partial atomic charges to the electrostatic potential using the i-RESP methodology. We validated these force field parameters by comparing the thermodynamic and dynamic properties obtained from molecular dynamics simulations with those from experiments. We enhanced the sampling of the triad conformations with replica exchange molecular dynamics simulations. We characterized the effects of induced polarization on the structural stability of the triad by analyzing the free energy landscapes constructed with polarizable force fields. Furthermore, by using principal component analysis, we found that the molecular triad conformations adopted a small range of torsional angles with induced polarization. The triad conformation solvated in polar solvent with a polarizable force field qualitatively agrees with that obtained from nuclear magnetic resonance spectroscopy.

Protein recognition and selection through conformational and mutually induced fit.

Protein-protein interactions drive most every biological process, but in many instances the domains mediating recognition are disordered. How specificity in binding is attained in the absence of defined structure contrasts with well-established experimental and theoretical work describing ligand binding to protein. The signaling protein calmodulin presents a unique opportunity to investigate mechanisms for target recognition given that it interacts with several hundred different targets. By advancing coarse-grained computer simulations and experimental techniques, mechanistic insights were gained in defining the pathways leading to recognition and in how target selectivity can be achieved at the molecular level. A model requiring mutually induced conformational changes in both calmodulin and target proteins was necessary and broadly informs how proteins can achieve both high affinity and high specificity.

Relative Cosolute Size Influences the Kinetics of Protein-Protein Interactions.

Protein signaling occurs in crowded intracellular environments, and while high concentrations of macromolecules are postulated to modulate protein-protein interactions, analysis of their impact at each step of the reaction pathway has not been systematically addressed. Potential cosolute-induced alterations in target association are particularly important for a signaling molecule like calmodulin (CaM), where competition among >300 targets governs which pathways are selectively activated. To explore how high concentrations of cosolutes influence CaM-target affinity and kinetics, we methodically investigated each step of the CaM-target binding mechanism under crowded or osmolyte-rich environments mimicked by ficoll-70, dextran-10, and sucrose. All cosolutes stabilized compact conformers of CaM and modulated association kinetics by affecting diffusion and rates of conformational change; however, the results showed that differently sized molecules had variable effects to enhance or impede unique steps of the association pathway. On- and off-rates were modulated by all cosolutes in a compensatory fashion, producing little change in steady-state affinity. From this work insights were gained on how high concentrations of inert crowding agents and osmolytes fit into a kinetic framework to describe protein-protein interactions relevant for cellular signaling.

Conformational frustration in calmodulin-target recognition.

Calmodulin (CaM) is a primary calcium (Ca(2+) )-signaling protein that specifically recognizes and activates highly diverse target proteins. We explored the molecular basis of target recognition of CaM with peptides representing the CaM-binding domains from two Ca(2+) -CaM-dependent kinases, CaMKI and CaMKII, by employing experimentally constrained molecular simulations. Detailed binding route analysis revealed that the two CaM target peptides, although similar in length and net charge, follow distinct routes that lead to a higher binding frustration in the CaM-CaMKII complex than in the CaM-CaMKI complex. We discovered that the molecular origin of the binding frustration is caused by intermolecular contacts formed with the C-domain of CaM that need to be broken before the formation of intermolecular contacts with the N-domain of CaM. We argue that the binding frustration is important for determining the kinetics of the recognition process of proteins involving large structural fluctuations.

Deciphering the Cofilin Oligomers via Intermolecular Disulfide Bond Formation: A Coarse-Grained Molecular Dynamics Approach to Understanding Cofilin's Regulation on Actin Filaments.

Cofilin, a key actin-binding protein, orchestrates the dynamics of the actomyosin network through its actin-severing activity and by promoting the recycling of actin monomers. Recent experiments suggest that cofilin forms functionally distinct oligomers via thiol post-translational modifications (PTMs) that promote actin nucleation and assembly. Despite these advances, the structural conformations of cofilin oligomers that modulate actin activity remain elusive because there are combinatorial ways to oxidize thiols in cysteines to form disulfide bonds rapidly. This study employs molecular dynamics simulations to investigate human cofilin 1 as a case study for exploring cofilin dimers via disulfide bond formation. Utilizing a biasing scheme in simulations, we focus on analyzing dimer conformations conducive to disulfide bond formation. Additionally, we explore potential PTMs arising from the examined conformational ensemble. Using the free energy profiling, our simulations unveil a range of probable cofilin dimer structures not represented in current Protein Data Bank entries. These candidate dimers are characterized by their distinct population distributions and relative free energies. Of particular note is a dimer featuring an interface between cysteines 139 and 147 residues, which demonstrates stable free energy characteristics and intriguingly symmetrical geometry. In contrast, the experimentally proposed dimer structure exhibits a less stable free energy profile. We also evaluate frustration quantification based on the energy landscape theory in the protein-protein interactions at the dimer interfaces. Notably, the 39-39 dimer configuration emerges as a promising candidate for forming cofilin tetramers, as substantiated by frustration analysis. Additionally, docking simulations with actin filaments further evaluate the stability of these cofilin dimer-actin complexes. Our findings thus offer a computational framework for understanding the role of thiol PTM of cofilin proteins in regulating oligomerization, and the subsequent cofilin-mediated actin dynamics in the actomyosin network.

Structure, function, and folding of phosphoglycerate kinase are strongly perturbed by macromolecular crowding.

We combine experiment and computer simulation to show how macromolecular crowding dramatically affects the structure, function, and folding landscape of phosphoglycerate kinase (PGK). Fluorescence labeling shows that compact states of yeast PGK are populated as the amount of crowding agents (Ficoll 70) increases. Coarse-grained molecular simulations reveal three compact ensembles: C (crystal structure), CC (collapsed crystal), and Sph (spherical compact). With an adjustment for viscosity, crowded wild-type PGK and fluorescent PGK are about 15 times or more active in 200 mg/ml Ficoll than in aqueous solution. Our results suggest a previously undescribed solution to the classic problem of how the ADP and diphosphoglycerate binding sites of PGK come together to make ATP: Rather than undergoing a hinge motion, the ADP and substrate sites are already located in proximity under crowded conditions that mimic the in vivo conditions under which the enzyme actually operates. We also examine T-jump unfolding of PGK as a function of crowding experimentally. We uncover a nonmonotonic folding relaxation time vs. Ficoll concentration. Theory and modeling explain why an optimum concentration exists for fastest folding. Below the optimum, folding slows down because the unfolded state is stabilized relative to the transition state. Above the optimum, folding slows down because of increased viscosity.

Experiment and Simulation Reveal Residue Details for How Target Binding Tunes Calmodulin's Calcium-Binding Properties.

We aim to elucidate the molecular mechanism of the reciprocal relation of calmodulin's (CaM) target binding and its affinity for calcium ions (Ca2+), which is central to decoding CaM-dependent Ca2+ signaling in a cell. We employed stopped-flow experiments and coarse-grained molecular simulations that learn the coordination chemistry of Ca2+ in CaM from first-principle calculations. The associative memories as part of the coarse-grained force fields built on known protein structures further influence CaM's selection of its polymorphic target peptides in the simulations. We modeled the peptides from the Ca2+/CaM-binding domain of Ca2+/CaM-dependent kinase II (CaMKII), CaMKIIp (293-310) and selected distinctive mutations at the N-terminus. Our stopped-flow experiments have shown that the CaM's affinity for Ca2+ in the bound complex of Ca2+/CaM/CaMKIIp decreased significantly when Ca2+/CaM bound to the mutant peptide (296-AAA-298) compared to that bound to the wild-type peptide (296-RRK-298). The coarse-grained molecular simulations revealed that the 296-AAA-298 mutant peptide destabilized the structures of Ca2+-binding loops at the C-domain of CaM (c-CaM) due to both loss of electrostatic interactions and differences in polymorphic structures. We have leveraged a powerful coarse-grained approach to advance a residue-level understanding of the reciprocal relation in CaM, that could not be possibly achieved by other computational approaches.