Robust biased Brownian dynamics for rate constant calculation

Dissecting the general mechanisms of protein cage self-assembly by coarse-grained simulations

The development of artificial protein cages has recently gained massive attention due to their promising application prospect as novel delivery vehicles for therapeutics. These nanoparticles are formed through a process called self-assembly, in which individual subunits spontaneously arrange into highly ordered patterns via non-covalent but specific interactions. Therefore, the first step toward the design of novel engineered protein cages is to understand the general mechanisms of their self-assembling dynamics. Here we have developed a new computational method to tackle this problem. Our method is based on a coarse-grained model and a diffusion-reaction simulation algorithm. Using a tetrahedral cage as test model, we showed that self-assembly of protein cage requires of a seeding process in which specific configurations of kinetic intermediate states are identified. We further found that there is a critical concentration to trigger self-assembly of protein cages. This critical concentration allows that cages can only be successfully assembled under a persistently high concentration. Additionally, phase diagram of self-assembly has been constructed by systematically testing the model across a wide range of binding parameters. Finally, our simulations demonstrated the importance of protein's structural flexibility in regulating the dynamics of cage assembly. In summary, this study throws lights on the general principles underlying self-assembly of large cage-like protein complexes and thus provides insights to design new nanomaterials.

Classification of protein-protein association rates based on biophysical informatics

BACKGROUND

Proteins form various complexes to carry out their versatile functions in cells. The dynamic properties of protein complex formation are mainly characterized by the association rates which measures how fast these complexes can be formed. It was experimentally observed that the association rates span an extremely wide range with over ten orders of magnitudes. Identification of association rates within this spectrum for specific protein complexes is therefore essential for us to understand their functional roles.

RESULTS

To tackle this problem, we integrate physics-based coarse-grained simulations into a neural-network-based classification model to estimate the range of association rates for protein complexes in a large-scale benchmark set. The cross-validation results show that, when an optimal threshold was selected, we can reach the best performance with specificity, precision, sensitivity and overall accuracy all higher than 70%. The quality of our cross-validation data has also been testified by further statistical analysis. Additionally, given an independent testing set, we can successfully predict the group of association rates for eight protein complexes out of ten. Finally, the analysis of failed cases suggests the future implementation of conformational dynamics into simulation can further improve model.

CONCLUSIONS

In summary, this study demonstrated that a new modeling framework that combines biophysical simulations with bioinformatics approaches is able to identify protein-protein interactions with low association rates from those with higher association rates. This method thereby can serve as a useful addition to a collection of existing experimental approaches that measure biomolecular recognition.

Understanding the Impacts of Conformational Dynamics on the Regulation of Protein-Protein Association by a Multiscale Simulation Method

Complexes formed among diverse proteins carry out versatile functions in nearly all physiological processes. Association rates which measure how fast proteins form various complexes are of fundamental importance to characterize their functions. The association rates are not only determined by the energetic features at binding interfaces of a protein complex but also influenced by the intrinsic conformational dynamics of each protein in the complex. Unfortunately, how this conformational effect regulates protein association has never been calibrated on a systematic level. To tackle this problem, we developed a multiscale strategy to incorporate the information on protein conformational variations from Langevin dynamic simulations into a kinetic Monte Carlo algorithm of protein-protein association. By systematically testing this approach against a large-scale benchmark set, we found the association of a protein complex with a relatively rigid structure tends to be reduced by its conformational fluctuations. With specific examples, we further show that higher degrees of structural flexibility in various protein complexes can facilitate the searching and formation of intermolecular interactions and thereby accelerate their associations. In general, the integration of conformational dynamics can improve the correlation between experimentally measured association rates and computationally derived association probabilities. Finally, we analyzed the statistical distributions of different secondary structural types on protein-protein binding interfaces and their preference to the change of association rates. Our study, to the best of our knowledge, is the first computational method that systematically estimates the impacts of protein conformational dynamics on protein-protein association. It throws lights on the molecular mechanisms of how protein-protein recognition is kinetically modulated.

Using Coarse-Grained Simulations to Characterize the Mechanisms of Protein-Protein Association

The formation of functionally versatile protein complexes underlies almost every biological process. The estimation of how fast these complexes can be formed has broad implications for unravelling the mechanism of biomolecular recognition. This kinetic property is traditionally quantified by association rates, which can be measured through various experimental techniques. To complement these time-consuming and labor-intensive approaches, we developed a coarse-grained simulation approach to study the physical processes of protein–protein association. We systematically calibrated our simulation method against a large-scale benchmark set. By combining a physics-based force field with a statistically-derived potential in the simulation, we found that the association rates of more than 80% of protein complexes can be correctly predicted within one order of magnitude relative to their experimental measurements. We further showed that a mixture of force fields derived from complementary sources was able to describe the process of protein–protein association with mechanistic details. For instance, we show that association of a protein complex contains multiple steps in which proteins continuously search their local binding orientations and form non-native-like intermediates through repeated dissociation and re-association. Moreover, with an ensemble of loosely bound encounter complexes observed around their native conformation, we suggest that the transition states of protein–protein association could be highly diverse on the structural level. Our study also supports the idea in which the association of a protein complex is driven by a “funnel-like” energy landscape. In summary, these results shed light on our understanding of how protein–protein recognition is kinetically modulated, and our coarse-grained simulation approach can serve as a useful addition to the existing experimental approaches that measure protein–protein association rates.

A Multiscale Computational Model for Simulating the Kinetics of Protein Complex Assembly

Proteins fulfill versatile biological functions by interacting with each other and forming high-order complexes. Although the order in which protein subunits assemble is important for the biological function of their final complex, this kinetic information has received comparatively little attention in recent years. Here we describe a multiscale framework that can be used to simulate the kinetics of protein complex assembly. There are two levels of models in the framework. The structural details of a protein complex are reflected by the residue-based model, while a lower-resolution model uses a rigid-body (RB) representation to simulate the process of complex assembly. These two levels of models are integrated together, so that we are able to provide the kinetic information about complex assembly with both structural details and computational efficiency.

Predicting Protein-protein Association Rates using Coarse-grained Simulation and Machine Learning

Protein–protein interactions dominate all major biological processes in living cells. We have developed a new Monte Carlo-based simulation algorithm to study the kinetic process of protein association. We tested our method on a previously used large benchmark set of 49 protein complexes. The predicted rate was overestimated in the benchmark test compared to the experimental results for a group of protein complexes. We hypothesized that this resulted from molecular flexibility at the interface regions of the interacting proteins. After applying a machine learning algorithm with input variables that accounted for both the conformational flexibility and the energetic factor of binding, we successfully identified most of the protein complexes with overestimated association rates and improved our final prediction by using a cross-validation test. This method was then applied to a new independent test set and resulted in a similar prediction accuracy to that obtained using the training set. It has been thought that diffusion-limited protein association is dominated by long-range interactions. Our results provide strong evidence that the conformational flexibility also plays an important role in regulating protein association. Our studies provide new insights into the mechanism of protein association and offer a computationally efficient tool for predicting its rate.

Dynamics theory for molecular liquids based on an interaction site model

Dynamics theories for molecular liquids based on an interaction site model have been developed over the past few decades and proved to be powerful tools to investigate various dynamical phenomena.

Analysis of Ligand-Receptor Association and Intermediate Transfer Rates in Multienzyme Nanostructures with All-Atom Brownian Dynamics Simulations

We present the second-generation GeomBD Brownian dynamics software for determining interenzyme intermediate transfer rates and substrate association rates in biomolecular complexes. Substrate and intermediate association rates for a series of enzymes or biomolecules can be compared between the freely diffusing disorganized configuration and various colocalized or complexed arrangements for kinetic investigation of enhanced intermediate transfer. In addition, enzyme engineering techniques, such as synthetic protein conjugation, can be computationally modeled and analyzed to better understand changes in substrate association relative to native enzymes. Tools are provided to determine nonspecific ligand-receptor association residence times, and to visualize common sites of nonspecific association of substrates on receptor surfaces. To demonstrate features of the software, interenzyme intermediate substrate transfer rate constants are calculated and compared for all-atom models of DNA origami scaffold-bound bienzyme systems of glucose oxidase and horseradish peroxidase. Also, a DNA conjugated horseradish peroxidase enzyme was analyzed for its propensity to increase substrate association rates and substrate local residence times relative to the unmodified enzyme. We also demonstrate the rapid determination and visualization of common sites of nonspecific ligand-receptor association by using HIV-1 protease and an inhibitor, XK263. GeomBD2 accelerates simulations by precomputing van der Waals potential energy grids and electrostatic potential grid maps, and has a flexible and extensible support for all-atom and coarse-grained force fields. Simulation software is written in C++ and utilizes modern parallelization techniques for potential grid preparation and Brownian dynamics simulation processes. Analysis scripts, written in the Python scripting language, are provided for quantitative simulation analysis. GeomBD2 is applicable to the fields of biophysics, bioengineering, and enzymology in both predictive and explanatory roles.

Effect of fluidic transport on the reaction kinetics in lectin microarrays

Lectins are the proteins which can distinguish glycosylation patterns. They are frequently used as biomarkers for progressions of several diseases including cancer. As the lectin microarray based prognosis devices miniaturize the process of glycoprofiling, it is anticipated that their performance can be augmented by integration with microfluidic framework. This is analogous to microfluidics based DNA arrays. However, unlike small oligonucleotide microarrays, it remains uncertain whether the binding reaction-kinetic parameters can be considered invariant of imposed hydrodynamics, for relatively larger and structure sensitive molecules such as lectins. Here we show, using two standard lectins namely Concanavalin A and Abrus Agglutinin, that the steady state binding efficiency unexpectedly declines beyond a critical shear rate magnitude. This observation can be explained only if the associated reaction constants are presumed to be functions of hydrodynamic parameters. We methodically deduce the shear rate dependence of association and dissociation constants from the comparison of experimental and model-simulation trends. The aforementioned phenomena are perceived to be the consequences of strong hydrodynamic perturbations, culminating into molecular structural distortion. The exploration, therefore, reveals a unique coupling between reaction kinetics and hydrodynamics for biomacromolecules and provides a generic scheme towards futuristic microfluidics-coupled biomedical assays.

Predicting kinetic constants of protein-protein interactions based on structural properties

Elucidating kinetic processes of protein-protein interactions (PPI) helps to understand how basic building blocks affect overall behavior of living systems. In this study, we used structure-based properties to build predictive models for kinetic constants of PPI. A highly diverse PPI dataset, protein-protein kinetic interaction data and structures (PPKIDS), was built. PPKIDS contains 62 PPI with complex structures and kinetic constants measured experimentally. The influence of structural properties on kinetics of PPI was studied using 35 structure-based features, describing different aspects of complex structures. Linear models for the prediction of kinetic constants were built by fitting with selected subsets of structure-based features. The models gave correlation coefficients of 0.801, 0.732, and 0.770 for k(off), k(on), and K(d), respectively, in leave-one-out cross validations. The predictive models reported here use only protein complex structures as input and can be generally applied in PPI studies as well as systems biology modeling. Our study confirmed that different properties play different roles in the kinetic process of PPI. For example, k(on) was affected by overall structural features of complexes, such as the composition of secondary structures, the change of translational and rotational entropy, and the electrostatic interaction; while k(off) was determined by interfacial properties, such as number of contacted atom pairs per 100 Ų. This information provides useful hints for PPI design.

Electrostatic rate enhancement and transient complex of protein-protein association

The association of two proteins is bounded by the rate at which they, via diffusion, find each other while in appropriate relative orientations. Orientational constraints restrict this rate to approximately 10(5)-10(6) M(-1) s(-1). Proteins with higher association rates generally have complementary electrostatic surfaces; proteins with lower association rates generally are slowed down by conformational changes upon complex formation. Previous studies (Zhou, Biophys J 1997;73:2441-2445) have shown that electrostatic enhancement of the diffusion-limited association rate can be accurately modeled by $k_{\bf D}$ = $k_{D}0\ {exp} ( - \langle U_{el} \rangle;{\star}/k_{B} T),$ where k(D) and k(D0) are the rates in the presence and absence of electrostatic interactions, respectively, U(el) is the average electrostatic interaction energy in a "transient-complex" ensemble, and k(B)T is the thermal energy. The transient-complex ensemble separates the bound state from the unbound state. Predictions of the transient-complex theory on four protein complexes were found to agree well with the experiment when the electrostatic interaction energy was calculated with the linearized Poisson-Boltzmann (PB) equation (Alsallaq and Zhou, Structure 2007;15:215-224). Here we show that the agreement is further improved when the nonlinear PB equation is used. These predictions are obtained with the dielectric boundary defined as the protein van der Waals surface. When the dielectric boundary is instead specified as the molecular surface, electrostatic interactions in the transient complex become repulsive and are thus predicted to retard association. Together these results demonstrate that the transient-complex theory is predictive of electrostatic rate enhancement and can help parameterize PB calculations.

Prediction of salt and mutational effects on the association rate of U1A protein and U1 small nuclear RNA stem/loop II

We have developed a computational approach for predicting protein-protein association rates (Alsallaq and Zhou, Structure 2007, 15, 215). Here we expand the range of applicability of this approach to protein-RNA binding and report the first results for protein-RNA binding rates predicted from atomistic modeling. The system studied is the U1A protein and stem/loop II of the U1 small nuclear RNA. Experimentally it was observed that the binding rate is significantly reduced by increasing salt concentration while the dissociation changes little with salt concentration, and charges distant from the binding site make marginal contribution to the binding rate. These observations are rationalized. Moreover, predicted effects of salt and charge mutations are found to be in quantitative agreement with experimental results.

Prediction of protein-protein association rates from a transition-state theory

We recently developed a theory for the rates of protein-protein association. The theory is based on the concept of a transition state, which separates the bound state, with numerous short-range interactions but restricted translational and rotational freedom, and the unbound state, with, at most, a small number of interactions but expanded configurational freedom. When not accompanied by large-scale conformational changes, protein-protein association becomes diffusion limited. The association rate is then predicted as k(a)=k(a)(0)exp(-DeltaG(el)(double dagger)/k(B)T), where DeltaG(el)(double dagger) is the electrostatic interaction free energy in the transition state, k(a)(0) is the rate in the absence of electrostatic interactions, and k(B)T is thermal energy. Here, this transition-state theory is used to predict the association rates of four protein complexes. The predictions for the wild-type complexes and 23 mutants are found to agree closely with experimental data over wide ranges of ionic strength.

Finite concentration effects on diffusion-controlled reactions

The algorithm by Northrup, Allison, and McCammon [J. Chem. Phys. 80, 1517 (1984)] has been used for two decades for calculating the diffusion-influenced rate-constants of enzymatic reactions. Although many interesting results have been obtained, the algorithm is based on the assumption that substrate-substrate interactions can be neglected. This approximation may not be valid when the concentration of the ligand is high. In this work, we constructed a simulation model that can take substrate-substrate interactions into account. We first validated the model by carrying out simulations in ways that could be compared to analytical theories. We then carried out simulations to examine the possible effects of substrate-substrate interactions on diffusion-controlled reaction rates. For a substrate concentration of 0.1 mM, we found that the diffusion-controlled reaction rates were not sensitive to whether substrate-substrate interactions were included. On the other hand, we observed significant influence of substrate-substrate interactions on calculated reaction rates at a substrate concentration of 0.1M. Therefore, a simulation model that takes substrate-substrate interactions into account is essential for reliably predicting diffusion-controlled reaction rates at high substrate concentrations, and one such simulation model is presented here.

Spatial modeling of dimerization reaction dynamics in the plasma membrane: Monte Carlo vs. continuum differential equations

Bimolecular reactions in the plasma membrane, such as receptor dimerization, are a key signaling step for many signaling systems. For receptors to dimerize, they must first diffuse until a collision happens, upon which a dimerization reaction may occur. Therefore, study of the dynamics of cell signaling on the membrane may require the use of a spatial modeling framework. Despite the availability of spatial simulation methods, e.g., stochastic spatial Monte Carlo (MC) simulation and partial differential equation (PDE) based approaches, many biological models invoke well-mixed assumptions without completely evaluating the importance of spatial organization. Whether one is to utilize a spatial or non-spatial simulation framework is therefore an important decision. In order to evaluate the importance of spatial effects a priori, i.e., without performing simulations, we have assessed the applicability of a dimensionless number, known as second Damköhler number (Da), defined here as the ratio of time scales of collision and reaction, for 2-dimensional bimolecular reactions. Our study shows that dimerization reactions in the plasma membrane with Da approximately >0.1 (tested in the receptor density range of 10(2)-10(5)/microm(2)) require spatial modeling. We also evaluated the effective reaction rate constants of MC and simple deterministic PDEs. Our simulations show that the effective reaction rate constant decreases with time due to time dependent changes in the spatial distribution of receptors. As a result, the effective reaction rate constant of simple PDEs can differ from that of MC by up to two orders of magnitude. Furthermore, we show that the fluctuations in the number of copies of signaling proteins (noise) may also depend on the diffusion properties of the system. Finally, we used the spatial MC model to explore the effect of plasma membrane heterogeneities, such as receptor localization and reduced diffusivity, on the dimerization rate. Interestingly, our simulations show that localization of epidermal growth factor receptor (EGFR) can cause the diffusion limited dimerization rate to be up to two orders of magnitude higher at higher average receptor densities reported for cancer cells, as compared to a normal cell.

Finite element solution of the steady-state Smoluchowski equation for rate constant calculations

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.