Rate theories for biologists

Signal execution modes emerge in biochemical reaction networks calibrated to experimental data

Mathematical models of biomolecular networks are commonly used to study cellular processes; however, their usefulness to explain and predict dynamic behaviors is often questioned due to the unclear relationship between parameter uncertainty and network dynamics. In this work, we introduce PyDyNo (Python dynamic analysis of biochemical networks), a non-equilibrium reaction-flux based analysis to identify dominant reaction paths within a biochemical reaction network calibrated to experimental data. We first show, in a simplified apoptosis execution model, that despite the thousands of parameter vectors with equally good fits to experimental data, our framework identifies the dynamic differences between these parameter sets and outputs three dominant execution modes, which exhibit varying sensitivity to perturbations. We then apply our methodology to JAK2/STAT5 network in colony-forming unit-erythroid (CFU-E) cells and provide previously unrecognized mechanistic explanation for the survival responses of CFU-E cell population that would have been impossible to deduce with traditional protein-concentration based analyses.

Synergism in the Molecular Crowding of Ligand-Induced Riboswitch Folding: Kinetic/Thermodynamic Insights from Single-Molecule Spectroscopy

Conformational dynamics in riboswitches involves ligand binding and folding of RNA, each of which can be influenced by excluded volume effects under "crowded" in vivo cellular conditions and thus incompletely characterized by in vitro studies under dilute buffer conditions. In this work, temperature-dependent single-molecule fluorescence resonance energy transfer (FRET) spectroscopy is used to characterize the thermodynamics of (i) cognate ligand and (ii) molecular crowders (PEG, polyethylene glycol) on folding of the B. subtilis LysC lysine riboswitch. With the help of detailed kinetic analysis, we isolate and study the effects of PEG on lysine binding and riboswitch folding steps individually, from which we find that PEG crowding facilitates riboswitch folding primarily via a surprising increase in affinity for the cognate ligand. This is furthermore confirmed by temperature-dependent studies, which reveal that PEG crowding is not purely entropic and instead significantly impacts both enthalpic and entropic contributions to the free energy landscape for folding. The results indicate that PEG molecular crowding/stabilization of the lysine riboswitch is more mechanistically complex and requires extension beyond the conventional picture of purely repulsive solvent-solute steric interactions arising from excluded volume and entropy. Instead, the current experimental FRET data support an alternative multistep mechanism, whereby PEG first entropically crowds the unfolded riboswitch into a "pre-folded" conformation, which in turn greatly increases the ligand binding affinity and thereby enhances the overall equilibrium for riboswitch folding.

A general theory for temperature dependence in biology

At present, there is no simple, first principles-based, and general model for quantitatively describing the full range of observed biological temperature responses. Here we derive a general theory for temperature dependence in biology based on Eyring-Evans-Polanyi's theory for chemical reaction rates. Assuming only that the conformational entropy of molecules changes with temperature, we derive a theory for the temperature dependence of enzyme reaction rates which takes the form of an exponential function modified by a power law and that describes the characteristic asymmetric curved temperature response. Based on a few additional principles, our model can be used to predict the temperature response above the enzyme level, thus spanning quantum to classical scales. Our theory provides an analytical description for the shape of temperature response curves and demonstrates its generality by showing the convergence of all temperature dependence responses onto universal relationships-a universal data collapse-under appropriate normalization and by identifying a general optimal temperature, around 25 ^∘C, characterizing all temperature response curves. The model provides a good fit to empirical data for a wide variety of biological rates, times, and steady-state quantities, from molecular to ecological scales and across multiple taxonomic groups (from viruses to mammals). This theory provides a simple framework to understand and predict the impact of temperature on biological quantities based on the first principles of thermodynamics, bridging quantum to classical scales.

Interplay of Affinity and Surface Tethering in Protein Recognition

Surface-tethered ligand-receptor complexes are key components in biological signaling and adhesion. They also find increasing utility in single-molecule assays and biotechnological applications. Here, we study the real-time binding kinetics between various surface-immobilized peptide ligands and their unrestrained receptors. A long peptide tether increases the association of ligand-receptor complexes, experimentally proving the fly casting mechanism where the disorder accelerates protein recognition. On the other hand, a short peptide tether enhances the complex dissociation. Notably, the rate constants measured for the same receptor, but under different spatial constraints, are strongly correlated to one another. Furthermore, this correlation can be used to predict how surface tethering on a ligand-receptor complex alters its binding kinetics. Our results have immediate implications in the broad areas of biomolecular recognition, intrinsically disordered proteins, and biosensor technology.

Remarked suppression of Aβ42 protomer-protomer dissociation reaction elucidated by molecular dynamics simulation

Multimeric protein complexes are molecular apparatuses to regulate biological systems and often determine their fate. Among proteins forming such molecular assemblies, amyloid proteins have drawn attention over a half-century since amyloid fibril formation of these proteins is supposed to be a common pathogenic cause for neurodegenerative diseases. This process is triggered by the accumulation of fibril-like aggregates, while the microscopic mechanisms are mostly elusive due to technical limitation of experimental methodologies in individually observing each of diverse aggregate species in the aqueous solution. We then addressed this problem by employing atomistic molecular dynamics simulations for the paradigmatic amyloid protein, amyloid-β (Aβ₄₂ ). Seven different dimeric forms of oligomeric Aβ₄₂ fibril-like aggregate in aqueous solution, ranging from tetramer to decamer, were considered. We found additive effects of the size of these fibril-like aggregates on their thermodynamic stability and have clarified kinetic suppression of protomer-protomer dissociation reactions at and beyond the point of pentamer dimer formation. This observation was obtained from the specific combination of the Aβ₄₂ protomer structure and the physicochemical condition that we here examined, while it is worthwhile to recall that several amyloid fibrils take dimeric forms of their protomers. We could thus conclude that the stable formation of fibril-like protomer dimer should be involved in a turning point where rapid growth of amyloid fibrils is triggered.

Effects of Molecular Crowders on Single-Molecule Nucleic Acid Folding: Temperature-Dependent Studies Reveal True Crowding vs Enthalpic Interactions

Biomolecular folding in cells can be strongly influenced by spatial overlap/excluded volume interactions (i.e., "crowding") with intracellular solutes. As a result, traditional in vitro experiments with dilute buffers may not accurately recapitulate biomolecule folding behavior in vivo. In order to account for such ubiquitous excluded volume effects, biologically inert polyethylene glycol (PEG) and polysaccharides (dextran and Ficoll) are often used as in vitro crowding agents to mimic in vivo crowding conditions, with a common observation that high concentrations of these polymers stabilize the more compact biomolecule conformation. However, such an analysis can be distorted by differences in polymer interactions with the folded vs unfolded conformers, requiring temperature-dependent analysis of the thermodynamics to reliably assess competing enthalpic vs entropic contributions and thus the explicit role of excluded volume. In this work, temperature-controlled single-molecule fluorescence resonance energy transfer (smFRET) is used to characterize the thermodynamic interaction between nucleic acids and common polymer crowders PEG, dextran, and Ficoll. The results reveal that PEG promotes secondary and tertiary nucleic acid folding by simultaneously increasing the folding rate while decreasing the unfolding rate, with temperature-dependent studies confirming that the source of PEG stabilization is predominantly entropic and consistent with a true excluded volume crowding mechanism. By way of contrast, neither dextran nor Ficoll induces any significant concentration-dependent change in nucleic acid folding stability at room temperature, but instead, stabilization effects gradually appear with a temperature increase. Such a thermal response indicates that both folding enthalpies and entropies are impacted by dextran and Ficoll. A detailed thermodynamic analysis of the kinetics suggests that, instead of true entropic molecular crowding, dextran and Ficoll associate preferentially with the unfolded vs folded nucleic acid conformer as a result of larger solvent accessible surface area, thereby skewing the free energy landscapes through both significant entropic/enthalpic contributions that compete and fortuitously cancel near room temperature.

Surface-facilitated trapping by active sites: From catalysts to viruses

Trapping by active sites on surfaces plays important roles in various chemical and biological processes, including catalysis, enzymatic reactions, and viral entry into host cells. However, the mechanisms of these processes remain not well understood, mostly because the existing theoretical descriptions are not fully accounting for the role of the surfaces. Here, we present a theoretical investigation on the dynamics of surface-assisted trapping by specific active sites. In our model, a diffusing particle can occasionally reversibly bind to the surface and diffuse on it before reaching the final target site. An approximate theoretical framework is developed, and its predictions are tested by Brownian dynamics computer simulations. It is found that the surface diffusion can be crucial in mediating trapping by active sites. Our theoretical predictions work reasonably well as long as the area of the active site is much smaller than the overall surface area. Potential applications of our approach are discussed.

Influence of molecular rebinding on the reaction rate of complex formation

We simulated Brownian diffusion and reaction-diffusion processes to study the influence of molecular rebinding on the reaction rates of bimolecular reactions. We found that the number of rebinding events, N_reb, is proportional to the target's size and inversely proportional to the diffusion coefficient D and simulation time-step Δt. We found the proportionality constant close to π^-1/2. We confirmed that N_reb is defined as a ratio of the activation-limited rate constant k_a to the diffusion-limited rate constant, k_D. We provide the formula describing the reactivity coefficient κ, modelling the transient-native complex transition for the activation-controlled reaction rates. We show that κ is proportional to (D/Δt)^1/2. Finally, we apply our rebinding-including reaction rate model to the real reactions of photoacid dissociation and protein association. Based on literature data for both types of reactions, we found the Δt time-scale. We show that for the photodissociation of a proton, the Δt is equal to 171 ± 18 fs and the average number of rebinding events is approximately equal to 40. For proteins, Δt is of the order of 100 ps with around 20 rebinding events. In both cases the timescale is similar to the timescale of fluctuation of the solvent molecules surrounding the reactants; vibrations and bending in the case of photoacid dissociation and diffusional motion for proteins.

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.

Diffusive barrier crossing rates from variationally determined eigenvalues

Kramers' procedure for calculating the rate of activated processes involves partitioning space into reactant, barrier, and product regions by introducing two dividing surfaces. Then, a nonequilibrium steady state is established by injecting particles on one surface and removing them when they reach the other. The rate is obtained as the ratio of the steady-state flux between the surfaces and the population of the initial well. An alternative procedure that seems less artificial is to estimate the first non-zero eigenvalue of the operator that describes the dynamics and then equate its magnitude to the sum of the forward and backward rate constants. Here, we establish the relationship between these approaches for diffusive dynamics, starting with the variational principle for the eigenvalue of interest and then using a trial function involving two adjustable surfaces. We show how Kramers' flux-over-population expression for the rate constant can be obtained from our variationally determined eigenvalue in the special case where the reactant and product regions are separated by a high barrier. This work exploits the modern theory of activated rate processes where the committor (the probability of reaching one dividing surface before the other) plays a central role. Surprisingly, our upper bound for the eigenvalue can be expressed solely in terms of mean first-passage times and the mean transition-path time between the two dividing surfaces.

Kinetics of the multitasking high-affinity Win binding site of WDR5 in restricted and unrestricted conditions

Recent advances in quantitative proteomics show that WD40 proteins play a pivotal role in numerous cellular networks. Yet, they have been fairly unexplored and their physical associations with other proteins are ambiguous. A quantitative understanding of these interactions has wide-ranging significance. WD40 repeat protein 5 (WDR5) interacts with all members of human SET1/MLL methyltransferases, which regulate methylation of the histone 3 lysine 4 (H3K4). Here, using real-time binding measurements in a high-throughput setting, we identified the kinetic fingerprint of transient associations between WDR5 and 14-residue WDR5 interaction (Win) motif peptides of each SET1 protein (SET1Win). Our results reveal that the high-affinity WDR5-SET1Win interactions feature slow association kinetics. This finding is likely due to the requirement of SET1Win to insert into the narrow WDR5 cavity, also named the Win binding site. Furthermore, our explorations indicate fairly slow dissociation kinetics. This conclusion is in accordance with the primary role of WDR5 in maintaining the functional integrity of a large multisubunit complex, which regulates the histone methylation. Because the Win binding site is considered a key therapeutic target, the immediate outcomes of this study could form the basis for accelerated developments in medical biotechnology.

Assessing the Role of Calmodulin's Linker Flexibility in Target Binding

Calmodulin (CaM) is a highly-expressed Ca2+ binding protein known to bind hundreds of protein targets. Its binding selectivity to many of these targets is partially attributed to the protein’s flexible alpha helical linker that connects its N- and C-domains. It is not well established how its linker mediates CaM’s binding to regulatory targets yet. Insights into this would be invaluable to understanding its regulation of diverse cellular signaling pathways. Therefore, we utilized Martini coarse-grained (CG) molecular dynamics simulations to probe CaM/target assembly for a model system: CaM binding to the calcineurin (CaN) regulatory domain. The simulations were conducted assuming a ‘wild-type’ calmodulin with normal flexibility of its linker, as well as a labile, highly-flexible linker variant to emulate structural changes that could be induced, for instance, by post-translational modifications. For the wild-type model, 98% of the 600 simulations across three ionic strengths adopted a bound complex within 2 μs of simulation time; of these, 1.7% sampled the fully-bound state observed in the experimentally-determined crystallographic structure. By calculating the mean-first-passage-time for these simulations, we estimated the association rate to be ka= 8.7 × 108 M−1 s−1, which is similar to the diffusion-limited, experimentally-determined rate of 2.2 × 108 M−1 s−1. Furthermore, our simulations recapitulated its well-known inverse relationship between the association rate and the solution ionic strength. In contrast, although over 97% of the labile linker simulations formed tightly-bound complexes, only 0.3% achieved the fully-bound configuration. This effect appears to stem from a difference in the ensembles of extended and collapsed states which are controlled by the linker flexibility. Therefore, our simulations suggest that variations in the CaM linker’s propensity for alpha helical secondary structure can modulate the kinetics of target binding.

Smaller molecules crowd better: Crowder size dependence revealed by single-molecule FRET studies and depletion force modeling analysis

The cell is an extremely crowded environment, which is known to have a profound impact on the thermodynamics, functionality, and conformational stability of biomolecules. Speculations from recent theoretical molecular dynamics studies suggest an intriguing size dependence to such purely entropic crowding effects, whereby small molecular weight crowders under constant enthalpy conditions are more effective than larger crowders on a per volume basis. If experimentally confirmed, this would be profoundly significant, as the cellular cytoplasm is also quite concentrated in smaller molecular weight solutes such as inorganic ions, amino acids, and various metabolites. The challenge is to perform such studies isolating entropic effects under isoenthalpic conditions. In this work, we first present results from single-molecule FRET spectroscopy (smFRET) on the molecular size-dependent crowding stabilization of a simple RNA tertiary motif (the GAAA tetraloop-tetraloop receptor), indeed providing evidence in support of the surprising notion in the crowding literature that "smaller is better." Specifically, systematic smFRET studies as a function of crowder solute size reveal that smaller molecules both significantly increase the RNA tertiary folding rate and, yet, simultaneously decrease the unfolding rate, predicting strongly size-dependent stabilization of RNA tertiary structures under crowded cellular conditions. The size dependence of these effects has been explored via systematic variation of crowder size over a broad range of molecular weights (90-3000 amu). Furthermore, corresponding temperature dependent studies indicate the systematic changes in the folding equilibrium to be predominantly entropic in origin, i.e., consistent with a fundamental picture of entropic molecular crowding without additional enthalpic interactions. Most importantly, all trends in the single-molecule crowding data can be quantitatively recapitulated by a simple analytic depletion force model, whereby excluded volume interactions represent the major thermodynamic driving force toward folding. Our study, thus, not only provides experimental evidence and theoretical support for small molecule crowding but also predicts further enhancement of crowding effects for even smaller molecules on a per volume basis.

Reaction Pathway Sampling and Free-Energy Analyses for Multimeric Protein Complex Disassembly by Employing Hybrid Configuration Bias Monte Carlo/Molecular Dynamics Simulation

Physicochemical characterization of multimeric biomacromolecule assembly and disassembly processes is a milestone to understand the mechanisms for biological phenomena at the molecular level. Mass spectroscopy (MS) and structural bioinformatics (SB) approaches have become feasible to identify subcomplexes involved in assembly and disassembly, while they cannot provide atomic information sufficient for free-energy calculation to characterize transition mechanism between two different sets of subcomplexes. To combine observations derived from MS and SB approaches with conventional free-energy calculation protocols, we here designed a new reaction pathway sampling method by employing hybrid configuration bias Monte Carlo/molecular dynamics (hcbMC/MD) scheme and applied it to simulate the disassembly process of serum amyloid P component (SAP) pentamer. The results we obtained are consistent with those of the earlier MS and SB studies with respect to SAP subcomplex species and the initial stage of SAP disassembly processes. Furthermore, we observed a novel dissociation event, ring-opening reaction of SAP pentamer. Employing free-energy calculation combined with the hcbMC/MD reaction pathway trajectories, we moreover obtained experimentally testable observations on (1) reaction time of the ring-opening reaction and (2) importance of Asp42 and Lys117 for stable formation of SAP oligomer.

Protein crystals as a key for deciphering macromolecular crowding effects on biological reactions

When placed in the same environment, biochemically unrelated macromolecules influence each other's biological function through macromolecular crowding (MC) effects. This has been illustrated in vitro by the effects of inert polymers on protein stability, protein structure, enzyme kinetics and protein aggregation kinetics. While a unified way to quantitatively characterize MC is still lacking, we show that the crystal solubility of lysozyme can be used to predict the influence of crowding agents on the catalytic efficiency of this enzyme. In order to capture general enthalpic effects, as well as hard entropic effects that are specific of large molecules, we tested sucrose and its cross-linked polymer Ficoll-70 as additives. Despite the different conditions of pH and ionic strength adopted, both the crystallization and the enzymatic assays point to an entropic contribution of approximately -1 kcal mol-1 caused by MC. Our results demonstrate that the thermodynamic activity of proteins is markedly increased by the reduction of accessible volume caused by the presence of macromolecular cosolutes. Unlike what is observed in protein folding studies, this MC effect cannot be reproduced using equivalent concentrations of monomeric crowding units. Applicable to any crystallizable protein, the thermodynamic interpretation of MC based on crystal solubility is expected to help in elucidating the full extent and importance of hard-type interactions in the crowded environment of the cell.

A Multiscale and Comparative Model for Receptor Binding of 2019 Novel Coronavirus and the Implication of its Life Cycle in Host Cells

The respiratory syndrome caused by a new type of coronavirus has been emerging from China and caused more than one million death globally since December 2019. This new virus, called severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) uses the same receptor called Angiotensin-converting enzyme 2 (ACE2) to attack humans as the coronavirus that caused the severe acute respiratory syndrome (SARS) seventeen years ago. Both viruses recognize ACE2 through the spike proteins (S-protein) on their surfaces. It was found that the S-protein from the SARS coronavirus (SARS-CoV) bind stronger to ACE2 than SARS-CoV-2. However, function of a bio-system is often under kinetic, rather than thermodynamic, control. To address this issue, we constructed a structural model for complex formed between ACE2 and the S-protein from SARS-CoV-2, so that the rate of their association can be estimated and compared with the binding of S-protein from SARS-CoV by a multiscale simulation method. Our simulation results suggest that the association of new virus to the receptor is slower than SARS, which is consistent with the experimental data obtained very recently. We further integrated this difference of association rate between virus and receptor into a mathematical model which describes the life cycle of virus in host cells and its interplay with the innate immune system. Interestingly, we found that the slower association between virus and receptor can result in longer incubation period, while still maintaining a relatively higher level of viral concentration in human body. Our computational study therefore provides, from the molecular level, one possible explanation that this new pandemic by far spread much faster than SARS.

Gradual compaction of the nascent peptide during cotranslational folding on the ribosome

Nascent polypeptides begin to fold in the constrained space of the ribosomal peptide exit tunnel. Here we use force-profile analysis (FPA) and photo-induced energy-transfer fluorescence correlation spectroscopy (PET-FCS) to show how a small α-helical domain, the N-terminal domain of HemK, folds cotranslationally. Compaction starts vectorially as soon as the first α-helical segments are synthesized. As nascent chain grows, emerging helical segments dock onto each other and continue to rearrange at the vicinity of the ribosome. Inside or in the proximity of the ribosome, the nascent peptide undergoes structural fluctuations on the µs time scale. The fluctuations slow down as the domain moves away from the ribosome. Mutations that destabilize the packing of the domain's hydrophobic core have little effect on folding within the exit tunnel, but abolish the final domain stabilization. The results show the power of FPA and PET-FCS in solving the trajectory of cotranslational protein folding and in characterizing the dynamic properties of folding intermediates.

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.

Interplay of reactive interference and crowding effects in the diffusion-influenced reaction kinetics

We investigate the interplay of reactive interference and crowding effects in the irreversible diffusion-influenced bimolecular reactions of the type A+B→P+B by using the Brownian dynamics simulation method. It is known that the presence of nonreactive crowding agents retards the reaction rate when the volume fraction of the crowding agents is large enough. On the other hand, a high concentration of B is known to increase the reaction rate more than expected from the mass action law, although the B's may also act as crowders. Therefore, it would be interesting to see which effect dominates when the number density of B as well as the number density of the crowders increases. We will present an approximate theory that provides a reasonable account for the Brownian dynamics simulation results.

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.

Co-localization and confinement of ecto-nucleotidases modulate extracellular adenosine nucleotide distributions

Nucleotides comprise small molecules that perform critical signaling roles in biological systems. Adenosine-based nucleotides, including adenosine tri-, di-, and mono-phosphate, are controlled through their rapid degradation by diphosphohydrolases and ecto-nucleotidases (NDAs). The interplay between nucleotide signaling and degradation is especially important in synapses formed between cells, which create signaling 'nanodomains'. Within these 'nanodomains', charged nucleotides interact with densely-packed membranes and biomolecules. While the contributions of electrostatic and steric interactions within such nanodomains are known to shape diffusion-limited reaction rates, less is understood about how these factors control the kinetics of nucleotidase activity. To quantify these factors, we utilized reaction-diffusion numerical simulations of 1) adenosine triphosphate (ATP) hydrolysis into adenosine monophosphate (AMP) and 2) AMP into adenosine (Ado) via two representative nucleotidases, CD39 and CD73. We evaluate these sequentially-coupled reactions in nanodomain geometries representative of extracellular synapses, within which we localize the nucleotidases. With this model, we find that 1) nucleotidase confinement reduces reaction rates relative to an open (bulk) system, 2) the rates of AMP and ADO formation are accelerated by restricting the diffusion of substrates away from the enzymes, and 3) nucleotidase co-localization and the presence of complementary (positive) charges to ATP enhance reaction rates, though the impact of these contributions on nucleotide pools depends on the degree to which the membrane competes for substrates. As a result, these contributions integratively control the relative concentrations and distributions of ATP and its metabolites within the junctional space. Altogether, our studies suggest that CD39 and CD73 nucleotidase activity within junctional spaces can exploit their confinement and favorable electrostatic interactions to finely control nucleotide signaling.

Single-molecule kinetic studies of DNA hybridization under extreme pressures

Pressure-responsive dynamics of DNA hairpin hybridization/dehybridization is directly visualized at the single molecule level.

Thermokinetic Analysis of Protein Subunit Exchange by Variable-Temperature Native Mass Spectrometry

Many protein complexes are assembled from a varying number of subunits, which are continuously exchanging with diverse time scales. This structural dynamics is considered to be important for many regulatory and sensory adaptation processes that occur in vivo. We have developed an accurate method for monitoring protein subunit exchange by using native electrospray ionization mass spectrometry (ESI-MS), exemplified here for an extremely stable Rad50 zinc hook (Hk) dimer assembly, Zn(Hk)₂. The method has two steps: appropriate protein/peptide mutation and native ESI-MS analysis using a variable-temperature sample inlet. In this work, two Hk mutants were produced, mixed with wild-type Hk, and measured at three different temperatures. A thermokinetic analysis of heterodimer formation allowed us to determine the enthalpy, entropy, and Gibbs free energy of activation for subunit exchange, showing that the reaction is slow and associated with a high enthalpic barrier, consistent with the exceptionally high stability of the Zn(Hk)₂ assembly.

Kinetics of Drug Binding and Residence Time

The kinetics of drug binding and unbinding is assuming an increasingly crucial role in the long, costly process of bringing a new medicine to patients. For example, the time a drug spends in contact with its biological target is known as residence time (the inverse of the kinetic constant of the drug-target unbinding, 1/ koff). Recent reports suggest that residence time could predict drug efficacy in vivo, perhaps even more effectively than conventional thermodynamic parameters (free energy, enthalpy, entropy). There are many experimental and computational methods for predicting drug-target residence time at an early stage of drug discovery programs. Here, we review and discuss the methodological approaches to estimating drug binding kinetics and residence time. We first introduce the theoretical background of drug binding kinetics from a physicochemical standpoint. We then analyze the recent literature in the field, starting from the experimental methodologies and applications thereof and moving to theoretical and computational approaches to the kinetics of drug binding and unbinding. We acknowledge the central role of molecular dynamics and related methods, which comprise a great number of the computational methods and applications reviewed here. However, we also consider kinetic Monte Carlo. We conclude with the outlook that drug (un)binding kinetics may soon become a go/no go step in the discovery and development of new medicines.

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.

Emergent Properties of Antiagglomerant Films Control Methane Transport: Implications for Hydrate Management

The relationship between collective properties and performance of antiagglomerants (AAs) used in hydrate management is handled using molecular dynamics simulations and enhanced sampling techniques. A thin film of AAs adsorbed at the interface between one flat sII methane hydrate substrate and a fluid hydrocarbon mixture containing methane and n-dodecane is studied. The AA considered is a surface-active compound with a complex hydrophilic head that contains both amide and tertiary ammonium cation groups and hydrophobic tails. At a sufficiently high AA density, the interplay between the surfactant layer and the liquid hydrocarbon excludes methane from the interfacial region. In this scenario, we combine metadynamics and umbrella sampling frameworks to study accurately the free-energy landscape and the equilibrium rates associated with the transport of one methane molecule across the AA film. We observe that the local configurational changes of the liquid hydrocarbon packed within the AA film are associated with high free-energy barriers for methane transport. The time scales estimated for the transport of methane across the AA film can be, in some cases, comparable to those reported in the literature for the growth of hydrates, suggesting that one possible mechanism by which AAs delay the formation of hydrate plugs could be providing a barrier to methane transport. Considering the interplay between the structural design and collective properties of AAs might be of relevance to improve their performance in flow assurance.

Analysis of Phosphoryl-Transfer Enzymes with QM/MM Free Energy Simulations

We discuss the application of quantum mechanics/molecular mechanics (QM/MM) free energy simulations to the analysis of phosphoryl transfers catalyzed by two enzymes: alkaline phosphatase and myosin. We focus on the nature of the transition state and the issue of mechanochemical coupling, respectively, in the two enzymes. The results illustrate unique insights that emerged from the QM/MM simulations, especially concerning the interpretation of experimental data regarding the nature of enzymatic transition states and coupling between global structural transition and catalysis in the active site. We also highlight a number of technical issues worthy of attention when applying QM/MM free energy simulations, and comment on a number of technical and mechanistic issues that require further studies.

Thermal decay of a metastable state: Influence of rescattering on the quasistationary dynamical rate

We study the effect of backscattering of the Brownian particles as they escape out of a metastable state overcoming the potential barrier. For this aim, we model this process numerically using the Langevin equations. This modeling is performed for the wide range of the friction constant covering both the energy and spatial diffusion regimes. It is shown how the influence of the descent stage on the quasistationary decay rate gradually disappears as the friction constant decreases. It is found that, in the energy diffusion regime, the rescattering absents and the descent stage does not influence the decay rate. As the value of friction increases, the descent alters the value of the rate by more than 50% for different values of thermal energy and different shapes of the potential. To study the influence of the backscattering on the decay rate, four potentials have been considered which coincide near the potential well and the barrier but differ beyond the barrier. It is shown that the potential for which the well and the barrier are described by two smoothly joined parabolas ("the parabolic potential") plays a role of a dividing range for the mutual layout of the quasistationary dynamical rate and the widely used in the literature Kramers rate. Namely, for the potentials with steeper tails, the Kramers rate R_{KM} underestimates the true quasistationary dynamical rate R_{D}, whereas for the less steep tails the opposite holds (inversion of R_{D}/R_{KM}). It is demonstrated that the mutual layout of the values of R_{D} for different potentials is explained by the rescattering of the particles from the potential tail.

Insight into the Role of the Hv1 C-Terminal Domain in Dimer Stabilization

The voltage-gated proton-selective channel (Hv1) conducts protons in response to changes in membrane potential. The Hv1 protein forms dimers in the membrane. Crystal structures of Hv1 channels have revealed that the primary contacts between the two monomers are in the C-terminal domain (CTD), which forms a coiled-coil structure. The role of Hv1-CTD in channel assembly and activity is not fully understood. Here, molecular dynamics (MD) simulations of full-length and truncated CTD models of human and mouse Hv1 channels reveal a strong contribution of the CTD to the packing of the transmembrane domains. Simulations of the CTD models highlight four fundamental interactions of the key residues contributing to dimer stability. These include salt bridges, hydrophobic interactions, hydrogen bonds, and a disulfide bond across the dimer interface. At neutral pH, salt-bridge interactions increase dimer stability and the dimer becomes less stable at acidic pH. Hydrophobic core packing of the heptad pattern is important for stability, as shown by favorable nonpolar binding free energies rather than by electrostatic components. Moreover, free-energy calculations indicate that a more uniform hydrophobic core in the coiled-coil structure of the Hv1-NIN, a channel carrying the triple mutation M234N-N235I-V236N, leads to an increase in dimer stability with respect to the wild-type. A Cys disulfide bond has a strong impact on dimer stability by holding the dimer together and facilitating the interactions described above. These results are consistent with dissociative temperatures and energy barriers of dimer dissociation obtained from the temperature-accelerated MD.

The dock-and-coalesce mechanism for the association of a WASP disordered region with the Cdc42 GTPase

Intrinsically disordered proteins (IDPs) play key roles in signaling and regulation. Many IDPs undergo folding upon binding to their targets. We have proposed that coupled folding and binding of IDPs generally follow a dock-and-coalesce mechanism, whereby a segment of the IDP, through diffusion, docks to its cognate subsite and, subsequently, the remaining segments coalesce around their subsites. Here, by a combination of experiment and computation, we determined the precise form of dock-and-coalesce operating in the association between the intrinsically disordered GTPase-binding domain (GBD) of the Wiskott-Aldrich Syndrome protein and the Cdc42 GTPase. The association rate constants (k_a ) were measured by stopped-flow fluorescence under various solvent conditions. k_a reached 10⁷ m^-1 ·s^-1 at physiological ionic strength and had a strong salt dependence, suggesting that an electrostatically enhanced, diffusion-controlled docking step may be rate limiting. Our computation, based on the transient-complex theory, identified the N-terminal basic region of the GBD as the docking segment. However, several other changes in solvent conditions provided strong evidence that the coalescing step also contributed to determining the magnitude of k_a . Addition of glucose and trifluoroethanol and an increase in temperature all produced experimental k_a values much higher than expected from the effects on the docking rate alone. Conversely, addition of urea led to k_a values much lower than expected if only the docking rate was affected. These results all pointed to k_a being approximately two-thirds of the docking rate constant under physiological solvent conditions.

Development of a multisite model for Ni(II) ion in solution from thermodynamic and kinetic data

Force-field parameters are developed for a multisite model of Ni(II) ions to be used in molecular dynamics simulations combined to enhanced sampling methods. The performances of two charge-partitioning schemes are validated by taking into account structural, thermodynamic, and kinetic observables. One of the two models, featuring partial charges on the dummy atoms only, matches both Ni(II) free energy of solvation and water exchange rates. Such model is particularly suited to study complexation events at a fully dynamic description. © 2017 Wiley Periodicals, Inc.

Torque-coupled thermodynamic model for F_{o}F_{1}-ATPase

F_{o}F_{1}-ATPase is a motor protein complex that utilizes transmembrane ion flow to drive the synthesis of adenosine triphosphate (ATP) from adenosine diphosphate (ADP) and phosphate (Pi). While many theoretical models have been proposed to account for its rotary activity, most of them focus on the F_{o} or F_{1} portions separately rather than the complex as a whole. Here, we propose a simple but new torque-coupled thermodynamic model of F_{o}F_{1}-ATPase. Solving this model at steady state, we find that the monotonic variation of each portion's efficiency becomes much more robust over a wide range of parameters when the F_{o} and F_{1} portions are coupled together, as compared to cases when they are considered separately. Furthermore, the coupled model predicts the dependence of each portion's kinetic behavior on the parameters of the other. Specifically, the power and efficiency of the F_{1} portion are quite sensitive to the proton gradient across the membrane, while those of the F_{o} portion as well as the related Michaelis constants for proton concentrations respond insensitively to concentration changes in the reactants of ATP synthesis. The physiological proton gradient across the membrane in the F_{o} portion is also shown to be optimal for the Michaelis constants of ADP and phosphate in the F_{1} portion during ATP synthesis. Together, our coupled model is able to predict key dynamic and thermodynamic features of the F_{o}F_{1}-ATPase in vivo semiquantitatively, and suggests that such coupling approach could be further applied to other biophysical systems.

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.

Rate Constants and Mechanisms of Protein-Ligand Binding

Whereas protein-ligand binding affinities have long-established prominence, binding rate constants and binding mechanisms have gained increasing attention in recent years. Both new computational methods and new experimental techniques have been developed to characterize the latter properties. It is now realized that binding mechanisms, like binding rate constants, can and should be quantitatively determined. In this review, we summarize studies and synthesize ideas on several topics in the hope of providing a coherent picture of and physical insight into binding kinetics. The topics include microscopic formulation of the kinetic problem and its reduction to simple rate equations; computation of binding rate constants; quantitative determination of binding mechanisms; and elucidation of physical factors that control binding rate constants and mechanisms.

Observation of the Global Dynamic Collectivity of a Hydration Shell around Apomyoglobin

Protein surface hydration is critical to the protein's structural properties and biological activities. However, it is still unknown whether the hydration shell is intrinsically connected and how its fluctuations dynamically interact with protein motion. Here, by selecting five site-specific locations with distinctly different environments around the surface of apomyoglobin, we used a tryptophan scan with femtosecond fluorescence spectroscopy and simultaneously detected hydration water dynamics and tryptophan side-chain relaxations with temperature dependence. We observed two types of relaxations for both interfacial hydration water and the tryptophan side chain. The former is always faster than the latter, and both motions show direct linear correlations with temperature changes, indicating one origin of their motions and hydration water driving of side-chain fluctuations. Significantly, we found the relaxation energy barriers are uniform across the entire protein surface, all less than 20 kJ/mol, strongly suggesting highly extended cooperative water networks and the nature of global dynamic collectivity of the entire hydration shell.

Mechanism of the Association Pathways for a Pair of Fast and Slow Binding Ligands of HIV-1 Protease

Equilibrium constants, together with kinetic rate constants of binding, are key factors in the efficacy and safety of drug compounds, informing drug design. However, the association pathways of protein-ligand binding, which contribute to their kinetic behaviors, are little understood. In this work, we used unbiased all-atom molecular dynamics (MD) simulations with an explicit solvent model to study the association processes of protein-ligand binding. Using the HIV protease (HIVp)-xk263 and HIVp-ritonavir protein-ligand systems as cases, we observed that ligand association is a multistep process involving diffusion, localization, and conformational rearrangements of the protein, ligand, and water molecules. Moreover, these two ligands preferred different routes of binding, which reflect two well-known binding mechanisms: induced-fit and conformation selection models. Our study shows that xk263 has a stronger capacity for desolvating surrounding water molecules, thereby inducing a semiopen conformation of the HIVp flaps (induced-fit model). In contrast, the slow dehydration characteristic of ritonavir allows for gradual association with the binding pocket of HIVp when the protein's flap conformation is fully open (conformation selection model). By studying the mechanism of ligand association and understanding the role of solvent molecules during the binding event, we can obtain a different perspective on the mechanism of macromolecule recognition, providing insights into drug discovery.

Additives for vaccine storage to improve thermal stability of adenoviruses from hours to months

Up to 80% of the cost of vaccination programmes is due to the cold chain problem (that is, keeping vaccines cold). Inexpensive, biocompatible additives to slow down the degradation of virus particles would address the problem. Here we propose and characterize additives that, already at very low concentrations, improve the storage time of adenovirus type 5. Anionic gold nanoparticles (10⁻⁸–10⁻⁶ M) or polyethylene glycol (PEG, molecular weight ∼8,000 Da, 10⁻⁷–10⁻⁴ M) increase the half-life of a green fluorescent protein expressing adenovirus from ∼48 h to 21 days at 37 °C (from 7 to >30 days at room temperature). They replicate the known stabilizing effect of sucrose, but at several orders of magnitude lower concentrations. PEG and sucrose maintained immunogenicity in vivo for viruses stored for 10 days at 37 °C. To achieve rational design of viral-vaccine stabilizers, our approach is aided by simplified quantitative models based on a single rate-limiting step.

Keeping viral vaccines cold from the manufacturers to patients is problematic and costly. Here, Krol and others show additives that can significantly improve at very low concentrations the storage of adenovirus type 5 at ambient and elevated temperature.

Precisely and Accurately Inferring Single-Molecule Rate Constants

The kinetics of biomolecular systems can be quantified by calculating the stochastic rate constants that govern the biomolecular state vs time trajectories (i.e., state trajectories) of individual biomolecules. To do so, the experimental signal vs time trajectories (i.e., signal trajectories) obtained from observing individual biomolecules are often idealized to generate state trajectories by methods such as thresholding or hidden Markov modeling. Here, we discuss approaches for idealizing signal trajectories and calculating stochastic rate constants from the resulting state trajectories. Importantly, we provide an analysis of how the finite length of signal trajectories restricts the precision of these approaches and demonstrate how Bayesian inference-based versions of these approaches allow rigorous determination of this precision. Similarly, we provide an analysis of how the finite lengths and limited time resolutions of signal trajectories restrict the accuracy of these approaches, and describe methods that, by accounting for the effects of the finite length and limited time resolution of signal trajectories, substantially improve this accuracy. Collectively, therefore, the methods we consider here enable a rigorous assessment of the precision, and a significant enhancement of the accuracy, with which stochastic rate constants can be calculated from single-molecule signal trajectories.

Kinetics of the Interactions between Copper and Amyloid-β with FAD Mutations and Phosphorylation at the N terminus

Mutations and post‐translational modifications of amyloid‐β (Aβ) peptide in its N terminus have been shown to increase fibril formation, yet the molecular mechanism is not clear. Here we investigated the kinetics of the interactions of copper with two Aβ peptides containing Familial Alzheimer's disease (FAD) mutations (English (H6R) and Tottori (D7N)), as well as with Aβ peptide phosphorylated at serine 8 (pS8). All three peptides bind to copper with a similar rate as the wild‐type (wt). The dissociation rates follow the order pS8>H6R>wt>D7N; the interconversion between the two coordinating species occurs 50 % faster for H6R and pS8, whereas D7N had only a negligible effect. Interestingly, the rate of ternary complex (copper‐bridged heterodimer) formation for the modified peptides was significantly faster than that for wt, thus leading us to propose that FAD and sporadic AD might share a kinetic origin for the enhanced oligomerisation of Aβ.

Theory of Crowding Effects on Bimolecular Reaction Rates

An analytical expression for the rate constant of a diffusion-influenced bimolecular reaction in a crowded environment is derived in the framework of a microscopic model that accounts for: (1) the slowdown of diffusion due to crowding and the dependence of the diffusivity on the distance between the reactants, (2) a crowding-induced attractive short-range potential of mean force, and (3) nonspecific reversible binding to the crowders. This expression spans the range from reaction to diffusion control. Crowding can increase the reaction-controlled rate by inducing an effective attraction between reactants but decrease the diffusion-controlled rate by reducing their relative diffusivity.

Multidimensional reaction rate theory with anisotropic diffusion

An analytical expression is derived for the rate constant that describes diffusive transitions between two deep wells of a multidimensional potential. The expression, in contrast to the Kramers-Langer formula for the rate constant, is valid even when the diffusion is highly anisotropic. Our approach is based on a variational principle for the reactive flux and uses a trial function for the splitting probability or commitor. The theoretical result is validated by Brownian dynamics simulations.

Theory of bimolecular reactions in a solution with linear traps: Application to the problem of target search on DNA

One-dimensional sliding along DNA as a means to accelerate protein target search is a well-known phenomenon occurring in various biological systems. Using a biomimetic approach, we have recently demonstrated the practical use of DNA-sliding peptides to speed up bimolecular reactions more than an order of magnitude by allowing the reactants to associate not only in the solution by three-dimensional (3D) diffusion, but also on DNA via one-dimensional (1D) diffusion [A. Turkin et al., Chem. Sci. (2015)]. Here we present a mean-field kinetic model of a bimolecular reaction in a solution with linear extended sinks (e.g., DNA) that can intermittently trap molecules present in a solution. The model consists of chemical rate equations for mean concentrations of reacting species. Our model demonstrates that addition of linear traps to the solution can significantly accelerate reactant association. We show that at optimum concentrations of linear traps the 1D reaction pathway dominates in the kinetics of the bimolecular reaction; i.e., these 1D traps function as an assembly line of the reaction product. Moreover, we show that the association reaction on linear sinks between trapped reactants exhibits a nonclassical third-order behavior. Predictions of the model agree well with our experimental observations. Our model provides a general description of bimolecular reactions that are controlled by a combined 3D+1D mechanism and can be used to quantitatively describe both naturally occurring as well as biomimetic biochemical systems that reduce the dimensionality of search.

Decoding the mechanical fingerprints of biomolecules

The capacity of biological macromolecules to act as exceedingly sophisticated and highly efficient cellular machines – switches, assembly factors, pumps, or motors – is realized through their conformational transitions, that is, their folding into distinct shapes and selective binding to other molecules. Conformational transitions can be induced, monitored, and manipulated by pulling individual macromolecules apart with an applied force. Pulling experiments reveal, for a given biomolecule, the relationship between applied force and molecular extension. Distinct signatures in the force–extension relationship identify a given biomolecule and thus serve as the molecule's ‘mechanical fingerprints’. But, how can these fingerprints be decoded to uncover the energy barriers crossed by the molecule in the course of its conformational transition, as well as the associated timescales? This review summarizes a powerful class of approaches to interpreting single-molecule force spectroscopy measurements – namely, analytically tractable approaches. On the fundamental side, analytical theories have the power to reveal the unifying principles underneath the bewildering diversity of biomolecules and their behaviors. On the practical side, analytical expressions that result from these theories are particularly well suited for a direct fit to experimental data, yielding the important parameters that govern biological processes at the molecular level.

Computational perspective and evaluation of plausible catalytic mechanisms of peptidyl-prolyl cis-trans isomerases

BACKGROUND

Peptidyl prolyl cis-trans isomerization of the protein backbone is involved in the regulation of many biological processes. Cis-trans isomerization is notoriously slow and is catalyzed by a family of cis-trans peptidyl prolyl isomerases (PPIases) that have been implicated in many diseases. A general consensus on how these enzymes speed up prolyl isomerization has not been reached after decades of both experimental and computational studies.

SCOPE OF REVIEW

Computational studies carried out to understand the catalytic mechanism of the prototypical FK506 binding protein 12, Cyclophilin A and peptidyl-prolyl cis-trans isomerase NIMA-interacting 1 (Pin1) are reviewed. A summary and an evaluation of the implications of the proposed mechanisms from computational studies are presented.

MAJOR CONCLUSIONS

The analysis of computational studies and evaluation of the proposed mechanisms provide a general consensus and a better understanding of PPIase catalysis. The speedup of the rate of peptidyl-prolyl isomerization by PPIases can be best described by a catalytic mechanism in which the substrate in transition state configuration is stabilized. The enzymes preferentially bind the transition state configuration of the substrate relative to the cis conformation, which in most cases is bound better than the trans conformation of the substrate. Stabilization of the transition state configuration of the substrate leads to a lower free energy barrier and a faster rate of isomerization when compared to the uncatalyzed isomerization reaction.

GENERAL SIGNIFICANCE

Fully understanding the catalytic mechanism of PPIases has broad implications for drug design, elucidation of the molecular basis of many diseases, protein engineering, and enzyme catalysis in general. This article is part of a Special Issue entitled Proline-directed Foldases: Cell Signaling Catalysts and Drug Targets.

Diffusion-influenced reactions in a hollow nano-reactor with a circular hole

The problem of diffusion to a reactive sphere inside a cavity through a hole is solved for arbitrary aperture values.