How do biomolecular systems speed up and regulate rates?

Statistics of Reaction Flux and Dynamical Activity Associated with a Diffusion-Influenced Ligand-Binding Reaction

In this paper, we consider a macromolecule with two competitive binding sites where a ligand can bind to and gives rise to a unicyclic reaction network consisting of four states-(i) a single state with both binding sites vacant, (ii) two states with one bound site and one free binding site, and (iii) an another single state with both sites occupied. We obtain probability densities of the time-integrated current along the clockwise direction and the dynamical activity or mean number of jumps between different states for finite times at a fast diffusion limit. On the other hand, in the diffusion-limited case, ligand diffusion between the two binding sites directly connects the mono-ligated states-changing the reaction scheme. Addition of the new reaction channel alters the precision of ligand occupancy to a single site, the mean dynamical activity, and the mean entropy production rate. All of these quantities are calculated with varying degrees of competition between the two sites for ligands, and we find that increase in the competition between the two sites decreases all above-mentioned additive functionals. The upper bound of precision associated with a single-site ligand occupancy for a diffusion-influenced reaction network is set by the mean dynamical activity (mean entropy production rate) at small (large) ligand concentrations at the steady-state limit.

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.

Diffusion-induced competitive two-site binding

The influence of diffusion on the kinetics of ligand binding to a macromolecule with two sites is considered for a simple model where, in the reaction-controlled limit, there is no cooperativity and hence the sites are independent. By applying our recently developed formalism to describe a network of coupled diffusion-influenced reactions, we show that the rate constants of chemical kinetics cannot just be renormalized. Rather a new reaction channel, which connects the two singly occupied states, must be introduced. The rate constants of this new channel depend on the committor or capture probability that a ligand that just dissociated from one site rebinds to the other. This result is rederived in an elementary way using the encounter complex model. Illustrative calculations are presented where the kinetics of the fractional saturation of one site is compared with that of a macromolecule that has only this site. If all sites are initially empty, then the second site slows down binding to the first due to competition between the sites. On the other hand, if the second site is initially occupied, the binding of the first site speeds up because of the direct diffusion-induced transitions between the two singly bound states.

Modulation of electrostatic interactions to reveal a reaction network unifying the aggregation behaviour of the Aβ42 peptide and its variants

The aggregation of the amyloid β peptide (Aβ42), which is linked to Alzheimer's disease, can be altered significantly by modulations of the peptide's intermolecular electrostatic interactions. Variations in sequence and solution conditions have been found to lead to highly variable aggregation behaviour. Here we modulate systematically the electrostatic interactions governing the aggregation kinetics by varying the ionic strength of the solution. We find that changes in the solution ionic strength induce a switch in the reaction pathway, altering the dominant mechanisms of aggregate multiplication. This strategy thereby allows us to continuously sample a large space of different reaction mechanisms and develop a minimal reaction network that unifies the experimental kinetics under a wide range of different conditions. More generally, this universal reaction network connects previously separate systems, such as charge mutants of the Aβ42 peptide, on a continuous mechanistic landscape, providing a unified picture of the aggregation mechanism of Aβ42.

The specificity of thioredoxins and glutaredoxins is determined by electrostatic and geometric complementarity

Thiol-disulfide oxidoreductases from the thioredoxin (Trx) family of proteins have a broad range of well documented functions and possess distinct substrate specificities. The mechanisms and characteristics that control these specificities are key to the understanding of both the reduction of catalytic disulfides as well as allosteric disulfides (thiol switches). Here, we have used the catalytic disulfide of E. coli 3'-phosphoadenosine 5'-phosphosulfate (PAPS) reductase (PR) that forms between the single active site thiols of two monomers during the reaction cycle as a model system to investigate the mechanisms of Trx and Grx protein specificity. Enzyme kinetics, ΔE'0 determination, and structural analysis of various Trx and Grx family members suggested that the redox potential does not determine specificity nor efficiency of the redoxins as reductant for PR. Instead, the efficiency of PR with various redoxins correlated strongly to the extent of a negative electric field of the redoxins reaching into the solvent outside the active site, and electrostatic and geometric complementary contact surfaces. These data suggest that, in contrast to common assumption, the composition of the active site motif is less important for substrate specificity than other amino acids in or even outside the immediate contact area.

A model study of sequential enzyme reactions and electrostatic channeling

We study models of two sequential enzyme-catalyzed reactions as a basic functional building block for coupled biochemical networks. We investigate the influence of enzyme distributions and long-range molecular interactions on reaction kinetics, which have been exploited in biological systems to maximize metabolic efficiency and signaling effects. Specifically, we examine how the maximal rate of product generation in a series of sequential reactions is dependent on the enzyme distribution and the electrostatic composition of its participant enzymes and substrates. We find that close proximity between enzymes does not guarantee optimal reaction rates, as the benefit of decreasing enzyme separation is countered by the volume excluded by adjacent enzymes. We further quantify the extent to which the electrostatic potential increases the efficiency of transferring substrate between enzymes, which supports the existence of electrostatic channeling in nature. Here, a major finding is that the role of attractive electrostatic interactions in confining intermediate substrates in the vicinity of the enzymes can contribute more to net reactive throughput than the directional properties of the electrostatic fields. These findings shed light on the interplay of long-range interactions and enzyme distributions in coupled enzyme-catalyzed reactions, and their influence on signaling in biological systems.

Importance of electrostatic interactions in the association of intrinsically disordered histone chaperone Chz1 and histone H2A.Z-H2B

Histone chaperones facilitate assembly and disassembly of nucleosomes. Understanding the process of how histone chaperones associate and dissociate from the histones can help clarify their roles in chromosome metabolism. Some histone chaperones are intrinsically disordered proteins (IDPs). Recent studies of IDPs revealed that the recognition of the biomolecules is realized by the flexibility and dynamics, challenging the century-old structure-function paradigm. Here we investigate the binding between intrinsically disordered chaperone Chz1 and histone variant H2A.Z-H2B by developing a structure-based coarse-grained model, in which Debye-Hückel model is implemented for describing electrostatic interactions due to highly charged characteristic of Chz1 and H2A.Z-H2B. We find that major structural changes of Chz1 only occur after the rate-limiting electrostatic dominant transition state and Chz1 undergoes folding coupled binding through two parallel pathways. Interestingly, although the electrostatic interactions stabilize bound complex and facilitate the recognition at first stage, the rate for formation of the complex is not always accelerated due to slow escape of conformations with non-native electrostatic interactions at low salt concentrations. Our studies provide an ionic-strength-controlled binding/folding mechanism, leading to a cooperative mechanism of “local collapse or trapping” and “fly-casting” together and a new understanding of the roles of electrostatic interactions in IDPs' binding.

Histone chaperones facilitate the assembly and disassembly of nucleosome by interacting with the corresponding histones or histone variants. As the biomolecules in nucleosome are highly charged, electrostatic interactions are particularly important in these processes. The experiments have explored that the histone chaperon Chz1 as an intrinsically disordered protein (IDP) can fold by binding to its histone variants H2A.Z-H2B. Here, we developed a molecular simulation program that treated electrostatic interactions with Debye-Hückel model to study the mechanism of the association. We found that the inter-chain electrostatic interactions facilitate the coupled folding and binding transitions, consistent with the kinetic experiments and microscopic structural perspectives. Furthermore, we show that the intra-chain electrostatic interactions collapse Chz1 and slow the binding rate. The collapsed structure in IDPs caused by intra-chain electrostatic interactions has been widely found in experiments and the effect in binding is well studied in our work. Our theoretical approach shed new light on the role of electrostatics on inter-chain and intra-chain interactions for IDPs' binding and is applicable to the binding/folding of other IDPs to their targets.

Elicitin-membrane interaction is driven by a positive charge on the protein surface: role of Lys13 residue in lipids loading and resistance induction

Elicitins are family of small proteins secreted by species of the pathogenic fungus Phytophthora inducing a defence reaction in plants. They contain a hydrophobic cavity capable of binding sterols and fatty acids, and on the basis of their pI they are classified as either α-elicitins or more necrotising β-elicitins. The residue Lys13 was previously identified as a key determinant of the necrotising activity of basic elicitins. In the present study we describe changes in the ability of cryptogein, a β-elicitin inducing a hypersensitive response in tobacco, to transfer sterols and fatty acids between micelles and liposomes upon Lys13Val mutation. We propose that the change in activity is influenced by the elimination of positive charge on the surface of cryptogein, which is significant for correct positioning of the protein during lipid loading, without adversely affecting the binding of sterol to the cavity of the protein. Compared to wild type cryptogein, mutation Lys13Val resulted in lowered expression of defence-related genes and compromised resistance to Phytophthora parasitica. Furthermore, resistance induced by Lys13Val mutant was similar to that induced by acidic elicitin capsicein containing at amino position 13 valine Determined results sustained a crucial role of positive lysine residues on the surface of basic elicitins and suggested their significant role in correct protein-membrane interaction and thus on their biological activity.

Interaction of the influenza A virus polymerase PB2 C-terminal region with importin alpha isoforms provides insights into host adaptation and polymerase assembly

In the adaptation of avian viruses to mammalian hosts, mutations in the viral polymerase, notably in the PB2 subunit, play an important role. A PB2 C-terminal domain rich in putative host adaptation residues has been shown to bind importin α nuclear import receptors. Adaptation has been proposed to involve binding of PB2 to importins of the new host. To date PB2-importin complexes have been characterized semiquantitatively with no precise measurement of binding parameters. To investigate the effects of adaptive mutations on importin interaction and selectivity, surface plasmon resonance was used to compare the binding rate constants and affinities of avian H5N1 and human H3N2 PB2 C-terminal variants with importin isoforms human α 1, 3, 5 and 7, and avian α 1. Using purified proteins eliminates host environment effects and permits measurement of intrinsic affinities and rates of complex formation and dissociation. Two effects were observed: first, adaptive mutations D701N, R702K, and S714R in the nuclear localization signal domain increased 2-4-fold the association rates with avian and human importins; second, measurement of different structural forms of the PB2 C terminus demonstrated that the upstream 627 domain reduced binding affinity, consistent with a steric clash predicted from crystal structures. From these kinetic data, structural analyses, and the data of others, a model is proposed in which an increase in charged surface residues during host adaptation increases the association rate of PB2 to cytoplasmic importins and where the C-terminal 627-nuclear localization signal domain may reorganize upon importin binding, consistent with a role in active polymerase assembly.

Nonnative interactions in coupled folding and binding processes of intrinsically disordered proteins

Proteins function by interacting with other molecules, where both native and nonnative interactions play important roles. Native interactions contribute to the stability and specificity of a complex, whereas nonnative interactions mainly perturb the binding kinetics. For intrinsically disordered proteins (IDPs), which do not adopt rigid structures when being free in solution, the role of nonnative interactions may be more prominent in binding processes due to their high flexibilities. In this work, we investigated the effect of nonnative hydrophobic interactions on the coupled folding and binding processes of IDPs and its interplay with chain flexibility by conducting molecular dynamics simulations. Our results showed that the free-energy profiles became rugged, and intermediate states occurred when nonnative hydrophobic interactions were introduced. The binding rate was initially accelerated and subsequently dramatically decreased as the strength of the nonnative hydrophobic interactions increased. Both thermodynamic and kinetic analysis showed that disordered systems were more readily affected by nonnative interactions than ordered systems. Furthermore, it was demonstrated that the kinetic advantage of IDPs (“fly-casting” mechanism) was enhanced by nonnative hydrophobic interactions. The relationship between chain flexibility and protein aggregation is also discussed.

Rate theories for biologists

Some of the rate theories that are most useful for modeling biological processes are reviewed. By delving into some of the details and subtleties in the development of the theories, the review will hopefully help the reader gain a more than superficial perspective. Examples are presented to illustrate how rate theories can be used to generate insight at the microscopic level into biomolecular behaviors. An attempt is made to clear up a number of misconceptions in the literature regarding popular rate theories, including the appearance of Planck's constant in the transition-state theory and the Smoluchowski result as an upper limit for protein-protein and protein-DNA association rate constants. Future work in combining the implementation of rate theories through computer simulations with experimental probes of rate processes, and in modeling effects of intracellular environments so that theories can be used for generating rate constants for systems biology studies is particularly exciting.

Kinetic advantage of intrinsically disordered proteins in coupled folding-binding process: a critical assessment of the "fly-casting" mechanism

Intrinsically disordered proteins (IDPs) are recognized to play important roles in many biological functions such as transcription and translation regulation, cellular signal transduction, protein phosphorylation, and molecular assemblies. The coupling of folding with binding through a "fly-casting" mechanism has been proposed to account for the fast binding kinetics of IDPs. In this article, experimental data from the literature were collated to verify the kinetic advantages of IDPs, while molecular simulations were performed to clarify the origin of the kinetic advantages. The phosphorylated KID-kinase-inducible domain interacting domain (KIX) complex was used as an example in the simulations. By modifying a coarse-grained model with a native-centric Gō-like potential, we were able to continuously tune the degree of disorder of the phosphorylated KID domain and thus investigate the intrinsic role of chain flexibility in binding kinetics. The simulations show that the "fly-casting" effect is not only due to the greater capture radii of IDPs. The coupling of folding with binding of IDPs leads to a significant reduction in binding free-energy barrier. Such a reduction accelerates the binding process. Although the greater capture radius has been regarded as the main factor in promoting the binding rate of IDPs, we found that this parameter will also lead to the slower translational diffusion of IDPs when compared with ordered proteins. As a result, the capture rate of IDPs was found to be slower than that of ordered proteins. The main origin of the faster binding for IDPs are the fewer encounter times required before the formation of the final binding complex. The roles of the interchain native contacts fraction (Q(b)) and the mass-center distance (DeltaR) as reaction coordinates are also discussed.

Binding mechanism of an SH3 domain studied by NMR and ITC

Complexes between Src-homology 3 domains and proline-rich target peptides can have lifetimes on the order of milliseconds, making them too short-lived for kinetic characterization by conventional methods. Nuclear magnetic resonance (NMR) dynamics experiments are ideally suited to study such rapid binding equilibria, and additionally provide information on partly bound intermediate states. We used NMR together with isothermal titration calorimetry (ITC) to characterize the interaction of the SH3 domain from the Fyn tyrosine kinase with a 12-residue peptide at temperatures between 10 and 50 degrees C. NMR data at all temperatures are consistent with an effectively two-state binding reaction, such that any intermediates are either very weakly populated or exchange extremely rapidly with the free or bound forms. Dissociation rate constants, determined by CPMG and ZZ-exchange NMR experiments, range from k(off)(10 degrees C) = 4.5 s(-1) to k(off)(50 degrees C) = 331 s(-1). ITC data at all temperatures follow a simple two-state interaction model. Binding is favored enthalpically, with a dissociation enthalpy, DeltaH(D)(30 degrees C) = 15.4 kcal mol(-1), and disfavored entropically, with a dissociation entropy, DeltaS(D)(30 degrees C) = 20.0 cal mol(-1) K(-1). The free protein and peptide have significantly higher heat capacity than the bound complex, DeltaC(p) = 352 cal mol(-1) K(-1), which is consistent with the largely hydrophobic character of the binding interface. An Eyring plot of k(off) values gives an activation enthalpy of dissociation, DeltaH(D)(double dagger)(30 degrees C) = 19.3 kcal mol(-1) and exhibits slight curvature consistent with the ITC-derived value of DeltaC(p). The curvature suggests that nonpolar residues of the hydrophobic interface are solvated in the transition state for dissociation. Association rate constants were calculated as k(on) = k(off)/K(D), and range from k(on)(10 degrees C) = 1.03 x 10(8) M(-1) s(-1) to k(on)(50 degrees C) = 2.0 x 10(8) M(-1) s(-1), with an apparent activation enthalpy, DeltaH(A)(double dagger) = 3.4 kcal mol(-1). Both the magnitudes and temperature dependence of k(on) values are consistent with a diffusion-limited association mechanism. The combination of NMR and ITC data sheds light on how the Fyn tyrosine kinase is activated by binding to proline-rich targets, and represents a powerful approach for characterizing transient protein/ligand interactions.

A minimum-reaction-flux solution to master-equation models of protein folding

Master equations are widely used for modeling protein folding. Here an approximate solution to such master equations is presented. The approach used may be viewed as a discrete variational transition-state theory. The folding rate constant kf is approximated by the outgoing reaction flux J, when the unfolded set of macrostates assumes an equilibrium distribution. Correspondingly the unfolding rate constant ku is calculated as Jpu(1-pu), where pu is the equilibrium fraction of the unfolded state. The dividing surface between the unfolded and folded states is chosen to minimize the reaction flux J. This minimum-reaction-flux surface plays the role of the transition-state ensemble and identifies rate-limiting steps. Test against exact results of master-equation models of Zwanzig [Proc. Natl. Acad. Sci. USA 92, 9801 (1995)] and Munoz et al. [Proc. Natl. Acad. Sci. USA 95, 5872 (1998)] shows that the minimum-reaction-flux solution works well. Macrostates separated by the minimum-reaction-flux surface show a gap in p(fold) values. The approach presented here significantly simplifies the solution of master-equation models and, at the same time, directly yields insight into folding mechanisms.

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.

Energy landscape and transition state of protein-protein association

Formation of a stereospecific protein complex is favored by specific interactions between two proteins but disfavored by the loss of translational and rotational freedom. Echoing the protein folding process, we have previously proposed a transition state for protein-protein association. Here we clarify the specification of the transition state by working with two types of toy models for protein association. A "hemisphere" model consists of two matching hemispheres as associating proteins, and a "crater" model consists of a spherical protein with a crater to which another spherical protein fits snugly. Short-range pairwise interactions between sites across the interface hold together the bound complex. Small relative translation and rotation between the subunits quickly destroy the interactions, leading to a sharp transition between the bound state with numerous short-range interactions but restricted translation and rotational freedom and the unbound state with, at most, a small number of interactions but expanded configurational freedom. This transition sets the outer boundary of the bound state as well as the transition state for association. The energy landscape is funnel-like, with the deep well of the bound state surrounded by a broad shallow basin. Calculations with the toy models suggest that mutational change in the interaction energy in the x-ray structure of a protein-protein complex, commonly used to approximate the effect on the association constant, overestimates the effect of mutation by 10-20%. With an eye toward specifying the transition states of actual protein complexes, we find that the total number of contacts between the subunits serves as a good surrogate of the interaction energy. This formalism of protein-protein association is applied to the barnase-barstar complex, reproducing the experimental results for the association rate over a wide range of ionic strength.

Similarity and difference in the unfolding of thermophilic and mesophilic cold shock proteins studied by molecular dynamics simulations

Molecular dynamics simulations were performed to unfold a homologous pair of thermophilic and mesophilic cold shock proteins at high temperatures. The two proteins differ in just 11 of 66 residues and have very similar structures with a closed five-stranded antiparallel beta-barrel. A long flexible loop connects the N-terminal side of the barrel, formed by three strands (beta1-beta3), with the C-terminal side, formed by two strands (beta4-beta5). The two proteins were found to follow the same unfolding pathway, but with the thermophilic protein showing much slower unfolding. Unfolding started with the melting of C-terminal strands, leading to exposure of the hydrophobic core. Subsequent melting of beta3 and the beta-hairpin formed by the first two strands then resulted in unfolding of the whole protein. The slower unfolding of the thermophilic protein could be attributed to ion pair formation of Arg-3 with Glu-46, Glu-21, and the C-terminal. These ion pairs were also found to be important for the difference in folding stability between the pair of proteins. Thus electrostatic interactions appear to play similar roles in the difference in folding stability and kinetics between the pair of proteins.

Kinesin's biased stepping mechanism: amplification of neck linker zippering

A physically motivated model of kinesin's motor function is developed within the framework of rectified Brownian motion. The model explains how the amplification of neck linker zippering arises naturally through well-known formulae for overdamped dynamics, thereby providing a means to understand how weakly-favorable zippering leads to strongly favorable plus-directed binding of a free kinesin head to microtubule. Additional aspects of kinesin's motion, such as head coordination and rate-limiting steps, are directly related to the force-dependent inhibition of ATP binding to a microtubule bound head. The model of rectified Brownian motion is presented as an alternative to power stroke models and provides an alternative interpretation for the significance of ATP hydrolysis in the kinesin stepping cycle.