Rate of Insulin Dimer Dissociation: Interplay between Memory Effects and Higher Dimensionality

Dimerization of model polymer chains under nonequilibrium conditions

Dimerization and subsequent aggregation of polymers and biopolymers often occur under nonequilibrium conditions. When the initial state of the polymer is not collapsed, or the final folded native state, the dynamics of dimerization can follow a course sensitive to both the initial conditions and the conformational dynamics. Here, we study the dimerization process by using computer simulations and analytical theory, where the two monomeric polymer chains are in the elongated state and are initially placed at a separation distance, d0. Subsequent dynamics lead to the concurrent processes of collapse, dimerization, and/or escape. We employ Langevin dynamics simulations with a coarse-grained model of the polymer to capture certain aspects of the dimerization process. At separations d0 much shorter than the length of the monomeric polymer, the dimerization could happen fast and irreversibly from the partly extended polymer state itself. At an initial separation larger than a critical distance, dc, the polymer collapse precedes dimerization, and a significant number of single polymers do not dimerize within the time scale of simulations. To quantify these competitions, we introduce several time-dependent order parameters, namely, (i) the time-dependent radius of gyration RG(t) of individual polymers describing the conformational state of the polymer, (ii) a center-to-center of mass distance parameter RMM, and (iii) a time dependent overlap function Q(t) between the two monomeric polymers, mimicking the contact order parameter popular in protein folding. In order to better quantify the findings, we perform a theoretical analysis to capture the stochastic processes of collapse and dimerization by using the dynamical disorder model.

Analysis of the Dynamics of a Complex, Multipathway Reaction: Insulin Dimer Dissociation

The protein hormone insulin forms a homodimer that must dissociate to bind to its receptor. Understanding the kinetics and mechanism of dissociation is essential for the rational design of therapeutic analogs. In addition to its physiological importance, this dissociation process serves as a paradigm for coupled (un)folding and (un)binding. Based on previous free energy simulations, insulin dissociation is thought to involve multiple pathways with comparable free energy barriers. Here, we analyze the mechanism of insulin dimer dissociation using a recently developed computational framework for estimating kinetic statistics from short-trajectory data. These statistics indicate that the likelihood of dissociation (the committor) closely tracks the decrease in the number of (native and nonnative) intermonomer contacts and the increase in the number of water contacts at the dimer interface; the transition state with equal likelihood of association and dissociation corresponds to an encounter complex with relatively few native contacts and many nonnative contacts. We identify four pathways out of the dimer state and quantify their contributions to the rate as well as their exchange by computing reactive fluxes. We show that both the pathways and their extents of exchange can be understood in terms of rotations around three axes of the dimer structure. Our results provide insights into the kinetics of insulin analogs and, more generally, how to characterize complex, multipathway processes.

Analysis of the dynamics of a complex, multipathway reaction: Insulin dimer dissociation

The protein hormone insulin forms a homodimer that must dissociate to bind to its receptor. Understanding the kinetics and mechanism of dissociation is essential for rational design of therapeutic analogs. In addition to its physiological importance, this dissociation process serves as a paradigm for coupled (un)folding and (un)binding. Based on previous free energy simulations, insulin dissociation is thought to involve multiple pathways with comparable free energy barriers. Here, we analyze the mechanism of insulin dimer dissociation using a recently developed computational framework for estimating kinetic statistics from short-trajectory data. These statistics indicate that the likelihood of dissociation (the committor) closely tracks the decrease in the number of (native and nonnative) intermonomer contacts and the increase in the number of water contacts at the dimer interface; the transition state with equal likelihood of association and dissociation corresponds to an encounter complex with relatively few native contacts and many nonnative contacts. We identify four pathways out of the dimer state and quantify their contributions to the rate, as well as their exchange, by computing reactive fluxes. We show that both the pathways and their extents of exchange can be understood in terms of rotations around three axes of the dimer structure. Our results provide insights into the kinetics of insulin analogues and, more generally, how to characterize complex, multipathway processes.

Diffusion in a two-dimensional energy landscape in the presence of dynamical correlations and validity of random walk model

Diffusion in a multidimensional energy surface with minima and barriers is a problem of importance in statistical mechanics and it also has wide applications, such as protein folding. To understand it in such a system, we carry out theory and simulations of a tagged particle moving on a two-dimensional periodic potential energy surface, both in the presence and absence of noise. Langevin dynamics simulations at multiple temperatures are carried out to obtain the diffusion coefficient of a solute particle. Friction is varied from zero to large values. Diffusive motion emerges in the limit of a long time, even in the absence of noise. Noise destroys the correlations and increases diffusion at small friction. Diffusion thus exhibits a nonmonotonic friction dependence at the intermediate value of the damping, ultimately converging to our theoretically predicted value. The latter is obtained using the well-established relationship between diffusion and random walk. An excellent agreement is obtained between theory and simulations in the high-friction limit but not so in the intermediate regime. We explain the deviation in the low- to intermediate-friction regime using the modified random walk theory. The rate of escape from one cell to another is obtained from the multidimensional rate theory of Langer. We find that enhanced dimensionality plays an important role. To quantify the effects of noise on the potential-imposed coherence on the trajectories, we calculate the Lyapunov exponent. At small friction values, the Lyapunov exponent mimics the friction dependence of the rate.

Counteractive Effects of Choline Geranate (CAGE) ILs and Ethanol on Insulin's Stability-A Leap Forward towards Oral Insulin Formulation

Choline geranate (CAGE) ionic liquids (ILs) stabilize insulin, thereby aiding its oral delivery, whereas ethanol (EtOH) affects its stability by disrupting the hydrophobic interactions. In this study, cognizance of the stabilization mechanism of insulin dimer in the presence of both CAGE ILs and EtOH mixtures is achieved through biased and unbiased molecular dynamics (MD) simulations. Here, two order parameters are employed to study the insulin dimer dissociation using well-tempered metadynamics (WT-MetaD). The stability of insulin is found to be strongly maintained until a 0.20 mole fraction of EtOH. Besides, higher concentrations of EtOH marginally affect the insulin stability. Moreover, geranate anions form a higher number of H-bonding interactions with water molecules, which aids insulin stabilization. Conversely, the addition of EtOH minimizes the water-mediated H-bonding interactions of geranate. Additionally, geranate traps the EtOH molecules, thereby preventing the interactions between insulin and EtOH. Furthermore, the free energy landscape (FEL) reveals the absence of dimer dissociation along with noticeable deviations in the distances R and the number of contacts Q. The dimerization free energy of insulin was calculated to be −16.1 kcal/mol at a 0.20 mole fraction of EtOH. Moreover, increments in mole fractions of EtOH effectuate a decrease in the insulin stability. Thus, the present study represents CAGE ILs as efficient insulin dimer stabilizes at low concentrations of EtOH.

Effects of Hydrodynamic Backflow on the Transmission Coefficient of a Barrier-Crossing Brownian Particle

The slow power law decay of the velocity autocorrelation function of a particle moving stochastically in a condensed-phase fluid is widely attributed to the momentum that fluid molecules displaced by the particle transfer back to it during the course of its motion. The forces created by this backflow effect are known as Basset forces, and they have been found in recent analytical work and numerical simulations to be implicated in a number of interesting dynamical phenomena, including boosted particle mobility in tilted washboard potentials. Motivated by these findings, the present paper is an investigation of the role of backflow in thermally activated barrier crossing, the governing process in essentially all condensed-phase chemical reactions. More specifically, it is an exact analytical calculation, carried out within the framework of the reactive-flux formalism, of the transmission coefficient κ(t) of a Brownian particle that crosses an inverted parabola under the influence of a colored noise process originating in the Basset force and a Markovian time-local friction. The calculation establishes that κ(t) is significantly enhanced over its backflow-free limit.

Progress in Simulation Studies of Insulin Structure and Function

Insulin is a peptide hormone known for chiefly regulating glucose level in blood among several other metabolic processes. Insulin remains the most effective drug for treating diabetes mellitus. Insulin is synthesized in the pancreatic β-cells where it exists in a compact hexameric architecture although its biologically active form is monomeric. Insulin exhibits a sequence of conformational variations during the transition from the hexamer state to its biologically-active monomer state. The structural transitions and the mechanism of action of insulin have been investigated using several experimental and computational methods. This review primarily highlights the contributions of molecular dynamics (MD) simulations in elucidating the atomic-level details of conformational dynamics in insulin, where the structure of the hormone has been probed as a monomer, dimer, and hexamer. The effect of solvent, pH, temperature, and pressure have been probed at the microscopic scale. Given the focus of this review on the structure of the hormone, simulation studies involving interactions between the hormone and its receptor are only briefly highlighted, and studies on other related peptides (e.g., insulin-like growth factors) are not discussed. However, the review highlights conformational dynamics underlying the activities of reported insulin analogs and mimetics. The future prospects for computational methods in developing promising synthetic insulin analogs are also briefly highlighted.

Non-Markovian rate theory on a multidimensional reaction surface: Complex interplay between enhanced configuration space and memory

A theory of barrier crossing rate on a multidimensional reaction energy surface is presented. The theory is a generalization of the earlier theoretical schemes to higher dimensions, with the inclusion of non-Markovian friction along both the reactive and the nonreactive coordinates. The theory additionally includes the bilinear coupling between the reactive and the nonreactive modes at the Hamiltonian level. Under suitable conditions, we recover the rate expressions of Langer and Hynes and establish a connection with the rate treatment of Pollak. Within the phenomenology of generalized Langevin equation description, our formulation provides an improvement over the existing ones because we explicitly include both the non-Markovian effects along the reaction coordinate and the bilinear coupling at the Hamiltonian level. At intermediate-to-large friction, an increase in dimensionality by itself tends to reduce the rate, while the inclusion of the memory effects increases the rate. The theory predicts an increase in rate when off-diagonal friction terms are included. We present a model calculation to study isomerization of a stilbene-like molecule using the prescription of Hochstrasser and co-workers on a two-dimensional reaction energy surface, employing Zwanzig–Bixon hydrodynamic theory of frequency-dependent friction. The calculated rate shows a departure from the predictions of Langer’s theory and also from the two-dimensional transition state theory.

Tug-of-War between Internal and External Frictions and Viscosity Dependence of Rate in Biological Reactions

The role of water in biological processes is studied in three reactions, namely, the Fe-CO bond rupture in myoglobin, GB1 unfolding, and insulin dimer dissociation. We compute both internal and external components of friction on relevant reaction coordinates. In all of the three cases, the cross-correlation between forces from protein and water is found to be large and negative that serves to reduce the total friction significantly, increase the calculated reaction rate, and weaken solvent viscosity dependence. The computed force spectrum reveals bimodal 1/f noise, suggesting the use of a non-Markovian rate theory.

Nonlinear Reaction Coordinate of an Enzyme Catalyzed Proton Transfer Reaction

We present an in-depth study on the theoretical calculation of an optimum reaction coordinate as a linear or nonlinear combination of important collective variables (CVs) sampled from an ensemble of reactive transition paths for an intramolecular proton transfer reaction catalyzed by the enzyme human carbonic anhydrase (HCA) II. The linear models are optimized by likelihood maximization for a given number of CVs. The nonlinear models are based on an artificial neural network with the same number of CVs and optimized by minimizing the root-mean-square error in comparison to a training set of committor estimators generated for the given transition. The nonlinear reaction coordinate thus obtained yields the free energy of activation and rate constant as 9.46 kcal mol^-1 and 1.25 × 10⁶ s^-1, respectively. These estimates are found to be in quantitative agreement with the known experimental results. We have also used an extended autoencoder model to show that a similar analysis can be carried out using a single CV only. The resultant free energies and kinetics of the reaction slightly overestimate the experimental data. The implications of these results are discussed using a detailed microkinetic scheme of the proton transfer reaction catalyzed by HCA II.

Particle dynamics in viscoelastic media: Effects of non-thermal white noise on barrier crossing rates

The growing interest in the dynamics of self-driven particle motion has brought increased attention to the effects of non-thermal noise on condensed phase diffusion. Thanks to data recently collected by Ferrer et al. on activated dynamics in the presence of memory [Phys. Rev. Lett. 126, 108001 (2021)], some of these effects can now be characterized quantitatively. In the present paper, the data collected by Ferrer et al. are used to calculate the extent to which non-thermal white noise alters the time taken by single micron-sized silica particles in a viscoelastic medium to cross the barrier separating the two wells of an optically created bistable potential. The calculation-based on a generalized version of Kramers's flux-over-population approach-indicates that the added noise causes the barrier crossing rate (compared to the noise-free case) to first increase as a function of the noise strength and then to plateau to a constant value. The precise degree of rate enhancement may depend on how the data from the experiments conducted by Ferrer et al. are used in the flux-over-population approach. As claimed by Ferrer et al., this approach predicts barrier crossing times for the original silica-fluid system that agree almost perfectly with their experimental counterparts. However, this near-perfect agreement between theory and experiment is only achieved if the theoretical crossing times are obtained from the most probable values of a crossing time distribution constructed from the distributions of various parameters in Kramers's rate expression. If the mean values of these parameters are used in the expression instead, as would be commonly done, the theoretical crossing times are found to be as much as 1.5 times higher than the experimental values. However, these times turn out to be consistent with an alternative model of viscoelastic barrier crossing based on a mean first passage time formalism, which also uses mean parameter values in its rate expression. The rate enhancements predicted for barrier crossing under non-thermal noise are based on these mean parameter values and are open to experimental verification.

Structural Stability of Insulin Oligomers and Protein Association-Dissociation Processes: Free Energy Landscape and Universal Role of Water

Association and dissociation of proteins are important biochemical events. In this Feature Article, we analyze the available studies of these processes for insulin oligomers in aqueous solution. We focus on the solvation of the insulin monomer in water, stability and dissociation of its dimer, and structural integrity of the hexamer. The intricate role of water in solvation of the dimer- and hexamer-forming surfaces, in long-range interactions between the monomers and the stability of the oligomers, is discussed. Ten water molecules inside the central cavity stabilize the structure of the insulin hexamer. We discuss how different order parameters can be used to understand the dissociation of the insulin dimer. The calculation of the rate using a recently computed multidimensional free energy provides considerable insight into the interplay between protein and water dynamics.