Optimal specificity and function for flexible biomolecular recognition

Biomolecular associations often accompanied by large conformational changes, sometimes folding and unfolding. By exploring an exactly solvable model, we constructed the free energy landscape and established a general framework for studying the biomolecular flexible binding process. We derived an optimal criterion for the specificity and function for flexible biomolecular binding where the binding and conformational folding are coupled.

Optimal specificity and function for flexible biomolecular recognition

Biomolecular associations often accompanied by large conformational changes, sometimes folding and unfolding. By exploring an exactly solvable model, we constructed the free energy landscape and established a general framework for studying the biomolecular flexible binding process. We derived an optimal criterion for the specificity and function for flexible biomolecular binding where the binding and conformational folding are coupled.

Configuration-dependent diffusion can shift the kinetic transition state and barrier height of protein folding

We show that diffusion can play an important role in protein-folding kinetics. We explicitly calculate the diffusion coefficient of protein folding in a lattice model. We found that diffusion typically is configuration- or reaction coordinate-dependent. The diffusion coefficient is found to be decreasing with respect to the progression of folding toward the native state, which is caused by the collapse to a compact state constraining the configurational space for exploration. The configuration- or position-dependent diffusion coefficient has a significant contribution to the kinetics in addition to the thermodynamic free-energy barrier. It effectively changes (increases in this case) the kinetic barrier height as well as the position of the corresponding transition state and therefore modifies the folding kinetic rates as well as the kinetic routes. The resulting folding time, by considering both kinetic diffusion and the thermodynamic folding free-energy profile, thus is slower than the estimation from the thermodynamic free-energy barrier with constant diffusion but is consistent with the results from kinetic simulations. The configuration- or coordinate-dependent diffusion is especially important with respect to fast folding, when there is a small or no free-energy barrier and kinetics is controlled by diffusion.Including the configurational dependence will challenge the transition state theory of protein folding.

Downhill kinetics of biomolecular interface binding: Globally connected scenario

We study the kinetics of the biomolecular binding process at the interface using energy landscape theory. The global kinetic connectivity case is considered for a downhill funneled energy landscape. By solving the kinetic master equation, the kinetic time for binding is obtained and shown to have a U-shape curve-dependence on the temperature. The kinetic minimum of the binding time monotonically decreases when the ratio of the underlying energy gap between native state and average non-native states versus the roughness or the fluctuations of the landscape increases. At intermediate temperatures,fluctuations measured by the higher moments of the binding time lead to non-Poissonian, non-exponential kinetics. At both high and very low temperatures, the kinetics is nearly Poissonian and exponential.

The complex kinetics of protein folding in wide temperature ranges

The complex protein folding kinetics in wide temperature ranges is studied through diffusive dynamics on the underlying energy landscape. The well-known kinetic chevron rollover behavior is recovered from the mean first passage time, with the U-shape dependence on temperature. The fastest folding temperature T-0 is found to be smaller than the folding transition temperature T-f. We found that the fluctuations of the kinetics through the distribution of first passage time show rather universal behavior, from high-temperature exponential Poissonian kinetics to the relatively low-temperature highly nonexponential kinetics. The transition temperature is at T-k and T-0, T-k, T-f. In certain low-temperature regimes, a power law behavior at long time emerges. At very low temperatures ( lower than trapping transition temperature T< T-0/(4&SIM;6)), the kinetics is an exponential Poissonian process again.

Probing the kinetics of single molecule protein folding

We propose an approach to integrate the theory, simulations, and experiments in protein-folding kinetics. This is realized by measuring the mean and high-order moments of the first-passage time and its associated distribution. The full kinetics is revealed in the current theoretical framework through these measurements. In the experiments, information about the statistical properties of first-passage times can be obtained from the kinetic folding trajectories of single molecule experiments ( for example, fluorescence). Theoretical/simulation and experimental approaches can be directly related. We study in particular the temperature-varying kinetics to probe the underlying structure of the folding energy landscape. At high temperatures, exponential kinetics is observed; there are multiple parallel kinetic paths leading to the native state. At intermediate temperatures, nonexponential kinetics appears, revealing the nature of the distribution of local traps on the landscape and, as a result, discrete kinetic paths emerge. At very low temperatures, exponential kinetics is again observed; the dynamics on the underlying landscape is dominated by a single barrier.

Quantifying the kinetic paths of flexible biomolecular recognition

Biomolecular recognition often involves large conformational changes, sometimes even local unfolding. The identification of kinetic pathways has become a central issue in understanding the nature of binding. A new approach is proposed here to study the dynamics of this binding-folding process through the establishment of a path-integral framework on the underlying energy landscape. The dominant kinetic paths of binding and folding can be determined and quantified. The significant coupling between the binding and folding of biomolecules often exists in many important cellular processes. In this case, the corresponding kinetic paths of binding are shown to be intimately correlated with those of folding and the dynamics becomes quite cooperative. This implies that binding and folding happen concurrently. When the coupling between binding and folding is weak (strong), the kinetic process usually starts with significant folding (binding) first, with the binding (folding) later proceeding to the end. The kinetic rate can be obtained through the contributions from the dominant paths. The rate is shown to have a bell-shaped dependence on temperature in the concentration-saturated regime consistent with experiment. The changes of the kinetics that occur upon changing the parameters of the underlying binding-folding energy landscape are studied.

Single-molecule dynamics reveals cooperative binding-folding in protein recognition

The study of associations between two biomolecules is the key to understanding molecular function and recognition. Molecular function is often thought to be determined by underlying structures. Here, combining a single-molecule study of protein binding with an energy-landscape-inspired microscopic model, we found strong evidence that biomolecular recognition is determined by flexibilities in addition to structures. Our model is based on coarse-grained molecular dynamics on the residue level with the energy function biased toward the native binding structure ( the Go model). With our model, the underlying free-energy landscape of the binding can be explored. There are two distinct conformational states at the free-energy minimum, one with partial folding of CBD itself and significant interface binding of CBD to Cdc42, and the other with native folding of CBD itself and native interface binding of CBD to Cdc42. This shows that the binding process proceeds with a significant interface binding of CBD with Cdc42 first, without a complete folding of CBD itself, and that binding and folding are then coupled to reach the native binding state.

First-passage time distribution and non-Markovian diffusion dynamics of protein folding

We study the kinetics of protein folding via statistical energy landscape theory. We concentrate on the local-connectivity case, where the configurational changes can only occur among neighboring states, with the folding progress described in terms of an order parameter given by the fraction of native conformations. The non-Markovian diffusion dynamics is analyzed in detail and an expression for the mean first-passage time (MFPT) from non-native unfolded states to native folded state is obtained. It was found that the MFPT has a V-shaped dependence on the temperature. We also find that the MFPT is shortened as one increases the gap between the energy of the native and average non-native folded states relative to the fluctuations of the energy landscape. The second- and higher-order moments are studied to infer the first-passage time distribution. At high temperature, the distribution becomes close to a Poisson distribution, while at low temperatures the distribution becomes a Levy-type distribution with power-law tails, indicating a nonself-averaging intermittent behavior of folding dynamics. We note the likely relevance of this result to single-molecule dynamics experiments, where a power law (Levy) distribution of the relaxation time of the underlined protein energy landscape is observed.

Temperature dependence of the distribution of the first passage time: Results from discontinuous molecular dynamics simulations of an all-atom model of the second beta-hairpin fragment of protein G

More than 22 000 folding kinetic simulations were performed to study the temperature dependence of the distribution of first passage time (FPT) for the folding of an all-atom Go-like model of the second beta-hairpin fragment of protein G. We find that the mean FPT (MFPT) for folding has a U (or V)-shaped dependence on the temperature with a minimum at a characteristic optimal folding temperature T-opt*. The optimal folding temperature T-opt* is located between the thermodynamic folding transition temperature and the solidification temperature based on the Lindemann criterion for the solid. Both the T-opt* and the MFPT decrease when the energy bias gap against nonnative contacts increases. The high-order moments are nearly constant when the temperature is higher than T-opt* and start to diverge when the temperature is lower than T-opt*. The distribution of FPT is close to a log-normal-like distribution at T* greater than or equal to T-opt*. At even lower temperatures, the distribution starts to develop long power-law-like tails, indicating the non-self-averaging intermittent behavior of the folding dynamics. It is demonstrated that the distribution of FPT can also be calculated reliably from the derivative of the fraction not folded (or fraction folded), a measurable quantity by routine ensemble-averaged experimental techniques at dilute protein concentrations.