Exploring the Origin of Power Law Distribution in Single-Molecule Conformation Dynamics: Energy Landscape Perspectives

We explored the origin of power law distribution observed in single-molecule conformational dynamics experiments. By establishing a kinetic master equation approach to study statistically the microscopic state dynamics, we show that the underlying landscape with exponentially distributed density of states leads to power law distribution of kinetics. The exponential density of states emerges when the system becomes glassy and landscape becomes rough with significant trapping.

Potential energy landscape and robustness of a gene regulatory network: Toggle switch

Finding a multidimensional potential landscape is the key for addressing important global issues, such as the robustness of cellular networks. We have uncovered the underlying potential energy landscape of a simple gene regulatory network: a toggle switch. This was realized by explicitly constructing the steady state probability of the gene switch in the protein concentration space in the presence of the intrinsic statistical fluctuations due to the small number of proteins in the cell. We explored the global phase space for the system. We found that the protein synthesis rate and the unbinding rate of proteins to the gene were small relative to the protein degradation rate; the gene switch is monostable with only one stable basin of attraction. When both the protein synthesis rate and the unbinding rate of proteins to the gene are large compared with the protein degradation rate, two global basins of attraction emerge for a toggle switch. These basins correspond to the biologically stable functional states. The potential energy barrier between the two basins determines the time scale of conversion from one to the other. We found as the protein synthesis rate and protein unbinding rate to the gene relative to the protein degradation rate became larger, the potential energy barrier became larger. This also corresponded to systems with less noise or the fluctuations on the protein numbers.

Quantifying robustness and dissipation cost of yeast cell cycle network: The funneled energy landscape perspectives

We study the origin of robustness of yeast cell cycle cellular network through uncovering its underlying energy landscape. This is realized from the information of the steady-state probabilities by solving a discrete set of kinetic master equations for the network. We discovered that the potential landscape of yeast cell cycle network is funneled toward the global minimum, G1 state. The ratio of the energy gap between G1 and average versus roughness of the landscape termed as robustness ratio ( RR) becomes a quantitative measure of the robustness and stability for the network. The funneled landscape is quite robust against random perturbations from the inherent wiring or connections of the network. There exists a global phase transition between the more sensitive response or less self-degradation phase leading to underlying funneled global landscape with large RR, and insensitive response or more self-degradation phase leading to shallower underlying landscape of the network with small RR. Furthermore, we show that the more robust landscape also leads to less dissipation cost of the network. Least dissipation and robust landscape might be a realization of Darwinian principle of natural selection at cellular network level. It may provide an optimal criterion for network wiring connections and design.

Dominant kinetic paths on biomolecular binding-folding energy landscape

The identification of kinetic pathways is a central issue in understanding the nature of flexible 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. 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. The kinetic time can be obtained through the contributions from the dominant paths and has a U-shape dependence on temperature.

Diffusion and single molecule dynamics on biomolecular interface binding energy landscape

We propose a new approach to study the diffusion dynamics on biomolecular interface binding energy landscape. The resulting mean first passage time (MFPT) has 'U'curve dependence on the temperature. It is shown that the large specificity ratio of gap to roughness of the underlying binding energy landscape not only guarantees the thermodynamic stability and the specificity [P.A. Rejto, G.M. Verkhivker, in: Proc. Natl. Acad. Sci. 93 (1996) 8945; C.J. Tsai, S. Kumar, B. Ma, R. Nussinov, Protein Sci. 8 (1999) 1181; G.A. Papoian, P.G. Wolynes, Biopolymers 68 (2003) 333; J. Wang, G.M. Verkhivker, Phys. Rev. Lett. 90 (2003) 198101] but also the kinetic accessibility. The complex kinetics and the associated fluctuations reflecting the structures of the binding energy landscape emerge upon temperature changes. The theory suggests a way of connecting the models/simulations with single molecule experiments by analysing the kinetic trajectories.

Energy landscape theory, funnels, specificity, and optimal criterion of biomolecular binding

We study the nature of biomolecular binding. We found that in general there exists several thermodynamic phases: a native binding phase, a non-native phase, and a glass or local trapping phase. The quantitative optimal criterion for the binding specificity is found to be the maximization of the ratio of the binding transition temperature versus the trapping transition temperature, or equivalently the ratio of the energy gap of binding between the native state and the average non-native states versus the dispersion or variance of the non-native states. This leads to a funneled binding energy landscape.

Diffusion dynamics, moments, and distribution of first-passage time on the protein-folding energy landscape, with applications to single molecules

We study the dynamics of protein folding via statistical energy-landscape theory. In particular, we concentrate on the local-connectivity case with the folding progress described by the fraction of native conformations. We found that the first passage-time (FPT) distribution undergoes a dynamic transition at a temperature below which the FPT distribution develops a power-law tail, a signature of the intermittent nonexponential kinetic phenomena for the folding dynamics. Possible applications to single-molecule dynamics experiments are discussed.

Potential Landscape and Flux Framework of Nonequilibrium Networks: Robustness, Dissipation, and Coherence of Biochemical Oscillations

We established a theoretical framework for studying nonequilibrium networks with two distinct natures essential for characterizing the global probabilistic dynamics: the underlying potential landscape and the corresponding curl flux. We applied the idea to a biochemical oscillation network and found that the underlying potential landscape for the oscillation limit cycle has a distinct closed ring valley (Mexican hat-like) shape when the fluctuations are small. This global landscape structure leads to attractions of the system to the ring valley.

Robustness and dissipation of mitogen-activated protein kinases signal transduction network: Underlying funneled landscape against stochastic fluctuations

We uncovered the underlying energy landscape of the mitogen-activated protein kinases signal transduction cellular network by exploring the statistical natures of the Brownian dynamical trajectories. We introduce a dimensionless quantity: The robustness ratio of energy gap versus local roughness to measure the global topography of the underlying landscape. A high robustness ratio implies funneled landscape. The landscape is quite robust against environmental fluctuations and variants of the intrinsic chemical reaction rates.

Robustness and Coherence of a Three-Protein Circadian Oscillator: Landscape and Flux Perspectives

Three-protein circadian oscillations in cyanobacteria sustain for weeks. To understand how cellular oscillations function robustly in stochastic fluctuating environments, we used a stochastic model to uncover two natures of circadian oscillation: the potential landscape related to steady-state probability distribution of protein concentrations; and the corresponding flux related to speed of concentration changes which drive the oscillations. The barrier height of escaping from the oscillation attractor on the landscape provides a quantitative measure of the robustness and coherence for oscillations against intrinsic and external fluctuations. The difference between the locations of the zero total driving force and the extremal of the potential provides a possible experimental probe and quantification of the force from curl flux. These results, correlated with experiments, can help in the design of robust oscillatory networks.

Funneled landscape leads to robustness of cellular networks: MAPK signal transduction

We uncover the underlying potential energy landscape for a cellular network. We find that the potential energy landscape of the mitogen-activated protein-kinase signal transduction network is funneled toward the global minimum. The funneled landscape is quite robust against random perturbations. This naturally explains robustness from a physical point of view. The ratio of slope versus roughness of the landscape becomes a quantitative measure of robustness of the network. Funneled landscape is a realization of the Darwinian principle of natural selection at the cellular network level. It provides an optimal criterion for network connections and design. Our approach is general and can be applied to other cellular networks.

Atomistic Origin Of Rate-Dependent Serrated Plastic Flow In Metallic Glasses

Nanoindentation simulations on a binary metallic glass were performed under various strain rates by using molecular dynamics. The rate-dependent serrated plastic flow was clearly observed, and the spatiotemporal behavior of its underlying irreversible atomic rearrangement was probed. Our findings clearly validate that the serration is a temporally inhomogeneous characteristic of such rearrangements and not directly dependent on the resultant shear-banding spatiality. The unique spatiotemporal distribution of shear banding during nanoindentation is highlighted in terms of the potential energy landscape (PEL) theory.

Intrinsic correlation between fragility and bulk modulus in metallic glasses

A systematic study on the available data of 26 metallic glasses shows that there is an intrinsic correlation between fragility of a liquid and bulk modulus of its glass. The underlying physics can be rationalized within the formalism of potential energy landscape thermodynamics. It is surprising to find that the linear correlation between the fragility and the bulk-shear modulus ratio exists strictly at either absolute zero temperature or very high frequency. Further analyses indicate that a real flow event in bulk metallic glasses is shear dominant, and fragility is in inverse proportion to shear-induced bulk dilatation. Finally, extension of these findings to nonmetallic glasses is discussed.

Kinetics and Statistical Distributions of Single-Molecule Conformational Dynamics

We developed a coarse-grained yet microscopic detailed model to study the statistical fluctuations of single-molecule protein conformational dynamics of adenylate kinase. We explored the underlying conformational energy landscape and found that the system has two basins of attractions, open and closed conformations connected by two separate pathways. The kinetics is found to be nonexponential, consistent with single-molecule conformational dynamics experiments. Furthermore, we found that the statistical distribution of the kinetic times for the conformational transition has a long power law tail, reflecting the exponential density of state of the underlying landscape. We also studied the joint distribution of the two pathways and found memory effects.

Quantifying intrinsic specificity: A potential complement to affinity in drug screening

We report here the investigation of a novel description of specificity in protein-ligand binding based on energy landscape theory. We define a new term, intrinsic specificity ratio (ISR), which describes the level of discrimination in binding free energies of the native basin for a protein-ligand complex from the weaker binding states of the same ligand. We discuss the relationship between the intrinsic specificity we defined here and the conventional definition of specificity. In a docking study of molecules with the enzyme COX-2, we demonstrate a statistical correspondence between ISR value and geometrical shapes of the small molecules binding to COX-2. We further observe that the known selective (nonselective) inhibitors of COX-2 have higher (lower) ISR values. We suggest that intrinsic specificity ratio may be a useful new criterion and a complement to affinity in drug screening and in searching for potential drug lead compounds.