Protein search for targets on DNA starts all major biological processes. Although significant experimental and theoretical efforts have been devoted to investigation of these phenomena, mechanisms of protein-DNA interactions during the search remain not fully understood. One of the most surprising observations is known as a speed-selectivity paradox. It suggests that experimentally observed fast findings of targets require smooth protein-DNA binding potentials, while the stability of the specific protein-DNA complex imposes a large energy gap which should significantly slow down the protein molecule. We developed a discrete-state stochastic approach that allowed us to investigate explicitly target search phenomena and to analyze the speed-selectivity paradox. A general dynamic phase diagram for different search regimes is constructed. The effect of the target position on search dynamics is investigated. Using experimentally observed parameters, it is found that slow protein diffusion on DNA does not lead to an increase in the search times. Thus, our theory resolves the speed-selectivity paradox by arguing that it does not exist. It is just an artifact of using approximate continuum theoretical models for analyzing protein search in the region of the parameter space beyond the range of validity of these models. In addition, the presented method, for the first time, provides an explanation for fast target search at the level of single protein molecules. Our theoretical predictions agree with all available experimental observations, and extensive Monte Carlo computer simulations are performed to support analytical calculations.

Formation of protrusions and protein segregation on the membrane is of a great importance for the functioning of the living cell. This is most evident in recent experiments that show the effects of the mechanical properties of the surrounding substrate on cell morphology. We propose a mechanism for the formation of membrane protrusions and protein phase separation, which may lay behind this effect. In our model, the fluid cell membrane has a mobile but constant population of proteins with a convex spontaneous curvature. Our basic assumption is that these membrane proteins represent small adhesion complexes, and also include proteins that activate actin polymerization. Such a continuum model couples the membrane and protein dynamics, including cell-substrate adhesion and protrusive actin force. Linear stability analysis shows that sufficiently strong adhesion energy and actin polymerization force can bring about phase separation of the membrane protein and the appearance of protrusions. Specifically, this occurs when the spontaneous curvature and aggregation potential alone (passive system) do not cause phase separation. Finite-size patterns may appear in the regime where the spontaneous curvature energy is a strong factor. Different instability characteristics are calculated for the various regimes, and are compared to various types of observed protrusions and phase separations, both in living cells and in artificial model systems. A number of testable predictions are proposed.

One of the most important features of biological systems that controls their functioning is the ability of protein molecules to find and recognize quickly specific target sites on DNA. Although these phenomena have been studied extensively, detailed mechanisms of protein-DNA interactions during the search are still not well understood. Experiments suggest that proteins typically find their targets fast by combining three-dimensional and one-dimensional motions, and most of the searching time proteins are non-specifically bound to DNA. However these observations are surprising since proteins diffuse very slowly on DNA, and it seems that the observed fast search cannot be achieved under these conditions for single proteins. Here we propose two simple mechanisms that might explain some of these controversial observations. Using first-passage time analysis, it is shown explicitly that the search can be accelerated by changing the location of the target and by effectively irreversible dissociations of proteins. Our theoretical predictions are supported by Monte Carlo computer simulations.

In multiple-front solutions of the Burgers equation, all the fronts, except for two, are generated through the inelastic interaction of exponential wave solutions of the Lax pair associated with the

This work proposes a new way for handling obstacles to asymptotic integrability in perturbed nonlinear PDE's within the method of Normal Forms (NF) for the case of multi-wave solutions. Instead of including the whole obstacle in the NF, only its resonant part (if one exists) is included in the NF, and the remainder is assigned to the homological equation. This leaves the NF integrable and its solutions retain the character of the solutions of the unperturbed equation.We exploit the freedom in the expansion to construct canonical obstacles which are confined to the interaction region of the waves. For soliton solutions (e.g., in the KdV equation), the interaction region is a finite domain around the origin; the canonical obstacles then do not generate secular terms in the homological equation. When the interaction region is infinite (or semi-infinite, e.g. in wavefront solutions of the Burgers equation), the obstacles may contain resonant terms.The obstacles generate waves of a new type which cannot be written as functionals of the solutions of the NF. When the obstacle contributes a resonant term to the NF, this leads to a non-standard update of the wave velocity.