Motivation: Protein structure alignment is one of the most important computational problems in molecular biology and plays a key role in protein structure prediction, fold family classification, motif finding, phylogenetic tree reconstruction and so on. From the viewpoint of computational complexity, a pairwise structure alignment is also a NP-hard problem, in contrast to the polynomial time algorithm for a pairwise sequence alignment. Results: We propose a method for solving the structure alignment problem in an accurate manner at the amino acid level, based on a mean field annealing technique. We define the structure alignment as a mixed integer-programming (MIP) problem. By avoiding complicated combinatorial computation and exploiting the special structure of the continuous partial problem, we transform the MIP into a reduced non-linear continuous optimization problem (NCOP) with a much simpler form. To optimize the reduced NCOP, a mean field annealing procedure is adopted with a modified Potts model, whose solution is generally identical to that of the MIP. There is no 'soft constraint' in our mean field model and all constraints are automatically satisfied throughout the annealing process, thereby not only making the optimization more efficient but also eliminating many unnecessary parameters that depend on problems and usually require careful tuning. A number of benchmark examples are tested by the proposed method with comparisons to several existing approaches.

All cell components exhibit intracellular noise due to random births and deaths of individual molecules, and extracellular noise due to environment perturbations. In particular, gene regulation is an inherently noisy process with transcriptional control, alternative splicing, translation, diffusion and chemical modification reactions, which all involve stochastic fluctuations. Such stochastic noises may not only affect the dynamics of the entire system but may also be exploited by living organisms to actively facilitate certain functions, such as cooperative behavior and communication. Results: We have provided a general model and an analytic tool to examine the cooperative behavior of a multi-cell system with both intracellular and extracellular stochastic fluctuations. A multi-cell system with a synthetic gene network is adopted to demonstrate the effects of noises and coupling on collective dynamics. These results establish not only a theoretical foundation but also a quantitative basis for understanding essential roles of noises on cooperative dynamics, such as synchronization and communication among cells.

An effective method of chaotification via time-delay feedback for a simple finite-dimensional continuous-time autonomous system is made rigorous in this paper. Some mathematical conditions are derived under which a nonchaotic system can be controlled to become chaotic, where the chaos so generated is in a rigorous mathematical sense of Li-Yorke in terms of the Marotto theorem. Numerical simulations are given to verify the theoretical analysis. Chaos has been found useful lately in various areas of science, engineering, and technology. Therefore, purposefully generating chaos (called chaotification, or anticontrol of chaos) has investigated rather intensively in the past few years. Recently, Wang, Chen, and Yu [Chaos 10, 771-779 (2000)] developed an anticontrol method via time-delay feedback for chaotifying a continuous-time dynamical system. The fundamental idea of this anticontrol method is correct and insightful, but a time-delay differential equation used therein is only approximated by a related discrete map, leaving some room for improvement. To present a mathematically rigorous approach, this paper adopts the same anticontrol idea but further improves its technical contents thereby deriving a similar yet rigorous design method for chaotificaiton. A rather general continuous-time system can be driven from nonchaotic to chaotic by using time-delay feedback perturbation on the system parameters or employing an exogenous time-delay state-feedback input, where the generated chaos is in a precise mathematical sense of Li-Yorke in terms of the Marotto theorem.

In-phase waves and echo waves are commonly existing in coupled systems. In Physica D 151 (2001) 199 we proved existence of echo waves in linearly coupled oregonators and showed that the echo waves may coexist with the stable homogeneous positive steady state. This Letter develops a different approach (so-called "reducing dimension") compared with our "mode decomposition" method of stability of in-phase waves in Phys. Lett. A 301 (2002) 231, and rigorously proves stability of the echo waves. The method can be easily applied to other similar systems.

Tyson [Ann. NY Acad. Sci. 316 (1979) 279] conjectured that the stable homogeneous positive steady state may coexist with stable echo wave (meaning anti-phase wave) in linearly coupled Oregonators (and thus gave a conjecture on the bifurcation diagram of this system). In this paper, we rigorously prove stability of the in-phase wave and existence of the anti-phase wave. Our proof procedure actually gives a general method (or line) to deal with the analogous problem. For instance, to prove stability of the in-phase wave, following our line one may decompose the corresponding variational equations (a four-dimensional system) into two independent planar systems; also for instance, existence of the anti-phase wave can be concluded as existence and uniqueness of limit cycle of the associate oscillator. In addition, according to parameter regimes of existence of the anti-phase wave and the stable homogeneous positive steady state, we give their coexistence regime and specify it, and in particular give the regime of the coupled coefficient. The specified results show that the theoretical results are in good accord with Tyson's numerical results.