Over the past three decades, investigators of nonlinear systems have
revealed that even the simplest laws of nature may lead to bewilderingly
complex dynamics. Therefore, to understand the underlying principles and
make predictions of a complex system, we need to develop and apply the
cutting-edge tools in mathematics and physics such as dynamical systems
theory and nonequilibrium statistical physics, not only to reveal the
universal features which are largely independent of the details of specific
systems and thus can be studied with common mathematical tools, but also
to disclose characteristic properties of each individual system which
do depend on the concrete form of interactions between components. Henceforth,
an effective study of complex systems requires both mastery of novel probing
tools and good knowledge of systems that are of interest.
1. Signal transduction, cell migration, systems biology.
With the advent of high throughput experiments, the analysis and understanding of enormous amount of accumulated data relies more and more on the mathematical modeling. Even when the function of each individual parts of a cell is clear, to get a complete picture and thus to exert effective control of a biological process or to design a cellular device with preferred properties, a study of component interactions and their overall manifestation is indispensable. Theoretical and experimental efforts in this direction are incorporated into one newly ermergent subject "systems biology".
I started research on these cellular processes by simulating stochastic biochemical reactions which could deviate significantly from the deterministic chemical kinetics. I have developed effective tools for a computational approach to stochastic signal transduction and as well discovered a novel phenomenon in noisy signal propagation - resonant signaling. More recently, I stochastically modeled the growth of a filopodium which is an important apparatus in cell migration. The model I proposed gives a full stochastic treatment of the actin monomer diffusion and polymerization of each individual actin filament subject to fluctuating membrane forces. The study justified those assumptions used in the previous deterministic models in the literature and identified the key parameters that control the filopodial growth.
From a theoretical point of view, the major difficulty of current modeling relies on the high heterogeneity and vast space and time scale separation involved in life activities of biological organisms. Qualitative analysis and quantitative computational tools are needed to break the complexity. For the stochastic modeling, the spatial consideration has just become a new challenge.
2. Graph theoretic analysis and evolution analysis of cell reaction networks
Current theoretical analysis of cell regulation networks mainly concentrates on small pathways separated from the intricate reaction web embedded inside a cell. However, analyzing realistic complex biochemical networks is a must to gain insights into the emergent regulation and control of cell life. Due to the continuous experimental effort of revealing details of cell regulation networks, it has become increasingly hard to identify the underlying characteristic structures and thus gain insights into the key mechanisms that shape the network function. How to decompose a large scale network into manipulatible parts and integrate modeling efforts of many different groups into one big picture is becoming a more and more urgent while challenging problem.
Graph theoretic analysis combined with dynamical systems tools can be efficiently applied to large-scale networks in chemistry and biology, and enable a hierarchical dissection of complex networks into subnetworks of different sizes. Future application of this modular analysis technique could go far beyond biological networks. Based on a production-control conception, I designed an automatic procedure to identify the key functional units of cell regulation networks through graph theoretic method and dynamical systems analysis and applied it to different cell regulatory networks. The detection of modular structures provides valuable insight into how a regulatory network works and thus gives clear indication of key protein species and key reactions in a cascade, which finds important applications in the drug design and synthetic biology.
3. Pattern formation, coherent structures and spatiotemporal chaos in fluids and complex systems.
A great variety of patterns are observed in nature and in people's everyday life. They constitute important features of different objects for our recognition and on the other hand display certain characteristics of internal dynamics of the objects. But their physical study started only recently. One old and interesting subject in which to look for patterns is fluid motion. Although Navier-Stokes equation describes many fluids well, its understanding and solution still haunts even the smartest heads of mathematicians and physicists. The recently developed dynamical systems theory may open a door for novel approaches. As first steps towards the Everest, many less formidable equations have been proposed for study.
To probe orbit structures of a spatially extended system in its phase space, I began with the Complex Ginzburg-Landau equation which is a generic amplitude equation in the investigation of instabilities of a spatiotemporally oscillating system. Using a novel reformulation, I proved the existence of the modulated amplitude waves (MAWs) and their stabilities, which is the very first step to unraveling its rich dynamics ranging from traveling waves to spatiotemporal chaos. Then 1-d Kuramoto-Sivashinsky equation is studied which is widely used in modeling combustion, fluid mixing, crystal growth and surface flows. I looked in depth into its solution in the weakly turbulent regime by establishing symbolic dynamics on the unstable manifolds of stationary points and studying the Poincare return maps on some carefully chosen hyper-surface. The work is not only substantial for understanding the dynamics of the Kuramoto-Sivashinsky system but also bears the significance of one important step to apply the low dimensional nonlinear theory to engineering or biological systems with many degrees of freedom. During my searching for recurrent patterns, I designed an innovative Newton descent method, based on the variational calculus. It is especially good for finding unstable periodic orbits or coherent structures in high-dimensional space like those appearing in the PDE pattern formation studies.
Challenges are all similar in these spatially extended systems: their phase spaces usually have very large dimension (infinite theoretically). For a qualitative understanding, it seems always necessary to persue a dimension reduction. Phase and amplitude equations work well when system size is not big. For large systems, to identify and characterize patterns and coherent structures is probably the best way to model. When system size approaches infinity and reaches the scaling regime, statistical description perhaps works best. By extracting the important coherent structures, from experimental or numerical time series, periodic orbit theory could be invoked to evaluate their contribution to important physical averages. Another interesting and important extension of the work is concerned with the noisy spatially extended systems.
4. Non-equilibrium statistical mechanics, stochastic processes, field theory, nonlinear and complex dynamics, semiclassical calculations.
Mathematically, all these different topics can be unified by the concept of dynamical systems. The statistical physics emphasizes the measure theoretic aspects of a system, which takes measurement either on an invariant set or in the long time or the large space limit. So the SRB measure and thermodynamic formalism serve as a bridge to connect dynamical sytems theory and physics. For the unknown or omitted degrees of freedom, sometimes a stochastic processes is used to take their effects into account, which irons out the singular or sharp distributions on the fractal sheets of strange attractors produced by many nonlinear systems. If the noise is weak, the deterministic orbit still sits at the ridge of the stochastic evolution and various perturbation techniques could be invoked to carry out effective computations. This is especially useful in the semiclassical approximation of quantum systems where the Planck constant is a natural parameter for expansions though here it is the probability amplitude that enters the Schodinger equation instead of the probability itself. In the large quantum number limit, classical orbits dominate the motion of the system and semiclassical approximation becomes very accurate. Surprisingly, in the periodic orbit theory calculation of the Helium atom, the spectrum is computed within several percent accuracy all the way down to the ground state, which provides the hope that semiclassical approximation probably can provide fast and precise computation for large quantum systems like macromolecules that perform important functions in cells. Spatial degrees of freedom involved in a stochastic process make a system equivalent to a quantum field in some sense. The effects of fluctuations in these systems are still in the fontier of research and more theoretical tools need to be developed to understand the spatial correlations and temporal evolution. A more realistic system that bears much similarity to the quantum field is large-scale networks with random dynamics taking place at each of its nodes. In fact, the type of dynamics could be much richer in a network since very heterogeneous coupling may be chosen among different nodes. The uncertainty propagation, the activation front spreading, the synchronization and the phase transition on these networks constitute a vast area for theoretical investigations and practical applications.
5. Exact solutions, renormalization group approach to nonlinear differential equations and complex systems.
There is an old saying: mathematics is the crown of natural science and number theory is the gem on the crown because mathematics provides a solid foundation for the precise characterization of all sciences and number theory identifies the most striking relations of the simple counting numbers - the integers. I would like to propose an analogue here: good insightful models make the crown of theoretical physics and exact solutions form the diamonds on the crown. Almost all the interesting theoretical discoveries are based on interesting models or idealized experements like the solar system model, hydrogen atom model, Ising model, harmonic oscillators, eight-fold way, Young-Mills model,... The exact solutions in these models provide us with a clear way to understand them and to make generalizations. In my opinion, the exact solution is a coincidence rather than a rule but it miraculously combines all the information coherently in a clean mathematical expression. It is a shortcut to an understanding if appropriately utilized and sometimes implies unexpected symmetries. I am very interested in finding exact solutions for differential equations or intesting physical models.
Very often, exact solutions come up hand in hand with symmetries or invariances. Renormalization group reveals how certain universal invariances emerge out of repeated transformations by coarsening interactions among agents. It is an important way to understand how a hierarchy of orders appear from elementary interactions and to analyze appearance and disappearance of order and disorder in complex systems, like the above mentioned large scale networked systems.
6. Algorithms, computer simulations, multi-scale analysis and computational biology and physics.
With the rapid increase of computer capacity, it is not surprising that numerical computation has become major means of scientific research. Good algorithms are magnifiers of computer power. Which methods are available for a particular problem, which is the best one among them and how a complex operation is decomposed to elementary steps are all interesting and important questions for computer simulation of physical or biological systems. In my conception, however, efficient numerical computation should reflect and lead to better physical understanding. For complex systems, multi-scale analysis is especially effective in this regard. It tries to understand each level individually and then puts them together to map out the overall effects. The difficulty lies in the partition of the levels and defining effective interactions between them in a natural way. The effort for searching invariance properties and achieving systematic reduction as explained above may nucleate a way for delimiting different levels and deducing effective interactions within each higher level and between levels. New computational tools will be tested on interesting biological or physical models.