The interactions between proteins, DNA, and RNA in living cells constitute molecular networks that govern various cellular functions. To investigate the global dynamical properties and stabilities of such networks, we studied the cell-cycle regulatory network of the budding yeast. With the use of a simple dynamical model, it was demonstrated that the cell-cycle network is extremely stable and robust for its function. The biological stationary state, the G1 state, is a global attractor of the dynamics. The biological pathway, the cell-cycle sequence of protein states, is a globally attracting trajectory of the dynamics. These properties are largely preserved with respect to small perturbations to the network. These results suggest that cellular regulatory networks are robustly designed for their functions.

We show that dynamical systems with spatial degrees of freedom naturally evolve into a self-organized critical point. Flicker noise, or l/f noise, can be identified with the dynamics of the critical state. This picture also yields insight into the origin of fractal objects.

Protein structures in nature often exhibit a high degree of regularity (for examlple, secondary structure and tertiary symmetries) that is absent from random compact conformations. With the use of a simple lattice model of protein folding, it was demonstrated that structural regularities are related to high "designability" and evolutionary stability. The designability of each compact structure is measured by the number of sequences that can design the structure-that is, sequences that possess the structure as their nondegenerate ground state. Compact structures differ markedly in terms of their designability; highly designable. These highly designable structures possess "proteinlike", secondary structure and even tertiary symmetries. In addition, they are thermodynamically more stable than other structures. These results suggest that protein structures are selected in nature because they are readily designed and stable against mutations, and that such a selection simultaneously leads to thermodymamic stability.

In a statistical approach to protein structure analysis, Miyazawa and Jernigan derived a 20×20 matrix of inter-residue contact energies between different types of amino acids. Using the method of eigenvalue decomposition, we find that the Miyazawa-Jernigan matrix can be accurately reconstructed from its first two principal component vectors as Mij=C0＋C1(qi＋qj)＋C2qiqj, with constant C's, and 20q values associated with the 20 amino acids. This regularity is due to hydrophobic interactions and a force of demixing, the latter obeying Hildebrand's solubility theory of simple liquids.

Protein structures are a very special class among all possible structures. It has been suggested that a "designability principle" plays a crucial role in nature's selection of protein sequences and structures. Here, we provide a theoretical base for such a selection principle, using a simple model of protein folding based on hydrophobic interactions. A structure is reduced to a string of 0s and 1s, which represent the surface and core sites, respectively, as the backbone is traced. Each structure is therefore associated with one point in a high dimensional space. Sequences are represented by strings of their hydrophobicities and thus can be mapped into the same space. A sequence that lies closer to a particular structure in this space than to any other structures will have that structure as its ground state. Atypical structures, namely those far away from other structures in the high dimensional space, have more sequences that fold into them and are thermodynamically more stable. We argue that the most common folds of proteins are the most atypical in the space of possible structures.