Cells generally adapt to environmental changes by first exhibiting an immediate response and then gradually returning to their original state to achieve homeostasis. Although simple network motifs consisting of a few genes have been shown to exhibit such adaptive dynamics, they do not reflect the complexity of real cells, where the expression of a large number of genes activates or represses other genes, permitting adaptive behaviors. Here, we investigated the responses of gene regulatory networks containing many genes that have undergone numerical evolution to achieve high fitness due to the adaptive response of only a single target gene; this single target gene responds to changes in external inputs and later returns to basal levels. Despite setting a single target, most genes showed adaptive responses after evolution. Such adaptive dynamics were not due to common motifs within a few genes; even without such motifs, almost all genes showed adaptation, albeit sometimes partial adaptation, in the sense that expression levels did not always return to original levels. The genes split into two groups: genes in the first group exhibited an initial increase in expression and then returned to basal levels, while genes in the second group exhibited the opposite changes in expression. From this model, genes in the first group received positive input from other genes within the first group, but negative input from genes in the second group, and vice versa. Thus, the adaptation dynamics of genes from both groups were consolidated. This cooperative adaptive behavior was commonly observed if the number of genes involved was larger than the order of ten. These results have implications in the collective responses of gene expression networks in microarray measurements of yeast Saccharomyces cerevisiae and the significance to the biological homeostasis of systems with many components.

Microorganisms often perform chemotaxis, (i.e., sensing and moving toward a region with a higher concentration of an attractive chemical) by changing the rate of tumbling for random walk. We studied several models with internal adaptive dynamics numerically to examine the validity of the condition for chemotaxis proposed by Oosawa and Nakaoka [1], which states that the time scale of tumbling frequency is smaller than that of adaptation and greater than that of sensing. Suitably renormalizing the timescales showed that the condition holds for a variety of environments and for both short-and long-term behavior.

Gene expression dynamics satisfying given input-output relationships were investigated by evolving the networks for an optimal response. We found three types of networks and corresponding dynamics, depending on the sensitivity of gene expression dynamics: direct response with straight paths, amplified response by a feed-forward network, and cooperative response with a complex network. When the sensitivity of each gene's response is low and expression dynamics is sloppy, the last type is selected, in which many genes respond collectively to inputs, with local-excitation and global-inhibition structures. The result provides an insight into how a reliable response is achieved with unreliable units, and on why complex networks with many genes are adopted in cells.

We have numerically studied chemotactic aggregation of microorganisms by introducing a model consisting of elements with intracellular dynamics, random walks with a state-dependent turnover rate, and secretion of attractant. Three phases with and without aggregation, as well as partial aggregation, were obtained as to the diffusion and degradation rates of the attractant, and conditions for cellular aggregation were analyzed. The size of aggregated clusters was shown to be independent of cell density, as is consistent with experiment.

Adaptation to environmental change is a common property of biological systems. Cells initially respond to external changes in the environment, but after some time, they regain their original state. By considering an element consisting of two variables that show such adaptation dynamics, we studied a coupled dynamical system containing such elements to examine the diverse dynamics in the system and classified the behaviors on the basis of the network structure that determined the interaction among elements. For a system with two elements, two types of behaviors, perfect adaptation and simple oscillation, were observed. For a system with three elements, in addition to these two types, novel types of dynamics, namely, rapid burst-type oscillation and a slow cycle, were discovered; depending on the initial conditions, these novel types of dynamics coexisted. These behaviors are a result of the characteristic dynamics of each element, i.e., fast response and slow adaptation processes. The behaviors depend on the network structure (in specific, a combination of positive or negative feedback among elements). Cooperativity among elements due to a positive feedback loop leads to simple oscillation, whereas frustration involving alternating positive and negative interactions among elements leads to the coexistence of rapid bursting oscillation and a slow cycle. These behaviors are classified on the basis of the frustration indices defined by the network structure. The period of the slow cycle is much longer than the original adaptation time scale, while the burst-type oscillation is a continued response that does not involve any adaptation. We briefly discuss the universal applicability of our results to a network of a larger number of elements and their possible relevance to biological systems.