Stuart A. Kauffman

Metabolic stability and epigenesis in randomly constructed genetic nets

Abstract Proto-organisms probably were randomly aggregated nets of chemical reactions. The hypothesis that contemporary organisms are also randomly constructed molecular automata is examined by modeling the gene as a binary (on-off) device and studying the behavior of large, randomly constructed nets of these binary “genes”. The results suggest that, if each “gene” is directly affected by two or three other “genes”, then such random nets: behave with great order and stability; undergo behavior cycles whose length predicts cell replication time as a function of the number of genes per cell; possess different modes of behavior whose number per net predicts roughly the number of cell types in an organism as a function of its number of genes; and under the stimulus of noise are capable of differentiating directly from any mode of behavior to at most a few other modes of behavior. Cellular differentation is modeled as a Markov chain among the modes of behavior of a genetic net. The possibility of a general theory of metabolic behavior is suggested.

Random Boolean network models and the yeast transcriptional network

The recently measured yeast transcriptional network is analyzed in terms of simplified Boolean network models, with the aim of determining feasible rule structures, given the requirement of stable solutions of the generated Boolean networks. We find that, for ensembles of generated models, those with canalyzing Boolean rules are remarkably stable, whereas those with random Boolean rules are only marginally stable. Furthermore, substantial parts of the generated networks are frozen, in the sense that they reach the same state, regardless of initial state. Thus, our ensemble approach suggests that the yeast network shows highly ordered dynamics.

Coevolution to the Edge of Chaos: Coupled Fitness Landscapes, Poised States, and Coevolutionary Avalanches

We introduce a broadened framework to study aspects of coevolution based on the NK class of statistical models of rugged fitness landscapes. In these models the fitness contribution of each of N genes in a genotype depends epistatically on K other genes. Increasing epistatic interactions increases the rugged multipeaked character of the fitness landscape. Coevolution is thought of, at the lowest level, as a coupling of landscapes such that adaptive moves by one player deform the landscapes of its immediate partners. In these models we are able to tune the ruggedness of landscapes, how richly intercoupled any two landscapes are, and how many other players interact with each player. All these properties profoundly alter the character of the coevolutionary dynamics. In particular, these parameters govern how readily coevolving ecosystems achieve Nash equilibria, how stable to perturbations such equilibria are, and the sustained mean fitness of coevolving partners. In turn, this raises the possibility that an evolutionary metadynamics due to natural selection may sculpt landscapes and their couplings to achieve coevolutionary systems able to coadapt well. The results suggest that sustained fitness is optimized when landscape ruggedness relative to couplings between landscapes is tuned such that Nash equilibria just tenuously form across the ecosystem. In this poised state, coevolutionary avalanches appear to propagate on all length scales in a power law distribution. Such avalanches may be related to the distribution of small and large extinction events in the record.

Genetic networks with canalyzing Boolean rules are always stable

We determine stability and attractor properties of random Boolean genetic network models with canalyzing rules for a variety of architectures. For all power law, exponential, and flat in-degree distributions, we find that the networks are dynamically stable. Furthermore, for architectures with few inputs per node, the dynamics of the networks is close to critical. In addition, the fraction of genes that are active decreases with the number of inputs per node. These results are based upon investigating ensembles of networks using analytical methods. Also, for different in-degree distributions, the numbers of fixed points and cycles are calculated, with results intuitively consistent with stability analysis; fewer inputs per node implies more cycles, and vice versa. There are hints that genetic networks acquire broader degree distributions with evolution, and hence our results indicate that for single cells, the dynamics should become more stable with evolution. However, such an effect is very likely compensated for by multicellular dynamics, because one expects less stability when interactions among cells are included. We verify this by simulations of a simple model for interactions among cells.

Autocatalytic replication of polymers

Abstract We construct a simplified model for the chemistry of molecules such as polypeptides or single stranded nucleic acids, whose reactions can be restricted to catalyzed cleavage and condensation. We use this model to study the spontaneous emergence of autocatalytic sets from an initial set of simple building blocks, for example short strands of amino acids or nucleotides. When the initial set exceeds a critical diversity, autocatalytic reactions generate large molecular species in abundance. Our results suggest that the critical diversity is not very large. Autocatalytic sets formed in this way can be regarded as primitive connected metabolisms, in which particular species are selected if their chemical properties are advantageous for the metabolism. Such autocatalytic sets may have played a crucial role in the origin of life, providing a bridge from simple molecular species to complex proteins and nucleic acids. Many of our results are experimentally testable.

Emergent properties in random complex automata

Abstract Studies of large, randomly assembled binary (Boolean) node automata have demonstrated that such systems can spontaneously exhibit enormously ordered dynamical behavior. An important approach to characterizing these behaviors has consisted of studying ensembles of automata. Ensemble specifications have been based on: 1) choice of Boolean function regulating each node in the automaton; 2) random or biased mappings of the 2 N automata states into themselves; 3) the numbers of inputs, K , per node. This article briefly reviews these approaches, reports new results on the expected behavior of threshold automata, and automata with a particular class of biased mappings. In addition, it discusses emergent, highly ordered dynamical behavior in random automata rich in a specific class of “canalizing” Boolean functions due to the crystallization of powerful subautomata called forcing structures. The ordered behaviors may be of importance in biological evolution, in physics, and in the design of adaptive automata.

Gene expression dynamics in the macrophage exhibit criticality

Cells are dynamical systems of biomolecular interactions that process information from their environment to mount diverse yet specific responses. A key property of many self-organized systems is that of criticality: a state of a system in which, on average, perturbations are neither dampened nor amplified, but are propagated over long temporal or spatial scales. Criticality enables the coordination of complex macroscopic behaviors that strike an optimal balance between stability and adaptability. It has long been hypothesized that biological systems are critical. Here, we address this hypothesis experimentally for system-wide gene expression dynamics in the macrophage. To this end, we have developed a method, based on algorithmic information theory, to assess macrophage criticality, and we have validated the method on networks with known properties. Using global gene expression data from macrophages stimulated with a variety of Toll-like receptor agonists, we found that macrophage dynamics are indeed critical, providing the most compelling evidence to date for this general principle of dynamics in biological systems.

Critical Dynamics in Genetic Regulatory Networks: Examples from Four Kingdoms

The coordinated expression of the different genes in an organism is essential to sustain functionality under the random external perturbations to which the organism might be subjected. To cope with such external variability, the global dynamics of the genetic network must possess two central properties. (a) It must be robust enough as to guarantee stability under a broad range of external conditions, and (b) it must be flexible enough to recognize and integrate specific external signals that may help the organism to change and adapt to different environments. This compromise between robustness and adaptability has been observed in dynamical systems operating at the brink of a phase transition between order and chaos. Such systems are termed critical. Thus, criticality, a precise, measurable, and well characterized property of dynamical systems, makes it possible for robustness and adaptability to coexist in living organisms. In this work we investigate the dynamical properties of the gene transcription networks reported for S. cerevisiae, E. coli, and B. subtilis, as well as the network of segment polarity genes of D. melanogaster, and the network of flower development of A. thaliana. We use hundreds of microarray experiments to infer the nature of the regulatory interactions among genes, and implement these data into the Boolean models of the genetic networks. Our results show that, to the best of the current experimental data available, the five networks under study indeed operate close to criticality. The generality of this result suggests that criticality at the genetic level might constitute a fundamental evolutionary mechanism that generates the great diversity of dynamically robust living forms that we observe around us.

The production recipes approach to modeling technological innovation: An application to learning by doing

We do two things in this paper. First, we put forward some elements of a microeconomic theory of technological evolution. This involves adding nascent (essentially undiscovered) technologies to the existing technologies of neoclassical production theory, and, more importantly, expanding the notion of the production plan to include the recipe---the complete description of the underlying engineering process. Second, we use the recipes approach in constructing a simple microeconomic model of shop-floor learning by doing. We simulate the dynamics of the model and report the effects of changes in the basic parameters on the resulting engineering experience curves. For correctly chosen values of these parameters, the predictions of the model match the observed data on experience curves. Journal of Economic Literature Classification Numbers: D20, D21, D24, D83, L23, O30. Submitted to Journal of Economic Dynamics and Control.

A General Modeling Strategy for Gene Regulatory Networks with Stochastic Dynamics

A stochastic genetic toggle switch model that consists of two identical, mutually repressive genes is built using the Gillespie algorithm with time delays as an example of a simple stochastic gene regulatory network. The stochastic kinetics of this model is investigated, and it is found that the delays for the protein productions can highly weaken the global fluctuations for the expressions of the two genes, making the two mutually repressive genes coexist for a long time. Starting from this model, we propose a practical modeling strategy for more complex gene regulatory networks. Unlike previous applications of the Gillespie algorithm to simulate specific genetic networks dynamics, this modeling strategy is proposed for an ensemble approach to study the dynamical properties of these networks. The model allows any combination of gene expression products, forming complex multimers, and each one of the multimers is assigned to a randomly chosen gene promoter site as an activator or inhibitor. In addition, each gene, although it has only one promoter site, can have multiple regulatory sites and distinct rates of translation and transcription. Also, different genes have different time delays for transcription and translation and all reaction constant rates are initially randomly chosen from a range of values. Therefore, the general strategy here proposed may be used to simulate real genetic networks.