Constraint and contingency in multifunctional gene regulatory circuits

Abstract

Gene regulatory circuits drive the development, physiology, and behavior of organisms from bacteria to humans. The phenotypes or functions of such circuits are embodied in the gene expression patterns they form. Regulatory circuits are typically multifunctional, forming distinct gene expression patterns in different embryonic stages, tissues, or physiological states. Any one circuit with a single function can be realized by many different regulatory genotypes. Multifunctionality presumably constrains this number, but we do not know to what extent. We here exhaustively characterize a genotype space harboring millions of model regulatory circuits and all their possible functions. As a circuit's number of functions increases, the number of genotypes with a given number of functions decreases exponentially but can remain very large for a modest number of functions. However, the sets of circuits that can form any one set of functions becomes increasingly fragmented. As a result, historical contingency becomes widespread in circuits with many functions. Whether a circuit can acquire an additional function in the course of its evolution becomes increasingly dependent on the function it already has. Circuits with many functions also become increasingly brittle and sensitive to mutation. These observations are generic properties of a broad class of circuits and independent of any one circuit genotype or phenotype.

Figure 1. Schematic illustration of the Boolean model of gene regulatory circuits.

(A) A Boolean circuit with genes (a,b,c), which are represented as open circles. Two genes are connected by a directed edge if the expression of gene b is regulated by the product of gene a. Gene expression is binary, such that genes are either expressed (1) or not (0). The signal-integration logic of each gene is shown as a lookup table that explicitly maps all possible input expression states to an output expression state, implicitly determining the circuit's topology. In the hypothetical circuit shown, the expression state of gene a is independent of the expression state of gene b, so is a non-existing regulatory interaction (gray arrow), whereas and are both existing regulatory interactions (black arrows). (B) The wiring diagram and signal-integration logic of the entire circuit can be represented by a single vector G that is constructed by concatenating the rightmost columns of the lookup tables of the individual genes in panel (A). The vector G corresponds to the circuit's genotype. (C) The circuit in (A) maps all of the possible initial states (gray brackets) onto two distinct stable equilibrium expression states (black brackets). This circuit therefore can have up to functions, and can express such a “bifunction” in different ways, since 6 initial states map to one equilibrium expression state and the other 2 initial states map to another equilibrium expression state. (D) In a genotype network, vertices represent circuits and two vertices share an edge if the genotypes G differ by a single element, yet have the same functions. Here, the genotype network corresponds to circuits with the bifunction , . For visual clarity, each circle only shows the first 8 binary digits of G, which represent the signal-integration logic of gene a. Note how changes in G may implicitly translate to changes in circuit topology.

Figure 2. Multifunctional regulatory circuits.

Each data point depicts the proportion and number of genotypes with k functions. The data include all k-functions. The line is provided as a visual guide. Note that there are more circuits with function than with functions, implying that a randomly selected circuit is more likely to be viable than not. Also note that any circuit with k functions will be included in the count of the number of circuits with between 1 and functions. The inset shows the number of observed combinations of functions (open circles) and the total number of possible combinations (solid line) of k functions. Note the logarithmic scale of all y-axes.

Figure 3. Robustness and multifunctionality.

(A) The robustness of a k-function is shown in relation to the number of functions k. Each data point corresponds to the genotype set of a specific combination of k functions. The data include all k-functions. The solid line depicts the average robustness of a k-function. The inset shows the proportion and number of genotypes in the genotype set of a k-function, as a function of k. Note the logarithmic scale of the y-axes. (B–D) Distributions of genotypic robustness for (B) , (C) , and (D) . For each k, we show data for a single genotype network.

Figure 4. Genotype set fragmentation.

(A) Each data point shows the number of genotype networks in the genotype set of a specific k-function. The data include all k-functions. The solid line depicts the average number of genotype networks per k-function. The inset shows the proportion of genotype sets that comprise a single genotype network, as a function of k. (B–D) The distributions of the number of genotype networks per genotype set for (B) , (C) , and (D) .

Figure 5. Historical contingency in multifunctional regulatory circuits.

Each data point shows the proportion of combinations of k functions that exhibit contingency, as a function of k. The line is provided as a visual guide. The inset shows the average proportion of the permutations of each combination of k functions that exhibit contingency. Error bars denote one standard deviation.