Use of game-theoretical methods in biochemistry and biophysics

Abstract

Evolutionary game theory can be considered as an extension of the theory of evolutionary optimisation in that two or more organisms (or more generally, units of replication) tend to optimise their properties in an interdependent way. Thus, the outcome of the strategy adopted by one species (e.g., as a result of mutation and selection) depends on the strategy adopted by the other species. In this review, the use of evolutionary game theory for analysing biochemical and biophysical systems is discussed. The presentation is illustrated by a number of instructive examples such as the competition between microorganisms using different metabolic pathways for adenosine triphosphate production, the secretion of extracellular enzymes, the growth of trees and photosynthesis. These examples show that, due to conflicts of interest, the global optimum (in the sense of being the best solution for the whole system) is not always obtained. For example, some yeast species use metabolic pathways that waste nutrients, and in a dense tree canopy, trees grow taller than would be optimal for biomass productivity. From the viewpoint of game theory, the examples considered can be described by the Prisoner's Dilemma, snowdrift game, Tragedy of the Commons and rock-scissors-paper game.

Fig. 1

Plot of the payoffs vs. the strategies of the two players in the Prisoner’s Dilemma (a) and snowdrift game (b). Strategies are here continuously varying between two extreme values, with the origin of coordinates corresponding to the strategy pair “cooperate, cooperate”. Grey (white) planes Payoffs of player A (B). Solid thin black arrows Temptation of leaving a point by player A or B. Solid thick black arrows Resultant effects. Dashed arrows Only possibilities to leave the Nash equilibria when switching alone. These are not chosen because they decrease the payoff. Filled circles Nash equilibria. In b, the point indicated by an empty circle is (in the two-player game) unstable with respect to small fluctuations

Fig. 2

Schematic picture showing the two different pathways of ATP formation from glucose. The thickness of arrows (schematically) represents ATP production rate, while the size of the symbol ATP (schematically) represents the amount of ATP produced per mole of glucose

Fig. 3

To digest sucrose, baker’s yeast secretes the enzyme invertase, which splits sucrose into glucose and fructose outside of the cytoplasm. Cheater cells (marked Defect) do not produce invertase and benefit from the glucose resulting from the enzyme secreted by other cells (Coop). As the synthesis of invertase implies metabolic costs, cheater cells can grow faster. However, at low cell densities, the glucose concentration around cheater cells becomes so low that they grow more slowly than cooperating cells (cf. [18])

Fig. 4

Illustration of the game of tree growth. a Schematic representation of a tree canopy in which all but one tree have reached an optimal height. The selfish outlier tree is taller and invests a higher percentage of biomass into supporting structure (stem). Although it can thus invest less into other parts, it has a competitive advantage because it absorbs sunlight (shown as two arrows) that would otherwise reach the neighbouring trees. Therefore, the other trees are forced to grow taller as well or are out-competed (cf. [16]). Δh Height difference. b Determining the evolutionarily stable tree height. The parabola-like curve shows the empirically determined dependence of productivity, p (biomass production per time) on tree height, h. Assuming that one tree is taller than all neighbouring trees by Δh, this gives it an advantage in productivity of Δp, which is approximately independent of h itself. The dashed straight lines have the slope Δp/Δh, that is, they depict the gain in productivity by growing taller. The solid straight lines represent the net effect. The evolutionarily stable height, h*, is reached when this net effect is zero due to investing more into supporting structures, that is, when the straight line is horizontal. Clearly, h* is larger than the optimal height, h_opt