Production and consumption of a <i>species-1</i> cell in a co-culture.

<p><b>(a)</b> Population equilibrium under autarky (no trade). The blue arrow represents the direction of increasing utility. Consumption of the metabolites in the ratio lead to growth. The <i>indifference curves</i> are contour lines of the utility function: any two consumption vectors on the same indifference curve lead to the same growth rate. The solid red line represents the possible production vectors of <i>metabolite-1</i> and <i>metabolite-2</i>. The intersection of the blue arrow with the consumption-set line is the equilibrium consumption vector for which utility is maximized. When there is no trade the consumption and production sets are equivalent. The cell can only consume what it produces. <b>(b)</b> Schematic of cells under autarky. Arrows represent the flow of resources during production and consumption of red and blue metabolites. The arrow width illustrates the utilization of each pathway. The gear size indicates pathway productivity. Cells need to allocate more input resources to produce metabolites for which it has lower productivity, given that utilization requirements are about equal. <b>(c)</b> Population equilibrium under trade where the consumption set is expanded (solid red line). The consumption set line shifts upward by the amount <i>species-1</i> imports from <i>species-2</i>, . The increased magnitude of the slope of the consumption set line indicates that a fraction of <i>metabolite-1</i> produced is exported to <i>species-2</i>. The slope is given by . The new equilibrium consumption vector and utility level is greater under trade than autarky implying higher growth rates. <b>(d)</b> Schematic of trading cells. A cell can allocate most of its input resources to produce metabolites for which it has high productivity when exchange occurs.