Humans make a number of choices when they walk, such as how fast and for how long. The preferred steady walking speed seems chosen to minimize energy expenditure per distance traveled. But the speed of actual walking bouts is not only steady, but rather a time-varying trajectory, which can also be modulated by task urgency or an individual’s movement vigor. Here we show that speed trajectories and durations of human walking bouts are explained better by an objective to minimize Energy and Time, meaning the total work or energy to reach destination, plus a cost proportional to bout duration. Applied to a computational model of walking dynamics, this objective predicts dynamic speed vs. time trajectories with inverted U shapes. Model and human experiment (N=10) show that shorter bouts are unsteady and dominated by the time and effort of accelerating, and longer ones are steadier and faster and dominated by steady-state time and effort. Individual-dependent vigor may be characterized by the energy one is willing to spend to save a unit of time, which explains why some may walk faster than others, but everyone may have similar-shaped trajectories due to similar walking dynamics. Tradeoffs between energy and time costs can predict transient, steady, and vigor-related aspects of walking.

Cycling of co-substrates, whereby a metabolite is converted among alternate forms via different reactions, is ubiquitous in metabolism. Several cycled co-substrates are well known as energy and electron carriers (e.g. ATP and NAD(P)H), but there are also other metabolites that act as cycled co-substrates in different parts of central metabolism. Here, we develop a mathematical framework to analyse the effect of co-substrate cycling on metabolic flux. In the cases of a single reaction and linear pathways, we find that co-substrate cycling imposes an additional flux limit on a reaction, distinct to the limit imposed by the kinetics of the primary enzyme catalysing that reaction. Using analytical methods, we show that this additional limit is a function of the total pool size and turnover rate of the cycled co-substrate. Expanding from this insight and using simulations, we show that regulation of these two parameters can allow regulation of flux dynamics in branched and coupled pathways. To support these theoretical insights, we analysed existing flux measurements and enzyme levels from the central carbon metabolism and identified several reactions that could be limited by the dynamics of co-substrate cycling. We discuss how the limitations imposed by co-substrate cycling provide experimentally testable hypotheses on specific metabolic phenotypes. We conclude that measuring and controlling co-substrate dynamics is crucial for understanding and engineering metabolic fluxes in cells.