Activation thresholds set the speed of biochemical trigger waves

Trigger waves are positive feedback-mediated traveling waves that propagate without slowing down or diminishing in amplitude, enabling biochemical coordination over distances where diffusion alone would be insufficient. Apoptosis can spread as trigger waves of caspase activity, as demonstrated in Xenopus eggs and egg extracts. We measured apoptotic wave speeds in extracts at two cell cycle states and found that waves propagated more slowly in M-phase arrested cytostatic factor (CSF) extracts than in interphase extracts (21 versus 27 m min-1). Luther's equation, v=2{surd}D{kappa}, predicts wave speed from the diffusion coefficient D and the positive-feedback doubling rate {kappa}. However, D and {kappa} were comparable between the two extracts and could not explain the speed difference. This discrepancy highlighted a key limitation of the classical equation: it does not account for an activation threshold in trigger wave initiation, and in fact the threshold required to trigger apoptosis was found to differ between cell cycle states. We therefore extended Luther's equation to include such a threshold, obtaining v{propto}{surd}(AD/Uc ), where A is the reaction rate and Uc is the activation threshold. Perturbing apoptotic sensitivity confirmed the predicted threshold-speed relationship. These findings establish the activation threshold as a key determinant of trigger wave speed and provide a broadly applicable scaling law for predicting the spread of switch-like biochemical signals.