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Finite time convergence to functionally important target states is a keycomponent of many biological processes. We previously found that the terminalapproach phase of such dynamics exhibits universal types of stochastic dynamicsthat differ qualitatively between noise-dominated and force-dominated regimesof the approach dynamics. While for the noise-dominated regime the approachdynamics is uninformative about the underlying force law, in theforce-dominated regime it enables the accurate inference of the underlyingdynamics. Biological systems often exhibit substantial parameter heterogeneity,for instance through copy number fluctuations of key molecules or variabilityin modulating factors. Here, we extend our theory of target state aligned (TSA)stochastic dynamics to investigate the impact of parameter heterogeneity in theunderlying stochastic dynamics. We examine the approach to target states for awide range of dynamical laws and additive as well as multiplicative noise. Wefind that the distinct regimes of noise-dominated and force-dominated dynamicsstrongly differ in their sensitivity to parameter heterogeneity. In thenoise-dominated regime, TSA ensembles are insensitive to parameterheterogeneity in the force law, but sensitive to sample to sample heterogeneityin the diffusion constant. For force-dominated dynamics, both parameterheterogeneity in the force law and diffusion constant change the behaviour ofthe non-stationary statistics and in particular the two-time-covariancefunctions. In this regime, TSA ensembles provide a sensitive readout ofparameter heterogeneity. Under natural conditions, parameter heterogeneity inmany biological systems cannot be experimentally controlled or eliminated. Ourresults provide a systematic theoretical foundation for the analysis of targetstate directed dynamics in a large class of systems with substantialheterogeneity.

Signatures of heterogeneity in the statistical structure of target state aligned ensembles