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The duration of path segments τ , the ‘damping fac- tor’ γ (from eqn. 1) and the target number of forward transitions per bin N must be set by the practitioner. For the method to be efficient, τ should be shorter than the shortest passage time of the rare event and longer than the timescale necessary to move from one λ -bin to another. If the dynamics can vary such that the time required to transit between λ -bins is often longer than τ , then a small value of γ (say between 0 . 01 and 0 . 5) can improve stability by stopping the adaptive sampling from attempting too many shots before it is possible that even one of them moves forward. On the other hand, in the case that τ should come near to the timescale over which the dynamics can evolve globally with respect to time then γ should be set close to 1, allowing the sampling rate to adjust rapidly. 

It is possible to dynamically vary both γ and τ if needed; for instance in a glassy material τ can be in- creased as the system arrests. If the system has a com- plex configurational space, with multiple substantially different reaction pathways all projecting onto the same values of λ , then ‘dead-end’ states may sometimes be reached, where n$_{i}$ grows out of control even for small γ and large τ . In this case, if a better λ is not available, a threshold should be defined for the largest acceptable n$_{i}$ given practical computational constraints. N should be set large enough such that it is larger than the typical fluctuations in n$_{i}$ . 

## F. Extraction of Observables and Statistics 

In order to compute expectation values of observables, we consider time slices throught the set of pathways. A configuration s from the set of configurations at time t is associated with a microscopic weight w t $_{s}$. All config- urations at the same timepoint in an S-PRES run have had the same history of external control parameters. The microscopic weight w t s of a configuration s is given by: 

using the same reasoning as eqn. 2. w t s is proportional to the estimated probability of occurrence of the config- uration at the same timepoint in a repeated experiment with the same control parameter history and distribution of initial configurations (but, obviously, with different in- dividual trajectories due to the stochastic nature of the dynamics). 

In order to extract the expected value of some observ- able x t it is required to take an average over all config- urations s at time t . The sum of the weights over all 

## E. Algorithm Parameters 

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configurations is 1. Then the time slice average is given by: 

This sample average will converge with a sufficiently large number of configurations. However, estimates of fluctuations require some care because the different con- figurations held at a given time are likely to have some de- gree of mutual information, having previously branched from a single parent or grandparent configuration, lead- ing to potentially dramatic underestimates of the vari- ance. The simplest solution to this problem is to run two completely independent calculations from different sets of starting configurations, calculating the variance at t in the second run based on the estimated mean at t from the first run (and vice-versa). Estimates of the error (as distinct from fluctuation) either in direct observables or in variances can then be achieved by making further independent runs. 

## G. Macroscopic Description of Time-Evolution 

In order to estimate the progress of the algorithm on- line, we consider the system dynamics in terms of the macroscopic coordinate λ . These dynamics will in gen- eral not be Markovian. For the following discussion, we borrow (with apologies) some mathematical notation from the language of Markov processes, but we do not imply that the dynamics in terms of λ are Markovian. 

We consider a time series of ‘macrostate vectors’ ⃗ d t of the estimated occupation probabilities of the bins, begin- ning from the initial distribution which has been chosen and progressing such that: ⃗ d ( t + τ ) = ( ⃗ d $^{t}$) $^{T}$M $^{t}$. 

We can define the entries of ⃗ d t and M t just like any other observables, as the sum of the associated weights (as from eqn. 3): 

where δ t i is 1 if the path occupies bin i at time t , oth- erwise 0. 

In the case of a Markov process, a marginalisation is carried out over ‘parent’ bins i such that d t j = ∑ i d t − τ i M t − τ i,j . Because we have time-varying control pa- rameters, and because we do not make the assumption of memorylessness, this marginalisation acquires caveats. Obviously, we must bear in mind that the estimated occu- pation probabilities depend on the history of control pa- rameters; and that they are by definition time-dependent. Less obviously, it does not hold that the marginalisation 

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