In this chapter, we develop the first consistent statistical theory that allows us to infer directed biological processes by target-state-alignment. We show whether and when such a dynamic can be represented by a single SDE and how spurious forces, which inevitably arise due to target-state-alignment can be separated from genuine biological forces. To provide intuition for the effect of target-state-alignment on data we derive the universal low-noise and short-term behavior of TSA ensembles. The biophysical applicability is demonstrated for a model of cytokinetic ring constriction where many complicating factors are present which might hamper the correct identification of directional biological dynamics.
In the literature target-state-alignment is typically avoided in favor of well established but less informative approaches based on hitting time distributions57. For this analysis only the last data point from a full trajectory is used and thus most of the information in the data is discarded. Its explicit dependence on (possibly ill defined) initial conditions makes this approach further error prone. Nevertheless, target-state-alignment has been used as means of data analysis
in various fields. It has been employed to infer entropic force differences of DNA in partially blocked spaces168, or to determine the dynamics that lead up to an irrevocable decision e.g.
represented by a saccadic eye movement65. The inevitable occurrence of spurious forces in TSA ensembles however has never been noted.
With the present TSA framework one dimensional powerlaw like biological forces in an uncorrelated mean free fluctuating environment can be identified. Erroneous assignments due to inevitable alignment forces can be avoided. It turns out that TSA ensembles fall into two broad classes. For power law exponentsα >0, the biological forces vanish at the boundary and directed dynamics become indistinguishable from the dynamics of a target-state aligned random search process with its analytically determinable moments. For power law exponents α < 0, directed dynamics are force induced and reverse-time statistics can be treated as random-walk like corrections to the deterministic solution. Expanding on these results, we present small noise expressions for reverse-time mean, variance and two-time correlation function valid for force induced target state convergence. The generic analysis of noise and force induced transitions can be used to guide the eye when searching for the driving phenomenological mechanism underlying target-state aligned data. For direct high precision inference of the true biological forces, we use a path ensemble maximum likelihood scheme adopted to reverse time dynamics. We demonstrate its applicability for different scenarios of cytokinetic ring constriction and point out possible force miss-assignments when ignoring the peculiarities of TSA ensembles.
The TSA approach allows for a new perspective on the design of future experiments. It renders the search for suitable (artificially induced) initial conditions superfluous and allows to study directed dynamics under as natural circumstances as possible using a biologically mo-tivated alignment point. Nevertheless, the identification of transition times, when dynamics change e.g. from an equilibrium state into target state directed dynamics such as during cy-tokinesis, remains a biologically highly relevant question. The here sketched path ensemble framework provides a possible route to this inference problem. A further generalization of the formalism at hand might include different types of fluctuating environments, the inclusion of external variables and an extension of the framework to more than one degree of freedom. In principle all these generalizations are mathematically straight forward. At least for the case of power law like multiplicative noise analytical extensions of the TSA-theory are possible.
Concluding, the here presented theory provides the mathematical foundations for the in-ference of target-state-aligned directed dynamics. It offers an intuitive understanding for the ensemble changing effects of target-state-alignment and provides a new tool for the study of non-stationary biological dynamics.
4
The mathematical theory of target state alignment
4.1 Stochastic dynamics setting
Markov processes
A time dependent stochastic process is defined by its probability to be at its current position Lbn at time tn and by the transitions it takes to get there starting at Lb0, t0. In this chapter Lb denotes processes in forward time t. Formally such as process can be written as the transition probability P(Lbn, tn|bLn−1, tn−1, . . . ,Lb0, t0). For simplicity we assume that the current state of a stochastic process does only depend on its last step and not on its full history, i.e. that the process is memory free and obeys the Markov property
P(Lbn, tn|bLn−1, tn−1, . . . ,Lb0, t0) =P(bLn, tn|bLn−1, tn−1) . (4.1) The Markov property implies that the Chapman Kolmogorov equation
P(Lbn, tn|bLn−2, tn−2) = Z
dbLn−1P(Lbn, tn|bLn−1, tn−1)P(Lbn−1, tn−1|bLn−2, tn−2) (4.2) is fulfilled which we will use repeatedly in the subsequent derivations.
SDE and Fokker Planck representations
The evolution of a dynamical system under the influence of random forcing can often be studied using a Langevin equation
dbL(t) =f(L)b dt+√
D dWt . (4.3)
The first term defines the deterministic drift f(bL) the second the strength D of random fluc-tuations. Here the term dWt denotes the Wiener process increment with zero mean hdWti = 0 and delta correlated covariance hD dWtdWt0i=Dδ(t−t0). Throughout this text we adopt the Ito-interpretation of Eq. (4.3) which assures that the random forcing is always applied at the beginning of a discrete time interval dt.
Using Ito’s Lemma55
dg(L(t)) =b f(L)b ∂g(L)b
∂Lb +D 2
∂2g(L)b
∂Lb2
! dt+
√
D∂g(L)b
∂Lb dWt , (4.4)
which shows how to apply a change of variables for quantities that are governed by Eq. (4.3), we can derive an evolution equation for its probability distribution P(L, t|b Lb0, t0). This evolution equation is called the (forward) Fokker-Planck equation (FPE) and is an equivalent description of the stochastic dynamics described in Eq. (4.3). A derivation can be found in Gardiner55. Averaging Eq. (4.4) with respect to P(L, t|bb L0, t0), twice integrating by parts and identifying g(L) with a delta-function then leads to the seeked evolution equationb 55. The resulting partial differential equation
∂
∂tP(L, t|bb L0, t0) =− ∂
∂Lbf(L)b P(L, t|bb L0, t0) +D 2
∂2
∂Lb2P(L, t|bb L0, t0) (4.5) is the seeked (forward) Fokker-Planck equation. The adjoint equation is called the backward Fokker-Planck equation
∂
∂tP(bLf, tf|bL, t) =−f(L)b ∂
∂LbP(Lbf, tf|bL, t)−D 2
∂2
∂Lb2P(Lbf, tf|bL, t) . (4.6) and defines the evolution from an initial conditionLb to a final stateLbf. It is important to note that after relabeling both equations yield the same transition probabilities and that it is only a different perspective to solve the same problem. The term ’backward’-Fokker-Planck equation is somewhat ill-posed and can not be exchanged for ’time-reversed’. In analogy to quantum mechanics, the forward Fokker-Planck equation describes the time evolution of the probability distribution and an average of any dynamical property is taken at a specific time point t like in the Schr¨odinger picture. Conversely, the backward Fokker-Planck equation corresponds to the Heisenberg picture where the focus lies on the time evolution of a dynamical observable and the averages are taken over initial conditions. Like in quantum mechanics both approaches are thus describing exactly the same physics. They are merely offering different perspectives and can be interchanged by a simple transformation. Ultimately both approaches hence refer to the same transition probability. In the next section I show how the concept of time-reversal is to be introduced into a stochastic framework.