Biological processes are often well represented within the mathematical framework of stochastic dynamics. In chapter 2, I provide a quick review on time-dependent stochastic processes and their inference from biological data. Due to a revolution in the fluorescence based imaging technology, today, the development of whole organisms can be imaged in 3d and in single cell resolution. After a brief review on fluorescence microscopy, I conclude the introductory chapter with a recapitulation on the stochastic dynamics literature on which my derivation of the TSA framework builds.
TSA ensemble analysis provides a powerful tool for model analysis, inference and distinction.
It allows to study directional dynamics irrespective of any unknown initial conditions and thus without perturbation of the system under study. In chapter 3, I introduce the key concepts nec-essary to understand the mathematical construction of TSA ensembles. I discuss the universal limit of force and noise driven dynamics close to the target state.
Chapter 4 can be read as an independent building block where the key results of reverse time TSA inference, introduced in chapter 3, are derived rigorously step by step. I provide analytical
results for reverse time TSA ensemble dynamics and compare them to forward simulations after target state alignment.
With the intuition formed by these results, I show in chapter 5 the applicability of this ap-proach to a simple model of cytokinetic ring constriction. I provide the mathematical framework to melt the TSA approach with an ensemble path inference scheme with empathis on numerical stability.
Chapter 6 starts with an introduction on cell intercalation as an important building block of the evolutionary preserved phenomenon of convergent extension. It is followed by a review on the current biological understanding of contact constriction during germband extension and how such dynamics are currently modeled using biophysical intuition. The intuition about TSA ensembles invoked in chapter 3 is then used to infer the most simple effective model of cell contact constriction during germband elongation in Drosophila morphogenesis. Guided by the experimental evidence the theory is stepwise extended to explore whether a multiplicative noise model, memory in the noise or heterogeneity in the dynamics is capable to explain the observed time-dependent statistics. This chapter concludes with preliminary results on myosin dynamics during contact constriction and how these dynamics can quantitatively be linked to the junction dynamics.
In its scope, chapter 7 goes beyond the results used in chapter 3 - 6. I show, that Langevin bridges, i.e. stochastic dynamics with delta initial and final conditions can equivalently be stud-ied in forward and reverse time. While identical as an ensemble they differ however in their dynamics. I further show how meander processes, i.e. processes, which up to time t do not return to their initial conditions, can be constructed additively from Langevin bridges. While probably known to some experts, both connections have to my knowledge not been linked in a publication. When ever possible, analytical results for reverse time TSA dynamics are obtained and compared to forward simulations after target state alignment.
2
Introduction
2.1 Stochastic dynamics in biological systems
Biological systems are comprised of a multitude of interacting components spanning several orders of magnitude. Starting from the very small scales of individual molecules, via protein complexes to cells, membranes and whole tissues, one observes noisy fluctuations. Rooted in thermal fluctuations on the small scales, are the cause of fluctuations on larger scale often active, energy consuming processes and the response of their environment. Independent of the details and the exact mechanisms, a prevalent observation for biological systems is a non negligible noise level on all biological relevant scales. In biological systems constitutive components are typically densely packed. Individual molecules and higher order structures do not move independently, but continuously interact with each other. These dynamical interactions lead to an almost instantaneous dissipation of accelerated dynamics at the typical time scales of observations.
Dynamics are thus almost exclusively observed in the overdamped limit.
This heuristic description of biological systems can be formalized in the framework of statis-tical physics. In this framework each particle is characterized with its position and momentum.
The whole system is summarized with its Hamiltonian, comprised of the kinetic and potential contributions of all particles. The Hamilton equations of motion describe the dynamics.
While a biological system with all its details is in principle described by its Hamiltonian, are the dynamics of interest, typically summarizable as a collective phenomenon, well represented by one or a few degrees of freedom. This idea has been exploited in a mathematically involved formalisms which separates the effective dynamics of interest from the residual system9,115,117,185. This residual degrees of freedom then act as a stochastic bath in which the effective dynamics evolve. For most biological systems it is furthermore well justified to assume that accelerating forces have dissipated on the time scale of the observation due to the dominant viscous damping of the surrounding. Assuming in a last simplifying step that the noisy fluctuations occur on a smaller timescale than the effective dynamics of interest, one arrives at a simple representation of effective stochastic biological dynamics. This equation is called an (overdamped) Langevin equation
dbL(t) =f(L)b dt+
√
D dWt , (2.1)
here stated for a single dynamical degree of freedom L. It is composed of a deterministic driftb f(L) and a stochastic forcingb dWtterm, which for simplicity, is independent ofL. The stochasticb forcing is called a Wiener increment dWt and defines a zero mean hdWti = 0, delta correlated hdWtdWt0i =δ(t−t0) Gaussian process. The noise strength is given by the diffusion constant D. With the Langevin equation, a sample path representation of stochastic biological dynamics, reduced to the core elements of an effective force and a stochastic component, is achieved.
In the limit of infinitely many sample paths, generated by Eq. (2.1) and equipped with a com-mon initial distribution, the resulting ensemble can equivalently be described by the (forward) Fokker-Planck equation (FPE)
∂
∂tP(L, t|bb L0, t0) =− ∂
∂Lbf(L)b P(bL, t|bL0, t0) +D 2
∂2
∂Lb2P(L, t|bb L0, t0) , (2.2) whereP(L, t|bb L0, t0) denotes a with time evolving probability density conditioned on the initial condition (bL0, t0). The first term denotes the drift and the second the diffusion. Both terms are in their effect identical to the respective terms in the Langevin equation. For example setting f(L) = 0 and assuming vanishing probabilities atb ±∞one finds
P(L, t|bb L0, t0) = 1
√
2πDte−(
L−bb L0)2
2Dt , (2.3)
which is the continuous representation of diffusion under Gaussian white noise.
The evolution of the transition probability can equivalently be obtained from 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) , (2.4) here denoted for transitions to the final state (Lbf, tf). It is important to note that solving the backward Fokker-Planck equation yields the very same transition probability as the forward Fokker-Planck equation. It does not(!) describe dynamics in reverse time.
In principle, both the Langevin and Fokker-Planck equation can capture the two recurrent themes of biological dynamics. That is, biological dynamics are typically (i) out of equilibrium and (ii) they change over time. To study the temporal evolution of an ensemble, within the Fokker-Planck approach a common initial reference time must be available to which all sample paths align. With well defined initial conditions temporal ensemble properties such as mean and variance can be determined. However, even in in-vitro systems, it is often a technical challenge to synchronize the observed dynamics (or find a common reference onset for independently recorded sample paths), to yield an ensemble of observations with identical temporal onset.
Giving up on a temporal description, but not on the non-equilibrium properties of biological systems, a natural simplifying assumption replaces the time dependent evolution of the FPE by a constant flux. The classical realization of this state is provided by constant flux boundary conditions, where the influx and outflux of the system are identical. A for the main text important generalization of this approach, lifts the confinement between two boundaries and replaces the insertion boundary by a source term with insertion probabilityPin(L). Under thisb assumption the FPE Eq. (2.2) with absorbing boundary atLbts simplifies to
−λPin(L) =b − ∂
∂Lb
f(L)Q(b L)b +D
2
∂2
∂Lb2Q(bL)
Q(Lbts) = 0, (2.5)
whereQ(L) denotes the non-equilibrium stationary state (NESS) density andb λ= D2 ∂
∂LbQ(Lbts) ensures that influx and outflux are balanced. The differential operator on the right hand side of Eq. (2.5) is identical to the right hand side of the forward FPE, while the temporal evolution on the left hand side is replaced by a constant in time. For the NESS densityQ(L) an analyticalb expression has been obtained182. We will recapitulate this solution and its derivation in the main text. An interesting interpretation of this approach is provided in the Langevin, that is
sample path picture. Sample paths which start at Aprogress over time until they terminate at B and are instantaneously reinserted at A. This approach thus only gives up on a global wall clock time to which all sample paths align but fully preserves the non-equilibrium character of many biological systems. In the next section, I will briefly discuss how the stochastic dynamics approach recapitulated in this section, underlies both classical and modern inference schemes used to characterize biological single trajectory dynamics.