Ensemble methods for stochastic networks with special reference to the biological clock of Neurospora crassa

Multimodal parameter spaces of a complex multi-channel neuron model

One of the most common types of models that helps us to understand neuron behavior is based on the Hodgkin-Huxley ion channel formulation (HH model). A major challenge with inferring parameters in HH models is non-uniqueness: many different sets of ion channel parameter values produce similar outputs for the same input stimulus. Such phenomena result in an objective function that exhibits multiple modes (i.e., multiple local minima). This non-uniqueness of local optimality poses challenges for parameter estimation with many algorithmic optimization techniques. HH models additionally have severe non-linearities resulting in further challenges for inferring parameters in an algorithmic fashion. To address these challenges with a tractable method in high-dimensional parameter spaces, we propose using a particular Markov chain Monte Carlo (MCMC) algorithm, which has the advantage of inferring parameters in a Bayesian framework. The Bayesian approach is designed to be suitable for multimodal solutions to inverse problems. We introduce and demonstrate the method using a three-channel HH model. We then focus on the inference of nine parameters in an eight-channel HH model, which we analyze in detail. We explore how the MCMC algorithm can uncover complex relationships between inferred parameters using five injected current levels. The MCMC method provides as a result a nine-dimensional posterior distribution, which we analyze visually with solution maps or landscapes of the possible parameter sets. The visualized solution maps show new complex structures of the multimodal posteriors, and they allow for selection of locally and globally optimal value sets, and they visually expose parameter sensitivities and regions of higher model robustness. We envision these solution maps as enabling experimentalists to improve the design of future experiments, increase scientific productivity and improve on model structure and ideation when the MCMC algorithm is applied to experimental data.

The macroscopic limit to synchronization of cellular clocks in single cells of Neurospora crassa

We determined the macroscopic limit for phase synchronization of cellular clocks in an artificial tissue created by a “big chamber” microfluidic device to be about 150,000 cells or less. The dimensions of the microfluidic chamber allowed us to calculate an upper limit on the radius of a hypothesized quorum sensing signal molecule of 13.05 nm using a diffusion approximation for signal travel within the device. The use of a second microwell microfluidic device allowed the refinement of the macroscopic limit to a cell density of 2166 cells per fixed area of the device for phase synchronization. The measurement of averages over single cell trajectories in the microwell device supported a deterministic quorum sensing model identified by ensemble methods for clock phase synchronization. A strong inference framework was used to test the communication mechanism in phase synchronization of quorum sensing versus cell-to-cell contact, suggesting support for quorum sensing. Further evidence came from showing phase synchronization was density-dependent.

Identifying a stochastic clock network with light entrainment for single cells of Neurospora crassa

Stochastic networks for the clock were identified by ensemble methods using genetic algorithms that captured the amplitude and period variation in single cell oscillators of Neurospora crassa. The genetic algorithms were at least an order of magnitude faster than ensemble methods using parallel tempering and appeared to provide a globally optimum solution from a random start in the initial guess of model parameters (i.e., rate constants and initial counts of molecules in a cell). The resulting goodness of fit [Formula: see text] was roughly halved versus solutions produced by ensemble methods using parallel tempering, and the resulting [Formula: see text] per data point was only [Formula: see text] = 2,708.05/953 = 2.84. The fitted model ensemble was robust to variation in proxies for "cell size". The fitted neutral models without cellular communication between single cells isolated by microfluidics provided evidence for only one Stochastic Resonance at one common level of stochastic intracellular noise across days from 6 to 36 h of light/dark (L/D) or in a D/D experiment. When the light-driven phase synchronization was strong as measured by the Kuramoto (K), there was degradation in the single cell oscillations away from the stochastic resonance. The rate constants for the stochastic clock network are consistent with those determined on a macroscopic scale of 10⁷ cells.

What is Phase in Cellular Clocks?

Four inter-related measures of phase are described to study the phase synchronization of cellular oscillators, and computation of these measures is described and illustrated on single cell fluorescence data from the model filamentous fungus, Neurospora crassa. One of these four measures is the phase shift ϕ in a sinusoid of the form x(t) = A(cos(ωt + ϕ), where t is time. The other measures arise by creating a replica of the periodic process x(t) called the Hilbert transform x̃(t), which is 90 degrees out of phase with the original process x(t). The second phase measure is the phase angle F^H(t) between the replica x̃(t) and x(t), taking values between -π and π. At extreme values the Hilbert Phase is discontinuous, and a continuous form F^C(t) of the Hilbert Phase is used, measuring time on the nonnegative real axis (t). The continuous Hilbert Phase F^C(t) is used to define the phase M^C(t₁,t₀) for an experiment beginning at time t₀ and ending at time t₁. In that phase differences at time t₀ are often of ancillary interest, the Hilbert Phase F^C(t₀) is subtracted from F^C(t₁). This difference is divided by 2π to obtain the phase M^C(t₁,t₀) in cycles. Both the Hilbert Phase F^C(t) and the phase M^C(t₁,t₀) are functions of time and useful in studying when oscillators phase-synchronize in time in signal processing and circadian rhythms in particular. The phase of cellular clocks is fundamentally different from circadian clocks at the macroscopic scale because there is an hourly cycle superimposed on the circadian cycle.