Limitations of the rate-distribution formalism in describing luminescence quenching in the presence of diffusion.

When encountering complex fluorescence decays that deviate from exponentiality, a very appealing approach is to use lifetime or rate constant distributions. These are related by Laplace transform to the sum of exponential functions, stretched exponentials, Becquerel's decay function, and others. However, the limitations of this approach have not been sufficiently discussed in the literature. In particular, the time-independent probability distributions of the rate constants or decay times are occasionally used to describe bimolecular quenching. We show that in such a case, this mathematical formalism has a clear physical interpretation only when the fluorophore and quencher molecules are immobile, as in the solid state. However, such an interpretation is no longer possible once we consider the motion of fluorophores with respect to quenchers. Therefore, for systems in which the relative motion of fluorophores and quenchers cannot be neglected, it is not appropriate to use the time-independent rate or decay time distributions to describe, fit, or rationalize experimental results on fluorescence decay.

Hill kinetics as a noise filter: the role of transcription factor autoregulation in gene cascades.

An intuition based on deterministic models of chemical kinetics is that population heterogeneity of transcription factor levels in cells is transmitted unchanged downstream to the target genes. We use a stochastic model of a two-gene cascade with a self-regulating upstream gene to show that, counter to the intuition, there is no simple mapping (bimodal to bimodal, unimodal to unimodal) between the shapes of the distributions of transcription factor numbers and target protein numbers in cells. Due to the presence of the two regulations, the system contains two nonlinear transfer functions, defined by the Hill kinetics of transcription factor binding. The transfer function of the regulator can "interfere" with the transfer function of the target, converting the bimodal input into a unimodal output or vice versa. We show that this effect can be predicted by a geometric construction. As an example application of the method, we present a case study of a system of several downstream genes of different sensitivities, controlled by a common transcription factor which also regulates its own transcription. We show that a single regulator can induce qualitatively different patterns (binary or graded) of responses to a signal in different downstream genes, depending on whether the sensitivity regions of the transfer functions of the upstream and downstream genes overlap or not. Alternatively, the same model can be interpreted as describing a single downstream gene that has different sensitivities in different cell lines due to mutations. Our model shows, therefore, a possible kinetic mechanism by which different genes can interpret the same biological signal in a different manner.

How good is the generalized Langevin equation to describe the dynamics of photo-induced electron transfer in fluid solution?

The dynamics of unimolecular photo-triggered reactions can be strongly affected by the surrounding medium for which a large number of theoretical descriptions have been used in the past. An accurate description of these reactions requires knowing the potential energy surface and the friction felt by the reactants. Most of these theories start from the Langevin equation to derive the dynamics, but there are few examples comparing it with experiments. Here we explore the applicability of a Generalized Langevin Equation (GLE) with an arbitrary potential and a non-Markovian friction. To this end, we have performed broadband fluorescence measurements with sub-picosecond time resolution of a covalently linked organic electron donor-acceptor system in solvents of changing viscosity and dielectric permittivity. In order to establish the free energy surface (FES) of the reaction, we resort to stationary electronic spectroscopy. On the other hand, the dynamics of a non-reacting substance, Coumarin 153, provide the calibrating tool for the non-Markovian friction over the FES, which is assumed to be solute independent. A simpler and computationally faster approach uses the Generalized Smoluchowski Equation (GSE), which can be derived from the GLE for pure harmonic potentials. Both approaches reproduce the measurements in most of the solvents reasonably well. At long times, some differences arise from the errors inherited from the analysis of the stationary solvatochromism and at short times from the excess excitation energy. However, whenever the dynamics become slow, the GSE shows larger deviations than the GLE, the results of which always agree qualitatively with the measured dynamics, regardless of the solvent viscosity or dielectric properties. The method applied here can be used to predict the dynamics of any other reacting system, given the FES parameters and solvent dynamics are provided. Thus no fitting parameters enter the GLE simulations, within the applicability limits found for the model in this work.

Influence of gene copy number on self-regulated gene expression.

Using an analytically solvable stochastic model, we study the properties of a simple genetic circuit consisting of multiple copies of a self-regulating gene. We analyse how the variation in gene copy number and the mutations changing the auto-regulation strength affect the steady-state distribution of protein concentration. We predict that one-reporter assay, an experimental method where the extrinsic noise level is inferred from the comparison of expression variance of a single and duplicated reporter gene, may give an incorrect estimation of the extrinsic noise contribution when applied to self-regulating genes. We also show that an imperfect duplication of an auto-activated gene, changing the regulation strength of one of the copies, may lead to a hybrid, binary+graded response of these genes to external signal. The analysis of relative changes in mean gene expression before and after duplication suggests that evolutionary accumulation of gene duplications may, at a given mean burst size, non-trivially depend on the inherent noisiness of a given gene, quantified by the inverse of the maximal mean frequency of bursts. Moreover, we find that the dependence of gene expression noise on gene copy number and auto-regulation strength may qualitatively differ, e.g. in monotonicity, depending on whether the noise is measured by Fano factor or coefficient of variation. Thus, experimentally-based hypotheses linking gene expression noise and evolutionary optimisation in the context of gene copy number variation may be ambiguous as they are dependent on the particular function chosen to quantify noise.

Generalization of Powell's results to population out of steady state.

Since the seminal work of Powell, the relationships between the population growth rate, the probability distributions of generation time, and the distribution of cell age have been known for the bacterial population in a steady state of exponential growth. Here we generalize these relationships to include an unsteady (transient) state for both the batch culture and the mother machine experiment. In particular, we derive a time-dependent Euler-Lotka equation (relating the generation-time distributions to the population growth rate) and a generalization of the inequality between the mean generation time and the population doubling time. To do this, we use a model proposed by Lebowitz and Rubinow, in which each cell is described by its age and generation time. We show that our results remain valid for a class of more complex models that use other state variables in addition to cell age and generation time, as long as the integration of these additional variables reduces the model to Lebowitz-Rubinow form. As an application of this formalism, we calculate the fitness landscapes for phenotypic traits (cell age, generation time) in a population that is not growing exponentially. We clarify that the known fitness landscape formula for the cell age as a phenotypic trait is an approximation to the exact time-dependent formula.

Contributions to the 'noise floor' in gene expression in a population of dividing cells.

Experiments with cells reveal the existence of a lower bound for protein noise, the noise floor, in highly expressed genes. Its origins are still debated. We propose a minimal model of gene expression in a proliferating bacterial cell population. The model predicts the existence of a noise floor and it semi-quantitatively reproduces the curved shape of the experimental noise vs. mean protein concentration plots. When the cell volume increases in a different manner than does the mean protein copy number, the noise floor level is determined by the cell population's age structure and by the dependence of the mean protein concentration on cell age. Additionally, the noise floor level may depend on a biological limit for the mean number of bursts in the cell cycle. In that case, the noise floor level depends on the burst size distribution width but it is insensitive to the mean burst size. Our model quantifies the contributions of each of these mechanisms to gene expression noise.

Exactly solvable model of gene expression in a proliferating bacterial cell population with stochastic protein bursts and protein partitioning.

Many of the existing stochastic models of gene expression contain the first-order decay reaction term that may describe active protein degradation or dilution. If the model variable is interpreted as the molecule number, and not concentration, the decay term may also approximate the loss of protein molecules due to cell division as a continuous degradation process. The seminal model of that kind leads to gamma distributions of protein levels, whose parameters are defined by the mean frequency of protein bursts and mean burst size. However, such models (whether interpreted in terms of molecule numbers or concentrations) do not correctly account for the noise due to protein partitioning between daughter cells. We present an exactly solvable stochastic model of gene expression in cells dividing at random times, where we assume description in terms of molecule numbers with a constant mean protein burst size. The model is based on a population balance equation supplemented with protein production in random bursts. If protein molecules are partitioned equally between daughter cells, we obtain at steady state the analytical expressions for probability distributions similar in shape to gamma distributions, yet with quite different values of mean burst size and mean burst frequency than would result from fitting of the classical continuous-decay model to these distributions. For random partitioning of protein molecules between daughter cells, we obtain the moment equations for the protein number distribution and thus the analytical formulas for the squared coefficient of variation.

Time-dependent solutions for a stochastic model of gene expression with molecule production in the form of a compound Poisson process.

We study a stochastic model of gene expression, in which protein production has a form of random bursts whose size distribution is arbitrary, whereas protein decay is a first-order reaction. We find exact analytical expressions for the time evolution of the cumulant-generating function for the most general case when both the burst size probability distribution and the model parameters depend on time in an arbitrary (e.g., oscillatory) manner, and for arbitrary initial conditions. We show that in the case of periodic external activation and constant protein degradation rate, the response of the gene is analogous to the resistor-capacitor low-pass filter, where slow oscillations of the external driving have a greater effect on gene expression than the fast ones. We also demonstrate that the nth cumulant of the protein number distribution depends on the nth moment of the burst size distribution. We use these results to show that different measures of noise (coefficient of variation, Fano factor, fractional change of variance) may vary in time in a different manner. Therefore, any biological hypothesis of evolutionary optimization based on the nonmonotonic dependence of a chosen measure of noise on time must justify why it assumes that biological evolution quantifies noise in that particular way. Finally, we show that not only for exponentially distributed burst sizes but also for a wider class of burst size distributions (e.g., Dirac delta and gamma) the control of gene expression level by burst frequency modulation gives rise to proportional scaling of variance of the protein number distribution to its mean, whereas the control by amplitude modulation implies proportionality of protein number variance to the mean squared.

Cluster-size distribution in the autocatalytic growth model.

We generalize the model of transition-metal nanocluster growth in aqueous solution, proposed recently [J. JÈ©drak, Phys. Rev. E 87, 022132 (2013)], by introducing a more complete description of chemical reactions. In order to model time evolution of the system, equations describing chemical reaction kinetics are combined with the Smoluchowski coagulation equation. In the absence of coagulation and fragmentation processes, the model equations are solved in two steps. First, we obtain the explicit analytical form of the i-mer concentration, ξ(i), as a function of ξ(1). This result allows us to reduce considerably the number of time-evolution equations. In the simplest situation, the remaining single kinetic equation for ξ(1)(t) is solved in quadratures. In a general case, we obtain a small system of time-evolution equations, which, although rarely analytically tractable, can be relatively easily solved numerically.

Exact solutions of kinetic equations in an autocatalytic growth model.

Kinetic equations are introduced for the transition-metal nanocluster nucleation and growth mechanism, as proposed by Watzky and Finke [J. Am. Chem. Soc. 119, 10382 (1997)]. Equations of this type take the form of Smoluchowski coagulation equations supplemented with the terms responsible for the chemical reactions. In the absence of coagulation, we find complete analytical solutions of the model equations for the autocatalytic rate constant both proportional to the cluster mass, and the mass-independent one. In the former case, ξ(k)=s(k)(ξ(1))[proportionality]ξ(1)(k)/k was obtained, while in the latter, the functional form of s(k)(ξ(1)) is more complicated. In both cases, ξ(1)(t)=h(µ)(M(µ)(t)) is a function of the moments of the mass distribution. Both functions, s(k)(ξ(1)) and h(µ)(M(µ)), depend on the assumed mechanism of autocatalytic growth and monomer production, and not on other chemical reactions present in a system.