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.

A depletion layer in polymer solutions at an interface oscillating at the subnano- to submicrometer scale.

The mobility of segments of the polymer mesh in a solution determines the dynamic response of the depletion layer (DL) to mechanical stimuli. This phenomenon can be used to vastly decrease the local viscosity experienced by any device performing periodic motion at the nano- and microscale in complex liquids. We refined the vibrating quartz tuning fork (QTF) method to probe the viscosity of model aqueous solutions of polyethylene glycol, covering a broad range of molecular weights (3 kDa to 1 MDa) and QTF oscillation amplitudes (50 pm to 100 nm). For semidilute solutions of PEGs of high molecular weight, we found a drop of local viscosity, up to two orders of magnitude below the bulk value. We propose a simple explanation based on the motion of the depletion layer, strongly supported by rheometry and dynamic light scattering results. We show that it is possible to directly probe the viscosity of the DL and increase its thickness far above the equilibrium value. The key role is played by the rate of relaxation of the entangled system. The relevance of this paradigm ranges from the basic research on dynamics of entangled systems to design of energy-efficient nanomachines operating in a crowded environment.

Taylor dispersion analysis in coiled capillaries at high flow rates.

Taylor Dispersion Analysis (TDA) has been performed for analytes moving at high flow rates in long, coiled capillaries. A thin injection zone of the analyte is stretched by the flow and final distribution of concentration of the analyte at the end of the capillary has the gaussian shape. The high flow rates in coiled capillary generate vortices. They convectively mix the analyte across the capillary. This mixing reduces the width of the gaussian distribution several times in comparison to the width obtained in a straight capillary in standard TDA. We have determined an empirical, scaling equation for the width as a function of the flow rate, molecular diffusion coefficient of the analyte, viscosity of the carrier phase, internal radius of the cylindrical capillary, and external radius of the coiled capillary. This equation can be used for different sizes of capillaries in a wide range of parameters without an additional calibration procedure. Our experimental results of flow in the coiled capillary could not be explained by current models based on approximate solutions of the Navier-Stokes equation. We applied the technique to determine the diffusion coefficients of the following analytes: salts, drugs, single amino acids, peptides (from dipeptides to hexapeptides), and proteins.

Bimodal gene expression in noncooperative regulatory systems.

Bimodality of gene expression, as a mechanism contributing to phenotypic diversity, enhances the survival of cells in a fluctuating environment. To date, the bimodal response of a gene regulatory system has been attributed to the cooperativity of transcription factor binding or to feedback loops. It has remained unclear whether noncooperative binding of transcription factors can give rise to bimodality in an open-loop system. We study a theoretical model of gene expression in a two-step cascade (a deterministically monostable system) in which the regulatory gene produces transcription factors that have a nonlinear effect on the activity of the target gene. We show that a unimodal distribution of transcription factors over the cell population can generate a bimodal steady-state output without cooperative transcription factor binding. We introduce a simple method of geometric construction that allows one to predict the onset of bimodality. The construction only involves the parameters of bursting of the regulatory gene and the dose-response curve of the target gene. Using this method, we show that the gene expression may switch between unimodal and bimodal as the concentration of inducers or corepressors is varied. These findings may explain the experimentally observed bimodal response of cascades consisting of a fluorescent protein reporter controlled by the tetracycline repressor. The geometric construction provides a useful tool for designing experiments and for interpretation of their results. Our findings may have important implications for understanding the strategies adopted by cell populations to survive in changing environments.

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.

Extrinsic noise passing through a Michaelis-Menten reaction: a universal response of a genetic switch.

The study of biochemical pathways usually focuses on a small section of a protein interactions network. Two distinct sources contribute to the noise in such a system: intrinsic noise, inherent in the studied reactions, and extrinsic noise generated in other parts of the network or in the environment. We study the effect of extrinsic noise entering the system through a nonlinear uptake reaction which acts as a nonlinear filter. Varying input noise intensity varies the mean of the noise after the passage through the filter, which changes the stability properties of the system. The steady-state displacement due to small noise is independent on the kinetics of the system but it only depends on the nonlinearity of the input function. For monotonically increasing and concave input functions such as the Michaelis-Menten uptake rate, we give a simple argument based on the small-noise expansion, which enables qualitative predictions of the steady-state displacement only by inspection of experimental data: when weak and rapid noise enters the system through a Michaelis-Menten reaction, then the graph of the system's steady states vs. the mean of the input signal always shifts to the right as noise intensity increases. We test the predictions on two models of lac operon, where TMG/lactose uptake is driven by a Michaelis-Menten enzymatic process. We show that as a consequence of the steady state displacement due to fluctuations in extracellular TMG/lactose concentration the lac switch responds in an asymmetric manner: as noise intensity increases, switching off lactose metabolism becomes easier and switching it on becomes more difficult.

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.

Noise-assisted spike propagation in myelinated neurons.

We consider noise-assisted spike propagation in myelinated axons within a multicompartment stochastic Hodgkin-Huxley model. The noise originates from a finite number of ion channels in each node of Ranvier. For the subthreshold internodal electric coupling, we show that (i) intrinsic noise removes the sharply defined threshold for spike propagation from node to node and (ii) there exists an optimum number of ion channels which allows for the most efficient signal propagation and it corresponds to the actual physiological values.

Modeling radiation-induced cell cycle delays.

Ionizing radiation is known to delay the cell cycle progression. In particular after particle exposure significant delays have been observed and it has been shown that the extent of delay affects the expression of damage, such as chromosome aberrations. Thus, to predict how cells respond to ionizing radiation and to derive reliable estimates of radiation risks, information about radiation-induced cell cycle perturbations is required. In the present study we describe and apply a method for retrieval of information about the time-course of all cell cycle phases from experimental data on the mitotic index only. We study the progression of mammalian cells through the cell cycle after exposure. The analysis reveals a prolonged block of damaged cells in the G2 phase. Furthermore, by performing an error analysis on simulated data valuable information for the design of experimental studies has been obtained. The analysis showed that the number of cells analyzed in an experimental sample should be at least 100 to obtain a relative error <20%.

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.

Predicting the asymmetric response of a genetic switch to noise.

We present a simple analytical tool which gives an approximate insight into the stationary behavior of nonlinear systems undergoing the influence of a weak and rapid noise from one dominating source, e.g. the kinetic equations describing a genetic switch with the concentration of one substrate fluctuating around a constant mean. The proposed method allows for predicting the asymmetric response of the genetic switch to noise, arising from the noise-induced shift of stationary states. The method has been tested on an example model of the lac operon regulatory network: a reduced Yildirim-Mackey model with fluctuating extracellular lactose concentration. We calculate analytically the shift of the system's stationary states in the presence of noise. The results of the analytical calculation are in excellent agreement with the results of numerical simulation of the noisy system. The simulation results suggest that the structure of the kinetics of the underlying biochemical reactions protects the bistability of the lactose utilization mechanism from environmental fluctuations. We show that, in the consequence of the noise-induced shift of stationary states, the presence of fluctuations stabilizes the behavior of the system in a selective way: Although the extrinsic noise facilitates, to some extent, switching off the lactose metabolism, the same noise prevents it from switching on.

Co-occurrence of resonant activation and noise-enhanced stability in a model of cancer growth in the presence of immune response.

We investigate a stochastic version of a simple enzymatic reaction which follows the generic Michaelis-Menten kinetics. At sufficiently high concentrations of reacting species, that represent here populations of cells involved in cancerous proliferation and cytotoxic response of the immune system, the overall kinetics can be approximated by a one-dimensional overdamped Langevin equation. The modulating activity of the immune response is here modeled as a dichotomous random process of the relative rate of neoplastic cell destruction. We discuss physical aspects of environmental noises acting in such a system, pointing out the possibility of coexistence of dynamical regimes where noise-enhanced stability and resonant activation phenomena can be observed together. We explain the underlying mechanisms by analyzing the behavior of the variance of first passage times as a function of the noise intensity.

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.

Transcriptional leakage versus noise: a simple mechanism of conversion between binary and graded response in autoregulated genes.

We study the response of an autoregulated gene to a range of concentrations of signal molecules. We show that transcriptional leakage and noise due to translational bursting have the opposite effects. In a positively autoregulated gene, increasing the noise converts the response from graded to binary, while increasing the leakage converts the response from binary to graded. Our findings support the hypothesis that, being a common phenomenon, leaky expression may be a relatively easy way for evolutionary tuning of the type of gene response without changing the type of regulation from positive to negative.

Motion of nanoprobes in complex liquids within the framework of the length-scale dependent viscosity model.

This paper deals with the recent phenomenological model of the motion of nanoscopic objects (colloidal particles, proteins, nanoparticles, molecules) in complex liquids. We analysed motion in polymer, micellar, colloidal and protein solutions and the cytoplasm of living cells using the length-scale dependent viscosity model. Viscosity monotonically approaches macroscopic viscosity as the size of the object increases and thus gives a single, coherent picture of motion at the nano and macro scale. The model includes interparticle interactions (solvent-solute), temperature and the internal structure of a complex liquid. The depletion layer ubiquitously occurring in complex liquids is also incorporated into the model. We also discuss the biological aspects of crowding in terms of the length-scale dependent viscosity model.

A universal model of restricted diffusion for fluorescence correlation spectroscopy.

Fluorescence correlation spectroscopy (FCS) is frequently used to study the processes of restricted diffusion. The most important quantity to determine is the size of the structures that hinder the Brownian motion of the molecules. We study three qualitatively different models of restricted diffusion, widely applied in biophysics and material science: Diffusion constrained by elastic force (i), walking confined diffusion (ii), and hop diffusion (iii). They cover the diversity of statistical behaviors, from purely Gaussian (i) to sharply non-Gaussian on intermediate time scales (ii) and, additionally, discrete (iii). We test whether one can use the Gaussian approximation of the FCS autocorrelation function to interpret the non-Gaussian data. We show that (i-iii) have approximately the same mean square displacements. Using simulations, we show that the FCS data suspected of restricted diffusion can be reliably interpreted using one archetypal model (i). Even if the underlying mechanism of the restriction is different or unknown, the accuracy of fitting the confinement size is excellent, and diffusion coefficients are also estimated with a good accuracy. This study gives a physical insight into the statistical behavior of different types of restricted diffusion and into the ability of fluorescence correlation spectroscopy to distinguish between them.