Andrew Mugler

Spectral solutions to stochastic models of gene expression with bursts and regulation

Signal-processing molecules inside cells are often present at low copy number, which necessitates probabilistic models to account for intrinsic noise. Probability distributions have traditionally been found using simulation-based approaches which then require estimating the distributions from many samples. Here we present in detail an alternative method for directly calculating a probability distribution by expanding in the natural eigenfunctions of the governing equation, which is linear. We apply the resulting spectral method to three general models of stochastic gene expression: a single gene with multiple expression states (often used as a model of bursting in the limit of two states), a gene regulatory cascade, and a combined model of bursting and regulation. In all cases we find either analytic results or numerical prescriptions that greatly outperform simulations in efficiency and accuracy. In the last case, we show that bimodal response in the limit of slow switching is not only possible but optimal in terms of information transmission.

Membrane Clustering and the Role of Rebinding in Biochemical Signaling

In many cellular signaling pathways, key components form clusters at the cell membrane. Although much work has focused on the mechanisms behind such cluster formation, the implications for downstream signaling remain poorly understood. Here, motivated by recent experiments, we use particle-based simulation to study a covalent modification network in which the activating component is either clustered or randomly distributed on the membrane. We find that whereas clustering reduces the response of a single-modification network, it can enhance the response of a double-modification network. The reduction is a bulk effect: a cluster presents a smaller effective target to a substrate molecule in the bulk. The enhancement, on the other hand, is a local effect: a cluster promotes the rapid rebinding and second activation of singly activated substrate molecules. As such, the enhancement relies on frequent collisions on a short timescale, leading to an optimal ratio of diffusion to association that agrees with typical measured rates. We complement simulation with analytic results at both the mean-field and first-passage distribution levels. Our results emphasize the importance of spatially resolved models, showing that significant effects of spatial correlations persist even in spatially averaged quantities such as response curves.

A stochastic spectral analysis of transcriptional regulatory cascades

The past decade has seen great advances in our understanding of the role of noise in gene regulation and the physical limits to signaling in biological networks. Here, we introduce the spectral method for computation of the joint probability distribution over all species in a biological network. The spectral method exploits the natural eigenfunctions of the master equation of birth – death processes to solve for the joint distribution of modules within the network, which then inform each other and facilitate calculation of the entire joint distribution. We illustrate the method on a ubiquitous case in nature: linear regulatory cascades. The efficiency of the method makes possible numerical optimization of the input and regulatory parameters, revealing design properties of, e.g., the most informative cascades. We find, for threshold regulation, that a cascade of strong regulations converts a unimodal input to a bimodal output, that multimodal inputs are no more informative than bimodal inputs, and that a chain of up-regulations outperforms a chain of down-regulations. We anticipate that this numerical approach may be useful for modeling noise in a variety of small network topologies in biology.

High-speed DNA-based rolling motors powered by RNase H

DNA-based machines that walk by converting chemical energy into controlled motion could be of use in applications such as next generation sensors, drug delivery platforms, and biological computing. Despite their exquisite programmability, DNA-based walkers are, however, challenging to work with due to their low fidelity and slow rates (~1 nm/min). Here, we report DNA-based machines that roll rather than walk, and consequently have a maximum speed and processivity that is three-orders of magnitude greater than conventional DNA motors. The motors are made from DNA-coated spherical particles that hybridise to a surface modified with complementary RNA; motion is achieved through the addition of RNase H, which selectively hydrolyses hybridised RNA. Spherical motors move in a self-avoiding manner, whereas anisotropic particles, such as dimerised particles or rod-shaped particles travel linearly without a track or external force. Finally, we demonstrate detection of single nucleotide polymorphism by measuring particle displacement using a smartphone camera.

Cell–cell communication enhances the capacity of cell ensembles to sense shallow gradients during morphogenesis

Significance What new properties may result from collective cell behavior, and how may these emerging capabilities influence shaping and function of tissues, in health and disease? Here, we explored these questions in the context of epithelial branching morphogenesis. We show experimentally that, whereas individual mammary epithelial cells are incapable of sensing extremely weak gradients of a growth factor, cellular collectives in organotypic cultures exhibit reliable, gradient-driven, directional growth. This underscores a critical importance of collective cell–cell communication and computation in gradient sensing. We develop and verify a biophysical theory of such communication and identify the mechanisms by which it is implemented in the mammary epithelium, quantitatively analyzing both advantages and limitations of biochemical cellular communication in collective decision making. Collective cell responses to exogenous cues depend on cell–cell interactions. In principle, these can result in enhanced sensitivity to weak and noisy stimuli. However, this has not yet been shown experimentally, and little is known about how multicellular signal processing modulates single-cell sensitivity to extracellular signaling inputs, including those guiding complex changes in the tissue form and function. Here we explored whether cell–cell communication can enhance the ability of cell ensembles to sense and respond to weak gradients of chemotactic cues. Using a combination of experiments with mammary epithelial cells and mathematical modeling, we find that multicellular sensing enables detection of and response to shallow epidermal growth factor (EGF) gradients that are undetectable by single cells. However, the advantage of this type of gradient sensing is limited by the noisiness of the signaling relay, necessary to integrate spatially distributed ligand concentration information. We calculate the fundamental sensory limits imposed by this communication noise and combine them with the experimental data to estimate the effective size of multicellular sensory groups involved in gradient sensing. Functional experiments strongly implicated intercellular communication through gap junctions and calcium release from intracellular stores as mediators of collective gradient sensing. The resulting integrative analysis provides a framework for understanding the advantages and limitations of sensory information processing by relays of chemically coupled cells.

An experimental study of micron-scale droplet aerosols produced via ultrasonic atomization

In the last 10 years, laser-driven fusion experiments performed on atomic clusters of deuterium have shown a surprisingly high neutron yield per joule of input laser energy. Results indicate that the optimal cluster size for maximizing fusion events should be in the 0.01‐1 mm diameter range, but an appropriate source of droplets of this size does not exist. In an attempt to meet this need, we use ultrasonic atomization to generate micron-scale droplet aerosols of high average density, and we have developed and refined a reliable droplet sizing technique based on Mie scattering. Harmonic excitation of the fluid in the MHz range yields an aerosol of droplets with diameters of a few microns. The droplet diameter distribution is well-peaked and the relationship between average droplet size and forcing frequency follows an inviscid scaling law, predictable by dimensional analysis and consistent with the linear theory for Faraday excitation of an infinitely deep fluid.

Mesoscopic statistical properties of multistep enzyme-mediated reactions

Enzyme-mediated reactions may proceed through multiple intermediate conformational states before creating a final product molecule, and one often wishes to identify such intermediate structures from observations of the product creation. In this study, the authors address this problem by solving the chemical master equations for various enzymatic reactions. A perturbation theory analogous to that used in quantum mechanics allows the determination of the first (n) and the second (σ2) cumulants of the distribution of created product molecules as a function of the substrate concentration and the kinetic rates of the intermediate processes. The mean product flux V=d(n)/dt (or dose-response curve) and the Fano factor F= σ2/(n) are both realistically measurable quantities, and whereas the mean flux can often appear the same for different reaction types, the Fano factor can be quite different. This suggests both qualitative and quantitative ways to discriminate between different reaction schemes, and the authors explore this possibility in the context of four sample multistep enzymatic reactions. Measuring both the mean flux and the Fano factor can not only discriminate between reaction types, but can also provide some detailed information about the internal, unobserved kinetic rates, and this can be done without measuring single-molecule transition events.

Limits to the precision of gradient sensing with spatial communication and temporal integration

Significance Knowing which way to move is crucial for many biological processes, from organismal development to migration of cancer cells and from motion of microbes to wound healing. To find their preferred directions, biological systems compare concentrations of a chemical cue at their different edges. The comparison requires information from these different locations to be communicated to the same place. However, all communication is noisy (just think of the childhood game “telephone”). This communication noise, as well as noise in the individual measurements themselves, sets the accuracy of the direction sensing. Here we quantify the importance of the communication noise and propose a mechanism that can improve the accuracy of direction sensing. Gradient sensing requires at least two measurements at different points in space. These measurements must then be communicated to a common location to be compared, which is unavoidably noisy. Although much is known about the limits of measurement precision by cells, the limits placed by the communication are not understood. Motivated by recent experiments, we derive the fundamental limits to the precision of gradient sensing in a multicellular system, accounting for communication and temporal integration. The gradient is estimated by comparing a “local” and a “global” molecular reporter of the external concentration, where the global reporter is exchanged between neighboring cells. Using the fluctuation–dissipation framework, we find, in contrast to the case when communication is ignored, that precision saturates with the number of cells independently of the measurement time duration, because communication establishes a maximum length scale over which sensory information can be reliably conveyed. Surprisingly, we also find that precision is improved if the local reporter is exchanged between cells as well, albeit more slowly than the global reporter. The reason is that whereas exchange of the local reporter weakens the comparison, it decreases the measurement noise. We term such a model “regional excitation–global inhibition.” Our results demonstrate that fundamental sensing limits are necessarily sharpened when the need to communicate information is taken into account.

Analytic Methods for Modeling Stochastic Regulatory Networks

Recent single-cell experiments have revived interest in the unavoidable or intrinsic noise in biochemical and genetic networks arising from the small number of molecules of the participating species. That is, rather than modeling regulatory networks in terms of the deterministic dynamics of concentrations, we model the dynamics of the probability of a given copy number of the reactants in single cells. Most of the modeling activity of the last decade has centered on stochastic simulation, i.e., Monte Carlo methods for generating stochastic time series. Here we review the mathematical description in terms of probability distributions, introducing the relevant derivations and illustrating several cases for which analytic progress can be made either instead of or before turning to numerical computation. Analytic progress can be useful both for suggesting more efficient numerical methods and for obviating the computational expense of, for example, exploring parametric dependence.

Using ultrasonic atomization to produce an aerosol of micron-scale particles

A device that uses ultrasonic atomization of a liquid to produce an aerosol of micron-scale droplets is described. This device represents a new approach to producing targets relevant to laser-driven fusion studies, and to rare studies of nonlinear optics in which wavelength-scale targets are irradiated. The device has also made possible tests of fluid dynamics models in a novel phase space. The distribution of droplet sizes produced by the device and the threshold power required for droplet production are shown to follow scaling laws predicted by fluid dynamics.