Information Thermodynamics and Reducibility of Large Gene Networks.

Gene regulatory networks (GRNs) control biological processes like pluripotency, differentiation, and apoptosis. Omics methods can identify a large number of putative network components (on the order of hundreds or thousands) but it is possible that in many cases a small subset of genes control the state of GRNs. Here, we explore how the topology of the interactions between network components may indicate whether the effective state of a GRN can be represented by a small subset of genes. We use methods from information theory to model the regulatory interactions in GRNs as cascading and superposing information channels. We propose an information loss function that enables identification of the conditions by which a small set of genes can represent the state of all the other genes in the network. This information-theoretic analysis extends to a measure of free energy change due to communication within the network, which provides a new perspective on the reducibility of GRNs. Both the information loss and relative free energy depend on the density of interactions and edge communication error in a network. Therefore, this work indicates that a loss in mutual information between genes in a GRN is directly coupled to a thermodynamic cost, i.e., a reduction of relative free energy, of the system.

The role of fluctuations in determining cellular network thermodynamics.

The steady state distributions of phenotypic responses within an isogenic population of cells result from both deterministic and stochastic characteristics of biochemical networks. A biochemical network can be characterized by a multidimensional potential landscape based on the distribution of responses and a diffusion matrix of the correlated dynamic fluctuations between N-numbers of intracellular network variables. In this work, we develop a thermodynamic description of biological networks at the level of microscopic interactions between network variables. The Boltzmann H-function defines the rate of free energy dissipation of a network system and provides a framework for determining the heat associated with the nonequilibrium steady state and its network components. The magnitudes of the landscape gradients and the dynamic correlated fluctuations of network variables are experimentally accessible. We describe the use of Fokker-Planck dynamics to calculate housekeeping heat from the experimental data by a method that we refer to as Thermo-FP. The method provides insight into the composition of the network and the relative thermodynamic contributions from network components. We surmise that these thermodynamic quantities allow determination of the relative importance of network components to overall network control. We conjecture that there is an upper limit to the rate of dissipative heat produced by a biological system that is associated with system size or modularity, and we show that the dissipative heat has a lower bound.

Nonparametric estimates of drift and diffusion profiles via Fokker-Planck algebra.

Diffusion processes superimposed upon deterministic motion play a key role in understanding and controlling the transport of matter, energy, momentum, and even information in physics, chemistry, material science, biology, and communications technology. Given functions defining these random and deterministic components, the Fokker-Planck (FP) equation is often used to model these diffusive systems. Many methods exist for estimating the drift and diffusion profiles from one or more identifiable diffusive trajectories; however, when many identical entities diffuse simultaneously, it may not be possible to identify individual trajectories. Here we present a method capable of simultaneously providing nonparametric estimates for both drift and diffusion profiles from evolving density profiles, requiring only the validity of Langevin/FP dynamics. This algebraic FP manipulation provides a flexible and robust framework for estimating stationary drift and diffusion coefficient profiles, is not based on fluctuation theory or solved diffusion equations, and may facilitate predictions for many experimental systems. We illustrate this approach on experimental data obtained from a model lipid bilayer system exhibiting free diffusion and electric field induced drift. The wide range over which this approach provides accurate estimates for drift and diffusion profiles is demonstrated through simulation.

Boltzmann's H-function and diffusion processes.

There exists a generalization of Boltzmann's H-function that allows for nonuniformly populated stationary states, which may exist far from thermodynamic equilibrium. Here we describe a method for obtaining a generalized or collective diffusion coefficient D directly from this H-function, the only constraints being that the relaxation process is Markov (short memory), continuous in the reaction coordinate, and local in the sense of a flux/force relationship. As an application of this H-function method, we simulate the self-consistent extraction of D via Langevin/Fokker-Planck (L/FP) dynamics on various potential energy landscapes. We observe that the initial epoch of relaxation, which is far removed from the stationary state, provides the most reliable estimates of D. The construction of an H-function that guarantees conformity with the second law of thermodynamics has been generalized to allow for diffusion coefficients that may depend on both the reaction coordinate and time, and the extension to an arbitrary number of reaction coordinates is straightforward. For this multidimensional case, the diffusion tensor must be positive definite in the sense that its eigenvalues must be real and positive. To illustrate the behavior of the proposed collective diffusion coefficient, we simulate the H-function for a variety of Langevin systems. In particular, the impacts on H and D of landscape shape, sample size, selection of an initial distribution, finite dynamic observation range, stochastic correlations, and short/long-term memory effects are examined.

Predicting rates of cell state change caused by stochastic fluctuations using a data-driven landscape model.

We develop a potential landscape approach to quantitatively describe experimental data from a fibroblast cell line that exhibits a wide range of GFP expression levels under the control of the promoter for tenascin-C. Time-lapse live-cell microscopy provides data about short-term fluctuations in promoter activity, and flow cytometry measurements provide data about the long-term kinetics, because isolated subpopulations of cells relax from a relatively narrow distribution of GFP expression back to the original broad distribution of responses. The landscape is obtained from the steady state distribution of GFP expression and connected to a potential-like function using a stochastic differential equation description (Langevin/Fokker-Planck). The range of cell states is constrained by a force that is proportional to the gradient of the potential, and biochemical noise causes movement of cells within the landscape. Analyzing the mean square displacement of GFP intensity changes in live cells indicates that these fluctuations are described by a single diffusion constant in log GFP space. This finding allows application of the Kramers' model to calculate rates of switching between two attractor states and enables an accurate simulation of the dynamics of relaxation back to the steady state with no adjustable parameters. With this approach, it is possible to use the steady state distribution of phenotypes and a quantitative description of the short-term fluctuations in individual cells to accurately predict the rates at which different phenotypes will arise from an isolated subpopulation of cells.

Cell volume distributions reveal cell growth rates and division times.

A population of cells in culture displays a range of phenotypic responses even when those cells are derived from a single cell and are exposed to a homogeneous environment. Phenotypic variability can have a number of sources including the variable rates at which individual cells within the population grow and divide. We have examined how such variations contribute to population responses by measuring cell volumes within genetically identical populations of cells where individual members of the population are continuously growing and dividing, and we have derived a function describing the stationary distribution of cell volumes that arises from these dynamics. The model includes stochastic parameters for the variability in cell cycle times and growth rates for individual cells in a proliferating cell line. We used the model to analyze the volume distributions obtained for two different cell lines and one cell line in the absence and presence of aphidicolin, a DNA polymerase inhibitor. The derivation and application of the model allows one to relate the stationary population distribution of cell volumes to extrinsic biological noise present in growing and dividing cell cultures.

Real-time measurement of spontaneous antigen-antibody dissociation.

We report observations in real time of thermally driven adhesion and dissociation between a monoclonal IgE antibody and its specific antigen N-epsilon-2,4-dinitrophenyl-L-lysine. Both molecules were attached to the surfaces of different polystyrene microspheres trapped by optical tweezers. Monitoring spontaneous successive attachment and detachment events allowed a direct determination of the reaction-limited detachment rate k(off) for a single bond and also for multiple bonds. We observed both positive and negative cooperativity between multiple bonds depending on whether the antigen was linked to the microsphere with or without a tether, respectively.