Emergent Criticality in Coupled Boolean Networks.

Early embryonic development involves forming all specialized cells from a fluid-like mass of identical stem cells. The differentiation process consists of a series of symmetry-breaking events, starting from a high-symmetry state (stem cells) to a low-symmetry state (specialized cells). This scenario closely resembles phase transitions in statistical mechanics. To theoretically study this hypothesis, we model embryonic stem cell (ESC) populations through a coupled Boolean network (BN) model. The interaction is applied using a multilayer Ising model that considers paracrine and autocrine signaling, along with external interventions. It is demonstrated that cell-to-cell variability can be interpreted as a mixture of steady-state probability distributions. Simulations have revealed that such models can undergo a series of first- and second-order phase transitions as a function of the system parameters that describe gene expression noise and interaction strengths. These phase transitions result in spontaneous symmetry-breaking events that generate new types of cells characterized by various steady-state distributions. Coupled BNs have also been shown to self-organize in states that allow spontaneous cell differentiation.

Self-Regulated Symmetry Breaking Model for Stem Cell Differentiation.

In conventional disorder-order phase transitions, a system shifts from a highly symmetric state, where all states are equally accessible (disorder) to a less symmetric state with a limited number of available states (order). This transition may occur by varying a control parameter that represents the intrinsic noise of the system. It has been suggested that stem cell differentiation can be considered as a sequence of such symmetry-breaking events. Pluripotent stem cells, with their capacity to develop into any specialized cell type, are considered highly symmetric systems. In contrast, differentiated cells have lower symmetry, as they can only carry out a limited number of functions. For this hypothesis to be valid, differentiation should emerge collectively in stem cell populations. Additionally, such populations must have the ability to self-regulate intrinsic noise and navigate through a critical point where spontaneous symmetry breaking (differentiation) occurs. This study presents a mean-field model for stem cell populations that considers the interplay of cell-cell cooperativity, cell-to-cell variability, and finite-size effects. By introducing a feedback mechanism to control intrinsic noise, the model can self-tune through different bifurcation points, facilitating spontaneous symmetry breaking. Standard stability analysis showed that the system can potentially differentiate into several cell types mathematically expressed as stable nodes and limit cycles. The existence of a Hopf bifurcation in our model is discussed in light of stem cell differentiation.

Modeling nanoscale hydrodynamics by smoothed dissipative particle dynamics.

Thermal fluctuation and hydrophobicity are two hallmarks of fluid hydrodynamics on the nano-scale. It is a challenge to consistently couple the small length and time scale phenomena associated with molecular interaction with larger scale phenomena. The development of this consistency is the essence of mesoscale science. In this study, we use a nanoscale fluid model based on smoothed dissipative particle dynamics that accounts for the phenomena associated with density fluctuations and hydrophobicity. We show consistency in the fluctuation spectrum across scales. In doing so, it is necessary to account for finite fluid particle size. Furthermore, we demonstrate that the present model can capture the void probability and solvation free energy of nonpolar hard particles of different sizes. The present fluid model is well suited for an understanding of emergent phenomena in nano-scale fluid systems.

Dynamic mesoscale model of dipolar fluids via fluctuating hydrodynamics.

Fluctuating hydrodynamics (FHD) is a general framework of mesoscopic modeling and simulation based on conservational laws and constitutive equations of linear and nonlinear responses. However, explicit representation of electrical forces in FHD has yet to appear. In this work, we devised an Ansatz for the dynamics of dipole moment densities that is linked with the Poisson equation of the electrical potential ϕ in coupling to the other equations of FHD. The resulting ϕ-FHD equations then serve as a platform for integrating the essential forces, including electrostatics in addition to hydrodynamics, pressure-volume equation of state, surface tension, and solvent-particle interactions that govern the emergent behaviors of molecular systems at an intermediate scale. This unique merit of ϕ-FHD is illustrated by showing that the water dielectric function and ion hydration free energies in homogeneous and heterogenous systems can be captured accurately via the mesoscopic simulation. Furthermore, we show that the field variables of ϕ-FHD can be mapped from the trajectory of an all-atom molecular dynamics simulation such that model development and parametrization can be based on the information obtained at a finer-grained scale. With the aforementioned multiscale capabilities and a spatial resolution as high as 5 Å, the ϕ-FHD equations represent a useful semi-explicit solvent model for the modeling and simulation of complex systems, such as biomolecular machines and nanofluidics.

Multilayered noise model for transport in complex environments.

Transport in complex fluidic environments often exhibits transient subdiffusive dynamics accompanied by non-Gaussian probability density profiles featuring a nonmonotonic non-Gaussian parameter. Such properties cannot be adequately explained by the original theory of Brownian motion. Based on an extension of kinetic theory, this study introduces a chain of hierarchically coupled random walks approach that effectively captures all these intriguing characteristics. If the environment consists of a series of independent white noise sources, then the problem can be expressed as a system of hierarchically coupled Ornstein-Uhlenbech equations. Due to the linearity of the system, the most essential transport properties have a closed analytical form.

Fluctuating hydrodynamics for multiscale modeling and simulation: energy and heat transfer in molecular fluids.

This work illustrates that fluctuating hydrodynamics (FHD) simulations can be used to capture the thermodynamic and hydrodynamic responses of molecular fluids at the nanoscale, including those associated with energy and heat transfer. Using all-atom molecular dynamics (MD) trajectories as the reference data, the atomistic coordinates of each snapshot are mapped onto mass, momentum, and energy density fields on Eulerian grids to generate a corresponding field trajectory. The molecular length-scale associated with finite molecule size is explicitly imposed during this coarse-graining by requiring that the variances of density fields scale inversely with the grid volume. From the fluctuations of field variables, the response functions and transport coefficients encoded in the all-atom MD trajectory are computed. By using the extracted fluid properties in FHD simulations, we show that the fluctuations and relaxation of hydrodynamic fields quantitatively match with those observed in the reference all-atom MD trajectory, hence establishing compatibility between the atomistic and field representations. We also show that inclusion of energy transfer in the FHD equations can more accurately capture the thermodynamic and hydrodynamic responses of molecular fluids. The results indicate that the proposed MD-to-FHD mapping with explicit consideration of finite molecule size provides a robust framework for coarse-graining the solution phase of complex molecular systems.

Fluctuating hydrodynamics for multiscale simulation of inhomogeneous fluids: mapping all-atom molecular dynamics to capillary waves.

We introduce a multiscale framework to simulate inhomogeneous fluids by coarse-graining an all-atom molecular dynamics (MD) trajectory onto sequential snapshots of hydrodynamic fields. We show that the field representation of an atomistic trajectory is quantitatively described by a dynamic field-theoretic model that couples hydrodynamic fluctuations with a Ginzburg-Landau free energy. For liquid-vapor interfaces of argon and water, the parameters of the field model can be adjusted to reproduce the bulk compressibility and surface tension calculated from the positions and forces of atoms in an MD simulation. These optimized parameters also enable the field model to reproduce the static and dynamic capillary wave spectra calculated from atomistic coordinates at the liquid-vapor interface. In addition, we show that a density-dependent gradient coefficient in the Ginzburg-Landau free energy enables bulk and interfacial fluctuations to be controlled separately. For water, this additional degree of freedom is necessary to capture both the bulk compressibility and surface tension emergent from the atomistic trajectory. The proposed multiscale framework illustrates that bottom-up coarse-graining and top-down phenomenology can be integrated with quantitative consistency to simulate the interfacial fluctuations in nanoscale transport processes.

Bridging fluctuating hydrodynamics and molecular dynamics simulations of fluids.

A new multiscale coarse-graining (CG) methodology is developed to bridge molecular and hydrodynamic models of a fluid. The hydrodynamic representation considered in this work is based on the equations of fluctuating hydrodynamics (FH). The essence of this method is a mapping from the position and velocity vectors of a snapshot of a molecular dynamics (MD) simulation to the field variables on Eulerian cells of a hydrodynamic representation. By explicit consideration of the effective lengthscale d(mol) that characterizes the volume of a molecule, the computed density fluctuations from MD via our mapping procedure have volume dependence that corresponds to a grand canonical ensemble of a cold liquid even when a small cell length (5-10 A) is used in a hydrodynamic representation. For TIP3P water at 300 K and 1 atm, d(mol) is found to be 2.4 A, corresponding to the excluded radius of a water molecule as revealed by its center-of-mass radial distribution function. By matching the density fluctuations and autocorrelation functions of momentum fields computed from solving the FH equations with those computed from MD simulation, the sound velocity and shear and bulk viscosities of a CG hydrodynamic model can be determined directly from MD. Furthermore, a novel staggered discretization scheme is developed for solving the FH equations of an isothermal compressive fluid in a three dimensional space with a central difference method. This scheme demonstrates high accuracy in satisfying the fluctuation-dissipation theorem. Since the causative relationship between field variables and fluxes is captured, we demonstrate that the staggered discretization scheme also predicts correct physical behaviors in simulating transient fluid flows. The techniques presented in this work may also be employed to design multiscale strategies for modeling complex fluids and macromolecules in solution.

Modeling the nanoscale viscoelasticity of fluids by bridging non-Markovian fluctuating hydrodynamics and molecular dynamics simulations.

A multiscale computational method is developed to model the nanoscale viscoelasticity of fluids by bridging non-Markovian fluctuating hydrodynamics (FHD) and molecular dynamics (MD) simulations. To capture the elastic responses that emerge at small length scales, we attach an additional rheological model parallel to the macroscopic constitutive equation of a fluid. The widely used linear Maxwell model is employed as a working choice; other models can be used as well. For a fluid that is Newtonian in the macroscopic limit, this approach results in a parallel Newtonian-Maxwell model. For water, argon, and an ionic liquid, the power spectrum of momentum field autocorrelation functions of the parallel Newtonian-Maxwell model agrees very well with those calculated from all-atom MD simulations. To incorporate thermal fluctuations, we generalize the equations of FHD to work with non-Markovian rheological models and colored noise. The fluctuating stress tensor (white noise) is integrated in time in the same manner as its dissipative counterpart and numerical simulations indicate that this approach accurately preserves the set temperature in a FHD simulation. By mapping position and velocity vectors in the molecular representation onto field variables, we bridge the non-Markovian FHD with atomistic MD simulations. Through this mapping, we quantitatively determine the transport coefficients of the parallel Newtonian-Maxwell model for water and argon from all-atom MD simulations. For both fluids, a significant enhancement in elastic responses is observed as the wave number of hydrodynamic modes is reduced to a few nanometers. The mapping from particle to field representations and the perturbative strategy of developing constitutive equations provide a useful framework for modeling the nanoscale viscoelasticity of fluids.

Pre-melting dynamics of DNA and its relation to specific functions.

We discuss connections between the nonlinear dynamics of double-stranded DNA, experimental findings, and specific DNA functions. We begin by discussing how thermally induced localized openings (bubbles) of the DNA double strand are important for interpreting dynamic force spectroscopy data. Then we demonstrate a correlation between a sequence-dependent propensity for pre-melting bubble formation and transcription initiation and other regulatory effects in viral DNA. Finally, we discuss the possibility of a connection between DNA dynamics and the ability of repair proteins to recognize ultraviolet (UV) radiation damage sites.

Opening rates of DNA hairpins: experiment and model.

We present single-molecule measurements of the opening rate of DNA hairpins under mechanical tension and compare with the results obtained from a reduced-degrees-of-freedom statistical mechanics model. We extract the apparent position of the transition state s and find that the model, with no fitting parameters, reproduces the experimental measurements surprisingly well. Our values for s are different from the ones obtained in previous experiments, where, however, the experimental conditions were different (different force fields, different salt concentrations). Thus it appears that the values of s measured for relatively short hairpins are strongly affected by these experimental conditions.

Allostery through protein-induced DNA bubbles.

Allostery through DNA is increasingly recognized as an important modulator of DNA functions. Here, we show that the coalescence of protein-induced DNA bubbles can mediate allosteric interactions that drive protein aggregation. We propose that such allostery may regulate DNA's flexibility and the assembly of the transcription machinery. Mitochondrial transcription factor A (TFAM), a dual-function protein involved in mitochondrial DNA (mtDNA) packaging and transcription initiation, is an ideal candidate to test such a hypothesis owing to its ability to locally unwind the double helix. Numerical simulations demonstrate that the coalescence of TFAM-induced bubbles can explain experimentally observed TFAM oligomerization. The resulting melted DNA segment, approximately 10 base pairs long, around the joints of the oligomers act as flexible hinges, which explains the efficiency of TFAM in compacting DNA. Since mitochondrial polymerase (mitoRNAP) is involved in melting the transcription bubble, TFAM may use the same allosteric interaction to both recruit mitoRNAP and initiate transcription.

Linking hydrophobicity and hydrodynamics by the hybrid fluctuating hydrodynamics and molecular dynamics methodologies.

The development of a hybrid fluctuating hydrodynamics (FHD) and molecular dynamics (MD) simulation method that combines the molecular dynamics of moving particles with the fluctuating hydrodynamics of solvent fields on Eulerian grid cells is presented. This method allows resolution of solute-solvent interfaces and realization of excluded volumes of particles in the presence of hydrodynamic coupling. With these capabilities, we show that the ubiquitous forces mediated by the solvent, hydrophobicity and hydrodynamics, can be linked in a mesoscopic simulation. The strategies we devise to overcome the numerical issues of mixing variables in the Eulerian and Lagrangian coordinate systems, i.e., using a pair of auxiliary fluids to realize the excluded volumes of particles and assigning collocating gridding systems on solutes to interface with solvent fields, are also presented. Simulation results show that the hybrid FHD and MD method can reproduce the solvation free energies and scaling laws of particles dynamics for hydrophobes of different sizes. The collapse of two hydrophobic particles was also simulated to illustrate that the hybrid FHD and MD method has the potential to be generally applied to study nanoscale self-assembly and dynamics-structure-function relationships of biomolecules.