Subharmonics and Chaos in Simple Periodically Forced Biomolecular Models

This article uncovers a remarkable behavior in two biochemical systems that commonly appear as components of signal transduction pathways in systems biology. These systems have globally attracting steady states when unforced, so they might have been considered uninteresting from a dynamical standpoint. However, when subject to a periodic excitation, strange attractors arise via a period-doubling cascade. Quantitative analyses of the corresponding discrete chaotic trajectories are conducted numerically by computing largest Lyapunov exponents, power spectra, and autocorrelation functions. To gain insight into the geometry of the strange attractors, the phase portraits of the corresponding iterated maps are interpreted as scatter plots for which marginal distributions are additionally evaluated. The lack of entrainment to external oscillations, in even the simplest biochemical networks, represents a level of additional complexity in molecular biology, which has previously been insufficiently recognized but is plausibly biologically important.     

Mathematical Models for Cell Migration with Real-Time Cell Cycle Dynamics

The fluorescent ubiquitination-based cell cycle indicator, also known as FUCCI, allows the visualization of the G1 and S/G2/M cell cycle phases of individual cells. FUCCI consists of two fluorescent probes, so that cells in the G1 phase fluoresce red and cells in the S/G2/M phase fluoresce green. FUCCI reveals real-time information about cell cycle dynamics of individual cells, and can be used to explore how the cell cycle relates to the location of individual cells, local cell density, and different cellular microenvironments. In particular, FUCCI is used in experimental studies examining cell migration, such as malignant invasion and wound healing. Here we present, to our knowledge, new mathematical models that can describe cell migration and cell cycle dynamics as indicated by FUCCI. The fundamental model describes the two cell cycle phases, G1 and S/G2/M, which FUCCI directly labels. The extended model includes a third phase, early S, which FUCCI indirectly labels. We present experimental data from scratch assays using FUCCI-transduced melanoma cells, and show that the predictions of spatial and temporal patterns of cell density in the experiments can be described by the fundamental model. We obtain numerical solutions of both the fundamental and extended models, which can take the form of traveling waves. These solutions are mathematically interesting because they are a combination of moving wavefronts and moving pulses. We derive and confirm a simple analytical expression for the minimum wave speed, as well as exploring how the wave speed depends on the spatial decay rate of the initial condition.     

Rotamer Dynamics: Analysis of Rotamers in Molecular Dynamics Simulations of Proteins

Given by χ torsional angles, rotamers describe the side-chain conformations of amino acid residues in a protein based on the rotational isomers (hence the word rotamer). Constructed rotamer libraries, based on either protein crystal structures or dynamics studies, are the tools for classifying rotamers (torsional angles) in a way that reflect their frequency in nature. Rotamer libraries are routinely used in structure modeling and evaluation. In this perspective article, we would like to encourage researchers to apply rotamer analyses beyond their traditional use. Molecular dynamics (MD) of proteins highlight the in silico behavior of molecules in solution and thus can identify favorable side-chain conformations. In this article, we used simple computational tools to study rotamer dynamics (RD) in MD simulations. First, we isolated each frame in the MD trajectories in separate Protein Data Bank files via the cpptraj module in AMBER. Then, we extracted torsional angles via the Bio3D module in R language. The classification of torsional angles was also done in R according to the penultimate rotamer library. RD analysis is useful for various applications such as protein folding, study of rotamer-rotamer relationship in protein-protein interaction, real-time correlation between secondary structures and rotamers, study of flexibility of side chains in binding site for molecular docking preparations, use of RD as guide in functional analysis and study of structural changes caused by mutations, providing parameters for improving coarse-grained MD accuracy and speed, and many others. Major challenges facing RD to emerge as a new scientific field involve the validation of results via easy, inexpensive wet-lab methods. This realm is yet to be explored.     

Dynamics of Posttranslational Modification Systems: Recent Progress and Future Directions

Posttranslational modification of proteins is important for signal transduction, and hence significant effort has gone toward understanding how posttranslational modification networks process information. This involves, on the theory side, analyzing the dynamical systems arising from such networks. Which networks are, for instance, bistable? Which networks admit sustained oscillations? Which parameter values enable such behaviors? In this Biophysical Perspective, we highlight recent progress in this area and point out some important future directions. Along the way, we summarize several techniques for analyzing general networks, such as eliminating variables to obtain steady-state parameterizations, and harnessing results on how incorporating intermediates affects dynamics.     

ECM Cross-Linking Regulates Invadopodia Dynamics

Invadopodia are membrane protrusions dynamically assembled by invasive cancer cells in contact with the extracellular matrix (ECM). Invadopodia are enriched by the structural proteins actin and cortactin as well as metalloproteases such as MT1-MMP, whose function is to degrade the surrounding ECM. During metastasis, invadopodia are necessary for cancer cell intravasation and extravasation. Although signaling pathways involved in the assembly and function of invadopodia are well studied, few studies address invadopodia dynamics and how the cell-ECM interactions contribute to cell invasion. Using iterative analysis based on time-lapse microscopy and mathematical modeling of invasive cancer cells, we found that cells oscillate between invadopodia presence and cell stasis—termed the “invadopodia state”—and invadopodia absence during cell translocation—termed the “migration state.” Our data suggest that β1-integrin-ECM binding and ECM cross-linking control the duration of each of the two states. By changing the concentration of cross-linkers in two-dimensional and three-dimensional cultures, we generate an ECM in which 0–0.92 of total lysine residues are cross-linked. Using an ECM with a range of cross-linking degrees, we demonstrate that the dynamics of invadopodia-related functions have a biphasic relationship to ECM cross-linking. At intermediate levels of ECM cross-linking (0.39), cells exhibit rapid invadopodia protrusion-retraction cycles and rapid calcium spikes, which lead to more frequent MT1-MMP delivery, causing maximal invadopodia-mediated ECM degradation. In contrast, both extremely high or low levels of cross-linking lead to slower invadopodia-related dynamics and lower ECM degradation. Additionally, β1-integrin inhibition modifies the dynamics of invadopodia-related functions as well as the length of time cells spend in either of the states. Collectively, these data suggest that β1-integrin-ECM binding nonlinearly translates small physical differences in the extracellular environment to differences in the dynamics of cancer cell behaviors. Understanding the conditions under which invadopodia can be reduced by subtle environment-targeting treatments may lead to combination therapies for preventing metastatic spread.     

Architectural Dynamics of CaMKII-Actin Networks

Calcium-calmodulin-dependent kinase II (CaMKII) has an important role in dendritic spine remodeling upon synaptic stimulation. Using fluorescence video microscopy and image analysis, we investigated the architectural dynamics of rhodamine-phalloidin stabilized filamentous actin (F-actin) networks cross-linked by CaMKII. We used automated image analysis to identify F-actin bundles and crossover junctions and developed a dimensionless metric to characterize network architecture. Similar networks were formed by three different CaMKII species with a 10-fold length difference in the linker region between the kinase domain and holoenzyme hub, implying linker length is not a primary determinant of F-actin cross-linking. Electron micrographs showed that at physiological molar ratios, single CaMKII holoenzymes cross-linked multiple F-actin filaments at random, whereas at higher CaMKII/F-actin ratios, filaments bundled. Light microscopy established that the random network architecture resisted macromolecular crowding with polyethylene glycol and blocked ATP-powered compaction by myosin-II miniature filaments. Importantly, the networks disassembled after the addition of calcium-calmodulin and were then spaced within 3 min into compacted foci by myosin motors or more slowly (30 min) aggregated by crowding. Single-molecule total internal reflection fluorescence microscopy showed CaMKII dissociation from surface-immobilized globular actin exhibited a monoexponential dwell-time distribution, whereas CaMKII bound to F-actin networks had a long-lived fraction, trapped at crossover junctions. Release of CaMKII from F-actin, triggered by calcium-calmodulin, was too rapid to measure with flow-cell exchange (<20 s). The residual bound fraction was reduced substantially upon addition of an N-methyl-D-aspartate receptor peptide analog but not ATP. These results provide mechanistic insights to CaMKII-actin interactions at the collective network and single-molecule level. Our findings argue that CaMKII-actin networks in dendritic spines maintain spine size against physical stress. Upon synaptic stimulation, CaMKII is disengaged by calcium-calmodulin, triggering network disassembly, expansion, and subsequent compaction by myosin motors with kinetics compatible with the times recorded for the poststimulus changes in spine volume.     

Lipid Configurations from Molecular Dynamics Simulations

The extent to which current force fields faithfully reproduce conformational properties of lipids in bilayer membranes, and whether these reflect the structural principles established for phospholipids in bilayer crystals, are central to biomembrane simulations. We determine the distribution of dihedral angles in palmitoyl-oleoyl phosphatidylcholine from molecular dynamics simulations of hydrated fluid bilayer membranes. We compare results from the widely used lipid force field of Berger et al. with those from the most recent C36 release of the CHARMM force field for lipids. Only the CHARMM force field produces the chain inequivalence with sn-1 as leading chain that is characteristic of glycerolipid packing in fluid bilayers. The exposure and high partial charge of the backbone carbonyls in Berger lipids leads to artifactual binding of Na⁺ ions reported in the literature. Both force fields predict coupled, near-symmetrical distributions of headgroup dihedral angles, which is compatible with models of interconverting mirror-image conformations used originally to interpret NMR order parameters. The Berger force field produces rotamer populations that correspond to the headgroup conformation found in a phosphatidylcholine lipid bilayer crystal, whereas CHARMM36 rotamer populations are closer to the more relaxed crystal conformations of phosphatidylethanolamine and glycerophosphocholine. CHARMM36 alone predicts the correct relative signs of the time-average headgroup order parameters, and reasonably reproduces the full range of NMR data from the phosphate diester to the choline methyls. There is strong motivation to seek further experimental criteria for verifying predicted conformational distributions in the choline headgroup, including the ³¹P chemical shift anisotropy and ¹⁴N and CD₃ NMR quadrupole splittings.     

A Distribution-Moment Approximation for Coupled Dynamics of the Airway Wall and Airway Smooth Muscle

Asthma is fundamentally a disease of airway constriction. Due to a variety of experimental challenges, the dynamics of airways are poorly understood. Of specific interest is the narrowing of the airway due to forces produced by the airway smooth muscle wrapped around each airway. The interaction between the muscle and the airway wall is crucial for the airway constriction that occurs during an asthma attack. Although cross-bridge theory is a well-studied representation of complex smooth muscle dynamics, and these dynamics can be coupled to the airway wall, this comes at significant computational cost—even for isolated airways. Because many phenomena of interest in pulmonary physiology cannot be adequately understood by studying isolated airways, this presents a significant limitation. We present a distribution-moment approximation of this coupled system and study the validity of the approximation throughout the physiological range. We show that the distribution-moment approximation is valid in most conditions, and we explore the region of breakdown. These results show that in many situations, the distribution-moment approximation is a viable option that provides an orders-of-magnitude reduction in computational complexity; not only is this valuable for isolated airway studies, but it moreover offers the prospect that rich ASM dynamics might be incorporated into interacting airway models where previously this was precluded by computational cost.     

Physical Properties of Bacterial Outer Membrane Models: Neutron Reflectometry & Molecular Simulation

The outer membrane (OM) of Gram-negative bacteria is an asymmetric bilayer having phospholipids in the inner leaflet and lipopolysaccharides in the outer leaflet. This unique asymmetry and the complex carbohydrates in lipopolysaccharides make it a daunting task to study the asymmetrical OM structure and dynamics, its interactions with OM proteins, and its roles in translocation of substrates, including antibiotics. In this study, we combine neutron reflectometry and molecular simulation to explore the physical properties of OM mimetics. There is excellent agreement between experiment and simulation, allowing experimental testing of the conclusions from simulations studies and also atomistic interpretation of the behavior of experimental model systems, such as the degree of lipid asymmetry, the lipid component (tail, head, and sugar) profiles along the bilayer normal, and lateral packing (i.e., average surface area per lipid). Therefore, the combination of both approaches provides a powerful new means to explore the biological and biophysical behavior of the bacterial OM.     

Advances in Micropipette Aspiration: Applications in Cell Biomechanics,Models, and Extended Studies

With five decades of sustained application, micropipette aspiration has enabled a wide range of biomechanical studies in the field of cell mechanics. Here, we provide an update on the use of the technique, with a focus on recent developments in the analysis of the experiments, innovative microaspiration-based approaches, and applications in a broad variety of cell types. We first recapitulate experimental variations of the technique. We then discuss analysis models focusing on important limitations of widely used biomechanical models, which underpin the urge to adopt the appropriate ones to avoid misleading conclusions. The possibilities of performing different studies on the same cell are also considered.     

Multimodal Measurements of Single-Molecule Dynamics Using FluoRBT

Single-molecule methods provide direct measurements of macromolecular dynamics, but are limited by the number of degrees of freedom that can be followed at one time. High-resolution rotor bead tracking (RBT) measures DNA torque, twist, and extension, and can be used to characterize the structural dynamics of DNA and diverse nucleoprotein complexes. Here, we extend RBT to enable simultaneous monitoring of additional degrees of freedom. Fluorescence-RBT (FluoRBT) combines magnetic tweezers, infrared evanescent scattering, and single-molecule FRET imaging, providing real-time multiparameter measurements of complex molecular processes. We demonstrate the capabilities of FluoRBT by conducting simultaneous measurements of extension and FRET during opening and closing of a DNA hairpin under tension, and by observing simultaneous changes in FRET and torque during a transition between right-handed B-form and left-handed Z-form DNA under controlled supercoiling. We discover unanticipated continuous changes in FRET with applied torque, and also show how FluoRBT can facilitate high-resolution FRET measurements of molecular states, by using a mechanical signal as an independent temporal reference for aligning and averaging noisy fluorescence data. By combining mechanical measurements of global DNA deformations with FRET measurements of local conformational changes, FluoRBT will enable multidimensional investigations of systems ranging from DNA structures to large macromolecular machines.