Networks of Causal Linkage Between Eigenmodes Characterize Behavioral Dynamics of Caenorhabditis elegans

Behavioral phenotyping of model organisms has played an important role in unravelling the complexities of animal behavior. Techniques for classifying behavior often rely on easily identified changes in posture and motion. However, such approaches are likely to miss complex behaviors that cannot be readily distinguished by eye (e.g., behaviors produced by high dimensional dynamics). To explore this issue, we focus on the model organism Caenorhabditis elegans, where behaviors have been extensively recorded and classified. Using a dynamical systems lens, we identify high dimensional, nonlinear causal relationships between four basic shapes that describe worm motion (eigenmodes, also called “eigenworms”). We find relationships between all pairs of eigenmodes, but the timescales of the interactions vary between pairs and across individuals. Using these varying timescales, we create “interaction profiles” to represent an individual’s behavioral dynamics. As desired, these profiles are able to distinguish well-known behavioral states: i.e., the profiles for foraging individuals are distinct from those of individuals exhibiting an escape response. More importantly, we find that interaction profiles can distinguish high dimensional behaviors among divergent mutant strains that were previously classified as phenotypically similar. Specifically, we find it is able to detect phenotypic behavioral differences not previously identified in strains related to dysfunction of hermaphrodite-specific neurons.     

Inferring Latent States and Refining Force Estimates via Hierarchical Dirichlet Process Modeling in Single Particle Tracking Experiments

Understanding the basis for intracellular motion is critical as the field moves toward a deeper understanding of the relation between Brownian forces, molecular crowding, and anisotropic (or isotropic) energetic forcing. Effective forces and other parameters used to summarize molecular motion change over time in live cells due to latent state changes, e.g., changes induced by dynamic micro-environments, photobleaching, and other heterogeneity inherent in biological processes. This study discusses limitations in currently popular analysis methods (e.g., mean square displacement-based analyses) and how new techniques can be used to systematically analyze Single Particle Tracking (SPT) data experiencing abrupt state changes in time or space. The approach is to track GFP tagged chromatids in metaphase in live yeast cells and quantitatively probe the effective forces resulting from dynamic interactions that reflect the sum of a number of physical phenomena. State changes can be induced by various sources including: microtubule dynamics exerting force through the centromere, thermal polymer fluctuations, and DNA-based molecular machines including polymerases and protein exchange complexes such as chaperones and chromatin remodeling complexes. Simulations aiming to show the relevance of the approach to more general SPT data analyses are also studied. Refined force estimates are obtained by adopting and modifying a nonparametric Bayesian modeling technique, the Hierarchical Dirichlet Process Switching Linear Dynamical System (HDP-SLDS), for SPT applications. The HDP-SLDS method shows promise in systematically identifying dynamical regime changes induced by unobserved state changes when the number of underlying states is unknown in advance (a common problem in SPT applications). We expand on the relevance of the HDP-SLDS approach, review the relevant background of Hierarchical Dirichlet Processes, show how to map discrete time HDP-SLDS models to classic SPT models, and discuss limitations of the approach. In addition, we demonstrate new computational techniques for tuning hyperparameters and for checking the statistical consistency of model assumptions directly against individual experimental trajectories; the techniques circumvent the need for “ground-truth” and/or subjective information.     

A mixed relaxed clock model

Over recent years, several alternative relaxed clock models have been proposed in the context of Bayesian dating. These models fall in two distinct categories: uncorrelated and autocorrelated across branches. The choice between these two classes of relaxed clocks is still an open question. More fundamentally, the true process of rate variation may have both long-term trends and short-term fluctuations, suggesting that more sophisticated clock models unfolding over multiple time scales should ultimately be developed. Here, a mixed relaxed clock model is introduced, which can be mechanistically interpreted as a rate variation process undergoing short-term fluctuations on the top of Brownian long-term trends. Statistically, this mixed clock represents an alternative solution to the problem of choosing between autocorrelated and uncorrelated relaxed clocks, by proposing instead to combine their respective merits. Fitting this model on a dataset of 105 placental mammals, using both node-dating and tip-dating approaches, suggests that the two pure clocks, Brownian and white noise, are rejected in favour of a mixed model with approximately equal contributions for its uncorrelated and autocorrelated components. The tip-dating analysis is particularly sensitive to the choice of the relaxed clock model. In this context, the classical pure Brownian relaxed clock appears to be overly rigid, leading to biases in divergence time estimation. By contrast, the use of a mixed clock leads to more recent and more reasonable estimates for the crown ages of placental orders and superorders. Altogether, the mixed clock introduced here represents a first step towards empirically more adequate models of the patterns of rate variation across phylogenetic trees.This article is part of the themed issue ‘Dating species divergences using rocks and clocks’.     

Thermal fluctuations biased for directional motion in molecular motors

Recently developed single molecule measurements have demonstrated that the mechanisms for numerous protein functions involve thermal fluctuation, or Brownian motion. Protein interactions bias the random thermal noise in a manner such that the protein can perform its given functions. This phenomenon has been observed in molecular motor unidirectional movement where Brownian motion is used to preferentially bind the motor heads in one direction causing directional motility. This is analogous to that used by proteins in which spontaneous structural fluctuations are used to switch function. Seeing that two very different systems implement similar mechanisms suggests there exists a general scheme applied by diverse proteins that exploits thermal fluctuations in order to achieve their respective functions.     

Fluctuations of Hi-Hat Timing and Dynamics in a Virtuoso Drum Track of a Popular Music Recording

Long-range correlated temporal fluctuations in the beats of musical rhythms are an inevitable consequence of human action. According to recent studies, such fluctuations also lead to a favored listening experience. The scaling laws of amplitude variations in rhythms, however, are widely unknown. Here we use highly sensitive onset detection and time series analysis to study the amplitude and temporal fluctuations of Jeff Porcaro’s one-handed hi-hat pattern in “I Keep Forgettin’”—one of the most renowned 16th note patterns in modern drumming. We show that fluctuations of hi-hat amplitudes and interbeat intervals (times between hits) have clear long-range correlations and short-range anticorrelations separated by a characteristic time scale. In addition, we detect subtle features in Porcaro’s drumming such as small drifts in the 16th note pulse and non-trivial periodic two-bar patterns in both hi-hat amplitudes and intervals. Through this investigation we introduce a step towards statistical studies of the 20th and 21st century music recordings in the framework of complex systems. Our analysis has direct applications to the development of drum machines and to drumming pedagogy.     

A comparison of metrics for estimating phylogenetic signal under alternative evolutionary models

Several metrics have been developed for estimating phylogenetic signal in comparative data. These may be important both in guiding future studies on correlated evolution and for inferring broad-scale evolutionary and ecological processes (e.g., phylogenetic niche conservatism). Notwithstanding, the validity of some of these metrics is under debate, especially after the development of more sophisticated model-based approaches that estimate departure from particular evolutionary models (i.e., Brownian motion). Here, two of these model-based metrics (Blomberg’s K-statistics and Pagel’s λ) are compared with three statistical approaches [Moran’s I autocorrelation coefficient, coefficients of determination from the autoregressive method (ARM), and phylogenetic eigenvector regression (PVR)]. Based on simulations of a trait evolving under Brownian motion for a phylogeny with 209 species, we showed that all metrics are strongly, although non-linearly, correlated to each other. Our analyses revealed that statistical approaches provide valid results and may be still particularly useful when detailed phylogenies are unavailable or when trait variation among species is difficult to describe by more standard Brownian or O-U evolutionary models.     

Heat Generation/Absorption Effects in a Boundary Layer Stretched Flow of Maxwell Nanofluid: Analytic and Numeric Solutions

Analysis has been done to investigate the heat generation/absorption effects in a steady flow of non-Newtonian nanofluid over a surface which is stretching linearly in its own plane. An upper convected Maxwell model (UCM) has been utilized as the non-Newtonian fluid model in view of the fact that it can predict relaxation time phenomenon which the Newtonian model cannot. Behavior of the relaxations phenomenon has been presented in terms of Deborah number. Transport phenomenon with convective cooling process has been analyzed. Brownian motion “D_b” and thermophoresis effects “D_t” occur in the transport equations. The momentum, energy and nanoparticle concentration profiles are examined with respect to the involved rheological parameters namely the Deborah number, source/sink parameter, the Brownian motion parameters, thermophoresis parameter and Biot number. Both numerical and analytic solutions are presented and found in nice agreement. Comparison with the published data is also made to ensure the validity. Stream lines for Maxwell and Newtonian fluid models are presented in the analysis.     

Anomalous versus Slowed-Down Brownian Diffusion in the Ligand-Binding Equilibrium

Measurements of protein motion in living cells and membranes consistently report transient anomalous diffusion (subdiffusion) that converges back to a Brownian motion with reduced diffusion coefficient at long times after the anomalous diffusion regime. Therefore, slowed-down Brownian motion could be considered the macroscopic limit of transient anomalous diffusion. On the other hand, membranes are also heterogeneous media in which Brownian motion may be locally slowed down due to variations in lipid composition. Here, we investigate whether both situations lead to a similar behavior for the reversible ligand-binding reaction in two dimensions. We compare the (long-time) equilibrium properties obtained with transient anomalous diffusion due to obstacle hindrance or power-law-distributed residence times (continuous-time random walks) to those obtained with space-dependent slowed-down Brownian motion. Using theoretical arguments and Monte Carlo simulations, we show that these three scenarios have distinctive effects on the apparent affinity of the reaction. Whereas continuous-time random walks decrease the apparent affinity of the reaction, locally slowed-down Brownian motion and local hindrance by obstacles both improve it. However, only in the case of slowed-down Brownian motion is the affinity maximal when the slowdown is restricted to a subregion of the available space. Hence, even at long times (equilibrium), these processes are different and exhibit irreconcilable behaviors when the area fraction of reduced mobility changes.     

Anomalous versus Slowed-Down Brownian Diffusion in the Ligand-Binding Equilibrium

Measurements of protein motion in living cells and membranes consistently report transient anomalous diffusion (subdiffusion) that converges back to a Brownian motion with reduced diffusion coefficient at long times after the anomalous diffusion regime. Therefore, slowed-down Brownian motion could be considered the macroscopic limit of transient anomalous diffusion. On the other hand, membranes are also heterogeneous media in which Brownian motion may be locally slowed down due to variations in lipid composition. Here, we investigate whether both situations lead to a similar behavior for the reversible ligand-binding reaction in two dimensions. We compare the (long-time) equilibrium properties obtained with transient anomalous diffusion due to obstacle hindrance or power-law-distributed residence times (continuous-time random walks) to those obtained with space-dependent slowed-down Brownian motion. Using theoretical arguments and Monte Carlo simulations, we show that these three scenarios have distinctive effects on the apparent affinity of the reaction. Whereas continuous-time random walks decrease the apparent affinity of the reaction, locally slowed-down Brownian motion and local hindrance by obstacles both improve it. However, only in the case of slowed-down Brownian motion is the affinity maximal when the slowdown is restricted to a subregion of the available space. Hence, even at long times (equilibrium), these processes are different and exhibit irreconcilable behaviors when the area fraction of reduced mobility changes.     

Are our phylomorphospace plots so terribly tangled? An investigation of disorder in data simulated under adaptive and nonadaptive models

Contemporary methods for visualizing phenotypic evolution, such as phylomorphospaces, often reveal patterns which depart strongly from a naïve expectation of consistently divergent branching and expansion. Instead, branches regularly crisscross as convergence, reversals, or other forms of homoplasy occur, forming patterns described as “birds’ nests”, “flies in vials”, or less elegantly, “a mess”. In other words, the phenotypic tree of life often appears highly tangled. Various explanations are given for this, such as differential degrees of developmental constraint, adaptation, or lack of adaptation. However, null expectations for the magnitude of disorder or “tangling” have never been established, so it is unclear which or even whether various evolutionary factors are required to explain messy patterns of evolution. I simulated evolution along phylogenies under a number of varying parameters (number of taxa and number of traits) and models (Brownian motion, Ornstein–Uhlenbeck (OU)-based, early burst, and character displacement (CD)] and quantified disorder using 2 measures. All models produce substantial amounts of disorder. Disorder increases with tree size and the number of phenotypic traits. OU models produced the largest amounts of disorder—adaptive peaks influence lineages to evolve within restricted areas, with concomitant increases in crossing of branches and density of evolution. Large early changes in trait values can be important in minimizing disorder. CD consistently produced trees with low (but not absent) disorder. Overall, neither constraints nor a lack of adaptation is required to explain messy phylomorphospaces—both stochastic and deterministic processes can act to produce the tantalizingly tangled phenotypic tree of life.     

Evolution of Conformity in Social Dilemmas

People often deviate from their individual Nash equilibrium strategy in game experiments based on the prisoner’s dilemma (PD) game and the public goods game (PGG), whereas conditional cooperation, or conformity, is supported by the data from these experiments. In a complicated environment with no obvious “dominant” strategy, conformists who choose the average strategy of the other players in their group could be able to avoid risk by guaranteeing their income will be close to the group average. In this paper, we study the repeated PD game and the repeated m-person PGG, where individuals’ strategies are restricted to the set of conforming strategies. We define a conforming strategy by two parameters, initial action in the game and the influence of the other players’ choices in the previous round. We are particularly interested in the tit-for-tat (TFT) strategy, which is the well-known conforming strategy in theoretical and empirical studies. In both the PD game and the PGG, TFT can prevent the invasion of non-cooperative strategy if the expected number of rounds exceeds a critical value. The stability analysis of adaptive dynamics shows that conformity in general promotes the evolution of cooperation, and that a regime of cooperation can be established in an AllD population through TFT-like strategies. These results provide insight into the emergence of cooperation in social dilemma games.     

Computational Analysis of Viscoelastic Properties of Crosslinked Actin Networks

Mechanical force plays an important role in the physiology of eukaryotic cells whose dominant structural constituent is the actin cytoskeleton composed mainly of actin and actin crosslinking proteins (ACPs). Thus, knowledge of rheological properties of actin networks is crucial for understanding the mechanics and processes of cells. We used Brownian dynamics simulations to study the viscoelasticity of crosslinked actin networks. Two methods were employed, bulk rheology and segment-tracking rheology, where the former measures the stress in response to an applied shear strain, and the latter analyzes thermal fluctuations of individual actin segments of the network. It was demonstrated that the storage shear modulus (G′) increases more by the addition of ACPs that form orthogonal crosslinks than by those that form parallel bundles. In networks with orthogonal crosslinks, as crosslink density increases, the power law exponent of G′ as a function of the oscillation frequency decreases from 0.75, which reflects the transverse thermal motion of actin filaments, to near zero at low frequency. Under increasing prestrain, the network becomes more elastic, and three regimes of behavior are observed, each dominated by different mechanisms: bending of actin filaments, bending of ACPs, and at the highest prestrain tested (55%), stretching of actin filaments and ACPs. In the last case, only a small portion of actin filaments connected via highly stressed ACPs support the strain. We thus introduce the concept of a ‘supportive framework,’ as a subset of the full network, which is responsible for high elasticity. Notably, entropic effects due to thermal fluctuations appear to be important only at relatively low prestrains and when the average crosslinking distance is comparable to or greater than the persistence length of the filament. Taken together, our results suggest that viscoelasticity of the actin network is attributable to different mechanisms depending on the amount of prestrain.     

Cytoplasmic diffusion: molecular motors mix it up

Random motion within the cytoplasm gives rise to molecular diffusion; this motion is essential to many biological processes. However, in addition to thermal Brownian motion, the cytoplasm also undergoes constant agitation caused by the activity of molecular motors and other nonequilibrium cellular processes. Here, we discuss recent work that suggests this activity can give rise to cytoplasmic motion that has the appearance of diffusion but is significantly enhanced in its magnitude and which can play an important biological role, particularly in cytoskeletal assembly.The cytoplasm of eukaryotic cells is a highly dynamic and out-of-equilibrium material that undergoes continual restructuring. This is largely driven by active processes such as polymerization of cytoskeletal filaments and forces generated by molecular motors. Such activity usually results in directed movement within cells; examples include the slow retrograde flow at the leading edge during cell crawling (Fisher et al., 1988; Waterman-Storer and Salmon, 1997; Cai et al., 2006) and the transport of motor-bound vesicles along cytoskeletal filaments (Vale, 2003). Given the intrinsically small, micrometer scales involved, the cytoplasm is also subject to the thermal agitation of Brownian motion (Brown, 1828; Einstein, 1905). Random though these thermal fluctuations may be, they are implicated in force generation by polymerization (Peskin et al., 1993; Mogilner and Oster, 1996), as well as the elastic response of cytoskeletal networks (MacKintosh et al., 1995; Gardel et al., 2004; Storm et al., 2005). Moreover, thermal fluctuations give rise to the diffusive transport of small molecules throughout the cell, without which molecular signaling would be impossible. However, it is becoming increasingly clear that nonthermal forces, such as those resulting from motor protein activity, can also lead to strongly fluctuating intracellular motion; this motion differs significantly from the directed motion commonly associated with motor activity (Caspi et al., 2000; Lau et al., 2003; Bursac et al., 2005). These active fluctuations remain poorly understood, and the relative contributions of thermal fluctuations compared with active fluctuations in living cells are only now being fully explored.

Particle-based probes of intracellular motion

To elucidate the nature of fluctuating motion in cells, many studies have analyzed the “passive” fluctuating motion of micrometer-sized spherical probe particles. If such particles were embedded in a viscous liquid driven by thermal Brownian fluctuations, they would exhibit random, diffusive motion, as shown schematically in Fig. 1 a (blue particle). Quantitatively, this means that the distance the particle has moved, Δx, after some time interval, τ, is described by , where the angled brackets indicate an average over many particles, and the diffusion coefficient, D, is given by the Stokes-Einstein equation:which depends on the viscosity η and particle radius a (Einstein, 1905). This reflects the fundamentally thermal origin of diffusion in an equilibrium liquid, depending as well on the temperature (T) and Boltzmann''s constant (k_B). This diffusive time dependence is shown schematically by the blue line in Fig. 1 a. Such motion is in stark contrast with steady particle motion in one direction with constant velocity v, illustrated by the black curve in Fig. 1 a; because , this motion is described by . In principle, one can track the motion of inert tracer particles in cells to determine whether their motion is diffusive with the appropriate diffusion coefficient. However, inside cells this picture is complicated by the fact that the cytoplasm is generally not a simple viscous liquid but rather a structured viscoelastic material (Luby-Phelps et al., 1987; Fabry et al., 2001). In a viscoelastic material, thermal fluctuations do not lead to ordinary diffusion, but rather “subdiffusive” motion, characterized by a different time-dependence: , where α < 1, as shown schematically by the red curve in Fig. 1 a. Thus, studies showing diffusive, or even “superdiffusive” motion (α > 1), within the viscoelastic cytoplasm suggested this random motion is not thermally induced (Caspi et al., 2000). However, other studies assume that random intracellular motion is thermally induced, and use this assumption to extract the mechanical properties of the cell from tracer particle motion (Tseng et al., 2002; Panorchan et al., 2006). To quantitatively clarify this, several recent studies have analyzed both measurements of the “passive” random motion of tracer particles, as well as direct, active measurements of the viscoelastic properties of the cytoplasm, for example, using a magnetic tweezer to pull on cells (Lau et al., 2003; Bursac et al., 2005). These studies have concluded that the random motion not only has an unexpected time dependence but is also significantly larger in magnitude than would be expected for purely thermal fluctuations, suggesting that biological activity can indeed give rise to random fluctuating motion that dominates over thermal fluctuations.Open in a separate windowFigure 1.Quantifying fluctuating motion. (a) Schematic showing different types of particle motion, characterized by the mean square displacement . The particle can exhibit directed (“ballistic”) motion, as shown by the black particle; here α = 2, as shown by the black curve. Particles can also exhibit random diffusive-like behavior, as shown by the blue particle, with α = 1 (blue line). Particles constrained by a viscoelastic network (red particle) often exhibit subdiffusive behavior, with α < 1 (red curve). (b) Data showing the bending motion of a fluctuating intracellular microtubule. The amplitude of a bend, ∝a_q, randomly fluctuates in time, as shown in the bottom inset. The mean squared amplitude difference as a function of lag time displays diffusive-like behavior (α = 1). (c) A CHO cell fixed and stained to reveal the nucleus (blue) and microtubules (green). Bottom inset shows schematically the variables used in the Fourier analysis of microtubule bending. Top inset shows fluctuations in a GFP-tubulin–transfected Cos7 cell over a time difference of 1.6 s. Microtubules that have fluctuated to a new position are in red and the earlier position is in green.Considerable new insight into the underlying physical origin of these motor-driven fluctuations was obtained from studies of a reconstituted actin network incorporating myosin II motors (Mizuno et al., 2007), oligomerized into processive motor assemblies similar to those found in the cellular cytoskeleton (Cai et al., 2006). The mechanical resistance of the actin network was precisely characterized using optical tweezers to actively pull on particles within the network. The fluctuating motion of the particles was also measured, both with and without motors present. By comparing these two measurements, the contribution of the thermal motion could be distinguished from that of the nonthermal motion, showing clearly that myosin motors can give rise to strong random fluctuating motion within the network. Interestingly, these random myosin-generated forces can also lead to a pronounced stiffening of the actin network, increasing the rigidity by as much as 100-fold (Mizuno et al., 2007; unpublished data).

Microtubule bending dynamics reflects active fluctuations

The random fluctuating motion arising from the activity of cytoskeletal motor proteins can have important biophysical consequences. In an attempt to directly measure the underlying fluctuating forces, a different but complementary approach has recently been developed. It uses endogenous cytoskeletal microtubules as probes. Microtubules are stiff biopolymer filaments that are present in almost every animal cell and are physically linked to other components of the cytoskeleton (Rodriguez et al., 2003; Rosales-Nieves et al., 2006). Thus, as with spherical probe particles, their motion reflects forces and fluctuations of the network (Waterman-Storer and Salmon, 1997; Odde et al., 1999). However, in contrast to spherical probes, microtubules also exhibit a local bending motion, whose amplitude can, like a simple elastic spring, be used to determine the applied force (Fig. 1 b). Microtubules in cells indeed appear highly bent, as can be seen, for example, in the fluorescence image of the microtubule network within an adherent CHO cell, shown in Fig. 1 c. These bends fluctuate dynamically in time, which can be seen by subtracting sequential images, as shown in the top inset of Fig. 1 c.This motion was studied in cells by tracking individual fluorescent microtubules, using both Cos7 and CHO cells (Brangwynne et al., 2007b). The curves defined by the microtubules are analyzed using a Fourier analysis technique developed to characterize the bending fluctuations of isolated biopolymers in thermal equilibrium (Gittes et al., 1993; Brangwynne et al., 2007a). This analysis is based on the fact that each curve can be represented as a sum of simpler sinusoidal curves, each of a different amplitude (a_q) and wavelength (λ). The wavelength is typically written in terms of its wave vector, q = 2π/λ, as sketched in the bottom inset of Fig. 1 c. Using this approach, intracellular motion was analyzed by calculating the mean-squared difference in amplitude using , as a function of lag time τ.This quantity is analogous to that calculated for fluctuating particles, , but here effectively measures the fluctuating motion of the microtubule at different wavelengths. In thermal equilibrium, the amplitude difference will grow with τ up to a maximum value given bywhere k_BT is the thermal energy scale and κ is the microtubule bending rigidity. The ratio of these defines the persistence length:which represents the length scale at which thermal fluctuations completely change the direction of the filament; for microtubules, l_p is on the order of 1 mm (Gittes et al., 1993). This establishes the maximum amplitude of bending fluctuations that can be induced by thermal agitation and allows thermal and motor-induced fluctuations within the cell to be distinguished.For microtubules in cells, the amplitude of the fluctuations was found to grow roughly linearly in time, the behavior expected for simple Brownian diffusion (Fig. 1 b). However, strikingly, the maximum bending amplitude in cells is much larger than that expected for thermally induced bends for l_p ≈ 1 mm. This is most apparent for small wavelength bends, q > 1 μm⁻¹, as shown by the blue triangles in Fig. 2 a, which are significantly above the maximum thermal amplitude shown by the solid line. Thus, although random intracellular motion can exhibit features similar to random Brownian motion, it appears inconsistent with a purely thermal origin.Open in a separate windowFigure 2.Microtubule bending in vivo and in vitro. (a) Thermal microtubules in an in vitro network of F-actin exhibit a roughly q⁻² spectrum of fluctuations (solid line), although wave vectors smaller than q ∼0.4 μm⁻¹ have not reached their maximum fluctuations on this time scale (τ = 2 s; red squares). In the presence of myosin II motors (blue squares), the bending fluctuations are significantly larger than thermal on short wavelengths (high q). Curves are means of 10 filaments. Intracellular microtubule fluctuations show similar behavior (blue triangles), with amplitudes larger than thermal on short wavelengths, as shown by the mean of 23 filaments from a CHO cell (τ = 2 s). (b) Microtubules embedded in an in vitro actin network in thermal equilibrium exhibit small fluctuations (inset, red squares, q ∼0.3 μm⁻¹), which evolve subdiffusively, i.e., , because of the elasticity of the surrounding actin network (red squares, mean of ∼10 filaments). In myosin-driven networks, the fluctuations are significantly larger and steplike (inset, blue squares, q ∼0.3 μm⁻¹). These large nonthermal fluctuations are diffusive in character, i.e., (blue squares, mean of ∼10 filaments). (c) The bending fluctuations of microtubules in vitro are highly localized and relax rapidly as shown in the top right inset (78 ms between each frame, top to bottom). These localized bends can be well fit to the expected shape resulting from transverse point forces (Landau and Lifshitz, 1986; Brangwynne et al., 2008). A localized bend with the fit to the theoretical form (red line) is shown in the top inset. From these fits, a distribution of localized force pulses with a mean magnitude of ∼10 pN is found (main plot).In these experiments in living cells, there are several unknown variables that could play a role. For example, the persistence length of microtubules may vary within the cell, caused by a possible length dependence (Pampaloni et al., 2006) or arising from the effects of microtubule-associated proteins (Felgner et al., 1997). A simplified in vitro cytoskeleton was therefore developed, incorporating purified microtubules in a model actin network (Brangwynne et al., 2008). In the absence of motor proteins or other sources of nonequilibrium activity, microtubules embedded in the actin network are subject to only thermal forces that result in small bending fluctuations; as with the motion of spherical probe particles, the viscoelasticity of the surrounding network leads to subdiffusive behavior of the bending amplitudes, , as shown in Fig. 2 b. These fluctuations display a small maximum amplitude corresponding to l_p ≈ 1 mm, as shown by the red squares in Fig. 2 a. When myosin II motor assemblies are added to the actin network, the embedded microtubules show dramatically different behavior: they bend significantly more, and the bends are highly localized. However, although this behavior is driven by processive motor activity, the microtubule bends fluctuate randomly in time, with localized bends growing and shrinking rapidly, as illustrated by the typical time series shown in the inset of Fig. 2 b; this behavior is similar to that found using spherical probe particles (Mizuno et al., 2007). These microtubule bends appear to result from localized transverse forces (Landau and Lifshitz, 1986), as sketched in the bottom inset of Fig. 2 c. Analogous to a simple spring (force proportional to displacement, F = kΔx), the maximum force was directly determined from the amplitude of these bends, revealing force pulses on the order of 10 pN, which is consistent with that of a few myosin motors acting together (Finer et al., 1994). Interestingly, short wavelength bends can also arise from compressive forces acting within cells (Brangwynne et al., 2006).The dynamic, localized microtubule bends observed in vitro lead to fluctuations in the bending amplitudes that are large and distinctly nonthermal at short wavelengths (large q), as shown by the comparison of thermal (Fig. 2 a, red squares) and motor-driven (blue squares) fluctuations for q ≥ 0.2 μm⁻¹. At longer wavelengths, however, the fluctuations are indistinguishable from those of thermally excited filaments, as expected for microtubules whose lateral motion is restricted by the surrounding elastic environment. Surprisingly, with added myosin motors, the actively driven bends fluctuate in a diffusive-like manner, , as shown in Fig. 2 b (blue squares). This nonthermal diffusive behavior can be understood in terms of steplike or on–off dynamics in the forces applied by individual motor assemblies as they bind and unbind to cytoskeletal filaments (Mizuno et al., 2007; MacKintosh and Levine, 2008). Thus, even processive motor activity has stochastic features that can give rise to diffusive-like motion of cytoplasmic components. Although it is likely that myosin II is not the only motor contributing to this behavior in cells, these experiments help elucidate the underlying biophysical processes.

Implications of active fluctuations for transport and cytoskeletal assembly

These random nonthermal force fluctuations appear to be a ubiquitous feature of living cells. They can, therefore, play an important role in a variety of cellular processes. For example, as a result of large microtubule bending fluctuations, the tips of growing microtubules also undergo large fluctuations in directional orientation, leading to highly curved microtubule shapes that appear to be “frozen-in” by the surrounding elastic cytoskeleton, as shown in the example in Fig. 3 a. These random fluctuations in tip orientation are analogous to random thermal fluctuations and give rise to microtubule bends with a thermal-like dependence on q, as shown in Fig. 3 b. However, the corresponding nonequilibrium persistence length is reduced to ∼30 μm, ∼100-fold less than the thermal persistence length. Thus, the microtubule network is more bent by ∼100-fold in these cells as compared with isolated microtubules in solution. Although the effects of microtubule binding proteins or defects in the tubulin lattice could also contribute, tip fluctuations alone are sufficient to explain these large bends. Driven fluctuations can thus play a significant role in determining cytoskeletal architecture.Open in a separate windowFigure 3.Nonequilibrium fluctuations affect the growth dynamics and final structure of the microtubule network. (a) A microtubule within a GFP-tubulin–transfected Cos7 cell, highlighted in red, can be seen growing toward the bottom right, in the direction indicated by the yellow line. In the second frame, the filament experiences a naturally occurring bending fluctuation caused by internal forces, indicated by the arrow. As a result, the orientation of the microtubule tip changes and the microtubule grows upward, giving rise to a long wavelength bend. Frame times are 0, 36, 46, 53, and 92 s, top to bottom. (b) Inset schematic shows how lateral bending fluctuations of microtubules will cause fluctuations in the microtubule tip during growth, giving rise to curved polymerization trajectories. The Fourier spectrum of a single representative microtubule in thermal equilibrium is shown by the green circles, calculated from the maximum variance of the fluctuations. This microtubule has a persistence length, l_p = 4 mm. The Fourier amplitude of an ensemble of microtubules in CHO cells, , is shown by the blue squares.The enhanced diffusive dynamics that result from motor activity may also affect the rates of biochemical reactions that take place on the cytoskeletal scaffold (Forgacs et al., 2004), as well as those usually thought to be limited by thermal diffusion. Thus, “active” cytoplasmic diffusion could represent a kind of microscopic mixing that enables rapid diffusion of vesicles and small molecules.Thermal fluctuations have been known to play a fundamental role in the behavior of all nonliving matter since Einstein''s seminal work in 1905 (Einstein, 1905), in a paper explaining how thermal forces give rise to the diffusive motion first observed by Brown in 1828 (Brown, 1828). We propose that active intracellular force fluctuations represent the biological analogue, controlling cell behavior while subject to biochemical regulation. Interestingly, this may actually be closer to Brown''s initial concept of a vital microscopic activity, in contrast to the nonliving, thermal motion that bears his name (Brown, 1828). This active motion is clearly a ubiquitous and important phenomenon in living cells; indeed, it may be that active processes contribute to virtually all randomly fluctuating, diffusive-like motion in cells.     

The Independent Evolution Method Is Not a Viable Phylogenetic Comparative Method

Phylogenetic comparative methods (PCMs) use data on species traits and phylogenetic relationships to shed light on evolutionary questions. Recently, Smaers and Vinicius suggested a new PCM, Independent Evolution (IE), which purportedly employs a novel model of evolution based on Felsenstein’s Adaptive Peak Model. The authors found that IE improves upon previous PCMs by producing more accurate estimates of ancestral states, as well as separate estimates of evolutionary rates for each branch of a phylogenetic tree. Here, we document substantial theoretical and computational issues with IE. When data are simulated under a simple Brownian motion model of evolution, IE produces severely biased estimates of ancestral states and changes along individual branches. We show that these branch-specific changes are essentially ancestor-descendant or “directional” contrasts, and draw parallels between IE and previous PCMs such as “minimum evolution”. Additionally, while comparisons of branch-specific changes between variables have been interpreted as reflecting the relative strength of selection on those traits, we demonstrate through simulations that regressing IE estimated branch-specific changes against one another gives a biased estimate of the scaling relationship between these variables, and provides no advantages or insights beyond established PCMs such as phylogenetically independent contrasts. In light of our findings, we discuss the results of previous papers that employed IE. We conclude that Independent Evolution is not a viable PCM, and should not be used in comparative analyses.     

Time Dependent MHD Nano-Second Grade Fluid Flow Induced by Permeable Vertical Sheet with Mixed Convection and Thermal Radiation

The aim of present paper is to study the series solution of time dependent MHD second grade incompressible nanofluid towards a stretching sheet. The effects of mixed convection and thermal radiation are also taken into account. Because of nanofluid model, effects Brownian motion and thermophoresis are encountered. The resulting nonlinear momentum, heat and concentration equations are simplified using appropriate transformations. Series solutions have been obtained for velocity, temperature and nanoparticle fraction profiles using Homotopy Analysis Method (HAM). Convergence of the acquired solution is discussed critically. Behavior of velocity, temperature and concentration profiles on the prominent parameters is depicted and argued graphically. It is observed that temperature and concentration profiles show similar behavior for thermophoresis parameter Νt but opposite tendency is noted in case of Brownian motion parameter Νb. It is further analyzed that suction parameter S and Hartman number Μ depict decreasing behavior on velocity profile.     

Nanoblinker: Brownian Motion Powered Bio-Nanomachine for FRET Detection of Phagocytic Phase of Apoptosis

We describe a new type of bio-nanomachine which runs on thermal noise. The machine is solely powered by the random motion of water molecules in its environment and does not ever require re-fuelling. The construct, which is made of DNA and vaccinia virus topoisomerase protein, can detect DNA damage by employing fluorescence. It uses Brownian motion as a cyclic motor to continually separate and bring together two types of fluorescent hairpins participating in FRET. This bio-molecular oscillator is a fast and specific sensor of 5′OH double-strand DNA breaks present in phagocytic phase of apoptosis. The detection takes 30 s in solution and 3 min in cell suspensions. The phagocytic phase is critical for the effective execution of apoptosis as it ensures complete degradation of the dying cells’ DNA, preventing release of pathological, viral and tumor DNA and self-immunization. The construct can be used as a smart FRET probe in studies of cell death and phagocytosis.     

Variation in Thermal Sensitivity and Thermal Tolerances in an Invasive Species across a Climatic Gradient: Lessons from the Land Snail Cornu aspersum

The ability of organisms to perform at different temperatures could be described by a continuous nonlinear reaction norm (i.e., thermal performance curve, TPC), in which the phenotypic trait value varies as a function of temperature. Almost any shift in the parameters of this performance curve could highlight the direct effect of temperature on organism fitness, providing a powerful framework for testing thermal adaptation hypotheses. Inter-and intraspecific differences in this performance curve are also reflected in thermal tolerances limits (e.g., critical and lethal limits), influencing the biogeographic patterns of species’ distribution. Within this context, here we investigated the intraspecific variation in thermal sensitivities and thermal tolerances in three populations of the invasive snail Cornu aspersum across a geographical gradient, characterized by different climatic conditions. Thus, we examined population differentiation in the TPCs, thermal-coma recovery times, expression of heat-shock proteins and standard metabolic rate (i.e., energetic costs of physiological differentiation). We tested two competing hypotheses regarding thermal adaptation (the “hotter is better” and the generalist-specialist trade-offs). Our results show that the differences in thermal sensitivity among populations of C. aspersum follow a latitudinal pattern, which is likely the result of a combination of thermodynamic constraints (“hotter is better”) and thermal adaptations to their local environments (generalist-specialist trade-offs). This finding is also consistent with some thermal tolerance indices such as the Heat-Shock Protein Response and the recovery time from chill-coma. However, mixed responses in the evaluated traits suggest that thermal adaptation in this species is not complete, as we were not able to detect any differences in neither energetic costs of physiological differentiation among populations, nor in the heat-coma recovery.     

Direct Observation of Single DNA Structural Alterations at Low Forces with Surface-Enhanced Raman Scattering

DNA experiences numerous mechanical events, necessitating single-molecule force spectroscopy techniques to provide insight into DNA mechanics as a whole system. Inherent Brownian motion limits current force spectroscopy methods from observing possible bond level structural changes. We combine optical trapping and surface-enhanced Raman scattering to establish a direct relationship between DNA’s extension and structure in the low force, entropic regime. A DNA molecule is trapped close to a surface-enhanced Raman scattering substrate to facilitate a detectable Raman signal. DNA Raman modes shift in response to applied force, indicating phosphodiester mechanical alterations. Molecular dynamic simulations confirm the local structural alterations and the Raman sensitive band identified experimentally. The combined Raman and force spectroscopy technique, to our knowledge, is a novel methodology that can be generalized to all single-molecule studies.     

A Coarse-Grained Model for Polyglutamine Aggregation Modulated by Amphipathic Flanking Sequences

The aggregation of proteins with expanded polyglutamine (polyQ) tracts is directly relevant to the formation of neuronal intranuclear inclusions in Huntington’s disease. In vitro studies have uncovered the effects of flanking sequences as modulators of the driving forces and mechanisms of polyQ aggregation in sequence segments associated with HD. Specifically, a seventeen-residue amphipathic stretch (N17) that is directly N-terminal to the polyQ tract in huntingtin decreases the overall solubility, destabilizes nonfibrillar aggregates, and accelerates fibril formation. Published results from atomistic simulations showed that the N17 module reduces the frequency of intermolecular association. Our reanalysis of these simulation results demonstrates that the N17 module also reduces interchain entanglements between polyQ domains. These two effects, which are observed on the smallest lengthscales, are incorporated into phenomenological pair potentials and used in coarse-grained Brownian dynamics simulations to investigate their impact on large-scale aggregation. We analyze the results from Brownian dynamics simulations using the framework of diffusion-limited cluster aggregation. When entanglements prevail, which is true in the absence of N17, small spherical clusters and large linear aggregates form on distinct timescales, in accord with in vitro experiments. Conversely, when entanglements are quenched and a barrier to intermolecular associations is introduced, both of which are attributable to N17, the timescales for forming small species and large linear aggregates become similar. Therefore, the combination of a reduction of interchain entanglements through homopolymeric polyQ and barriers to intermolecular associations appears to be sufficient for providing a minimalist phenomenological rationalization of in vitro observations regarding the effects of N17 on polyQ aggregation.