Inferring Diffusion Dynamics from FCS in Heterogeneous Nuclear Environments

Fluorescence correlation spectroscopy (FCS) is a noninvasive technique that probes the diffusion dynamics of proteins down to single-molecule sensitivity in living cells. Critical mechanistic insight is often drawn from FCS experiments by fitting the resulting time-intensity correlation function, G(t), to known diffusion models. When simple models fail, the complex diffusion dynamics of proteins within heterogeneous cellular environments can be fit to anomalous diffusion models with adjustable anomalous exponents. Here, we take a different approach. We use the maximum entropy method to show—first using synthetic data—that a model for proteins diffusing while stochastically binding/unbinding to various affinity sites in living cells gives rise to a G(t) that could otherwise be equally well fit using anomalous diffusion models. We explain the mechanistic insight derived from our method. In particular, using real FCS data, we describe how the effects of cell crowding and binding to affinity sites manifest themselves in the behavior of G(t). Our focus is on the diffusive behavior of an engineered protein in 1) the heterochromatin region of the cell’s nucleus as well as 2) in the cell’s cytoplasm and 3) in solution. The protein consists of the basic region-leucine zipper (BZip) domain of the CCAAT/enhancer-binding protein (C/EBP) fused to fluorescent proteins.     

Quantifying the Diffusion of Membrane Proteins and Peptides in Black Lipid Membranes with 2-Focus Fluorescence Correlation Spectroscopy

Protein diffusion in lipid membranes is a key aspect of many cellular signaling processes. To quantitatively describe protein diffusion in membranes, several competing theoretical models have been proposed. Among these, the Saffman-Delbrück model is the most famous. This model predicts a logarithmic dependence of a protein’s diffusion coefficient on its inverse hydrodynamic radius (D ∝ ln 1/R) for small radius values. For large radius values, it converges toward a D ∝ 1/R scaling. Recently, however, experimental data indicate a Stokes-Einstein-like behavior (D ∝ 1/R) of membrane protein diffusion at small protein radii. In this study, we investigate protein diffusion in black lipid membranes using dual-focus fluorescence correlation spectroscopy. This technique yields highly accurate diffusion coefficients for lipid and protein diffusion in membranes. We find that despite its simplicity, the Saffman-Delbrück model is able to describe protein diffusion extremely well and a Stokes-Einstein-like behavior can be ruled out.     

Neurokernel: An Open Source Platform for Emulating the Fruit Fly Brain

We have developed an open software platform called Neurokernel for collaborative development of comprehensive models of the brain of the fruit fly Drosophila melanogaster and their execution and testing on multiple Graphics Processing Units (GPUs). Neurokernel provides a programming model that capitalizes upon the structural organization of the fly brain into a fixed number of functional modules to distinguish between these modules’ local information processing capabilities and the connectivity patterns that link them. By defining mandatory communication interfaces that specify how data is transmitted between models of each of these modules regardless of their internal design, Neurokernel explicitly enables multiple researchers to collaboratively model the fruit fly’s entire brain by integration of their independently developed models of its constituent processing units. We demonstrate the power of Neurokernel’s model integration by combining independently developed models of the retina and lamina neuropils in the fly’s visual system and by demonstrating their neuroinformation processing capability. We also illustrate Neurokernel’s ability to take advantage of direct GPU-to-GPU data transfers with benchmarks that demonstrate scaling of Neurokernel’s communication performance both over the number of interface ports exposed by an emulation’s constituent modules and the total number of modules comprised by an emulation.     

Molecular Dynamics of Class A β-lactamases—Effects of Substrate Binding

The effects of substrate binding on class A β-lactamase dynamics were studied using molecular dynamics simulations of two model enzymes; 40 100-ns trajectories of the free and substrate-bound forms of TEM-1 (with benzylpenicillin) and PSE-4 (with carbenicillin) were recorded (totaling 4.0 μs). Substrates were parameterized with the CHARMM General Force Field. In both enzymes, the Ω loop exhibits a marked flexibility increase upon substrate binding, supporting the hypothesis of substrate gating. However, specific interactions that are formed or broken in the Ω loop upon binding differ between the two enzymes: dynamics are conserved, but not specific interactions. Substrate binding also has a global structuring effect on TEM-1, but not on PSE-4. Changes in TEM-1’s normal modes show long-range effects of substrate binding on enzyme dynamics. Hydrogen bonds observed in the active site are mostly preserved upon substrate binding, and new, transient interactions are also formed. Agreement between NMR relaxation parameters and our theoretical results highlights the dynamic duality of class A β-lactamases: enzymes that are highly structured on the ps-ns timescale, with important flexibility on the μs-ms timescale in regions such as the Ω loop.     

Evolution of social behavior by reciprocation

A general model is presented for the evolution of social behavior by reciprocation. The results of our model apply to social traits which are transmitted from one generation to the next by a process which guarantees that the frequency of the trait in one generation is directly related to its fitness in the preceding generation. The basic parameters of the model are α, the number of interactions per generation, and β, the number of these interactions which are with individuals who are perceived as strangers. It is shown that so long as α/β can be made large, social reciprocation may increase when arbitrarily rare even in the absence of population structure.This conclusion appears to be at odds with several recent investigations of Axelrod & Hamilton (1981) and Boorman & Levitt (1980). We use our model to reconcile these various approaches. By casting Axelrod & Hamilton's (1981) single-partner model in terms of the general parameters, α and β, we show that social reciprocation can increase when arbitrarily rare in a homogeneous population dominated by non-cooperators. Using a gene frequency approach, Boorman & Levitt (1973, 1980) demonstrated the existence of a selection threshold in frequency of the social trait, which must be surmounted for social reciprocation to increase. We show our analysis of reciprocation to be consistent with Boorman and Levitt's result, since for our general model the cost to the social individuals of learning the non-social's identity goes to zero as the ratio α/β gets large. We also use our general model to study two multi-partner models not considered elsewhere, which differ in regards to the memory capabilities assigned to the organism.Finally we use our model to compare directly the evolution of social behavior by reciprocation with the main alternate hypothesis, kin selection. We show that an act which accrues some cost ?c to the fitness of the donor while benefiting a recipient by b, will increase in frequency so long as c/b < Φ (equation (30)), where Φ is defined as the “coefficient of reciprocation” or probability that a cooperative act is reciprocated. By comparing the coefficient of reciprocation with the coefficient of relatedness of kin selection, direct comparisons of the two hypotheses may be made.     

Integrating Solid-State NMR and Computational Modeling to Investigate the Structure and Dynamics of Membrane-Associated Ghrelin

The peptide hormone ghrelin activates the growth hormone secretagogue receptor 1a, also known as the ghrelin receptor. This 28-residue peptide is acylated at Ser3 and is the only peptide hormone in the human body that is lipid-modified by an octanoyl group. Little is known about the structure and dynamics of membrane-associated ghrelin. We carried out solid-state NMR studies of ghrelin in lipid vesicles, followed by computational modeling of the peptide using Rosetta. Isotropic chemical shift data of isotopically labeled ghrelin provide information about the peptide’s secondary structure. Spin diffusion experiments indicate that ghrelin binds to membranes via its lipidated Ser3. Further, Phe4, as well as electrostatics involving the peptide’s positively charged residues and lipid polar headgroups, contribute to the binding energy. Other than the lipid anchor, ghrelin is highly flexible and mobile at the membrane surface. This observation is supported by our predicted model ensemble, which is in good agreement with experimentally determined chemical shifts. In the final ensemble of models, residues 8–17 form an α-helix, while residues 21–23 and 26–27 often adopt a polyproline II helical conformation. These helices appear to assist the peptide in forming an amphipathic conformation so that it can bind to the membrane.     

Inferring Epidemiological Dynamics with Bayesian Coalescent Inference: The Merits of Deterministic and Stochastic Models

Estimation of epidemiological and population parameters from molecular sequence data has become central to the understanding of infectious disease dynamics. Various models have been proposed to infer details of the dynamics that describe epidemic progression. These include inference approaches derived from Kingman’s coalescent theory. Here, we use recently described coalescent theory for epidemic dynamics to develop stochastic and deterministic coalescent susceptible–infected–removed (SIR) tree priors. We implement these in a Bayesian phylogenetic inference framework to permit joint estimation of SIR epidemic parameters and the sample genealogy. We assess the performance of the two coalescent models and also juxtapose results obtained with a recently published birth–death-sampling model for epidemic inference. Comparisons are made by analyzing sets of genealogies simulated under precisely known epidemiological parameters. Additionally, we analyze influenza A (H1N1) sequence data sampled in the Canterbury region of New Zealand and HIV-1 sequence data obtained from known United Kingdom infection clusters. We show that both coalescent SIR models are effective at estimating epidemiological parameters from data with large fundamental reproductive number R₀ and large population size S₀. Furthermore, we find that the stochastic variant generally outperforms its deterministic counterpart in terms of error, bias, and highest posterior density coverage, particularly for smaller R₀ and S₀. However, each of these inference models is shown to have undesirable properties in certain circumstances, especially for epidemic outbreaks with R₀ close to one or with small effective susceptible populations.     

Constructing functional models from biophysically-detailed neurons

Improving biological plausibility and functional capacity are two important goals for brain models that connect low-level neural details to high-level behavioral phenomena. We develop a method called “oracle-supervised Neural Engineering Framework” (osNEF) to train biologically-detailed spiking neural networks that realize a variety of cognitively-relevant dynamical systems. Specifically, we train networks to perform computations that are commonly found in cognitive systems (communication, multiplication, harmonic oscillation, and gated working memory) using four distinct neuron models (leaky-integrate-and-fire neurons, Izhikevich neurons, 4-dimensional nonlinear point neurons, and 4-compartment, 6-ion-channel layer-V pyramidal cell reconstructions) connected with various synaptic models (current-based synapses, conductance-based synapses, and voltage-gated synapses). We show that osNEF networks exhibit the target dynamics by accounting for nonlinearities present within the neuron models: performance is comparable across all four systems and all four neuron models, with variance proportional to task and neuron model complexity. We also apply osNEF to build a model of working memory that performs a delayed response task using a combination of pyramidal cells and inhibitory interneurons connected with NMDA and GABA synapses. The baseline performance and forgetting rate of the model are consistent with animal data from delayed match-to-sample tasks (DMTST): we observe a baseline performance of 95% and exponential forgetting with time constant τ = 8.5s, while a recent meta-analysis of DMTST performance across species observed baseline performances of 58 − 99% and exponential forgetting with time constants of τ = 2.4 − 71s. These results demonstrate that osNEF can train functional brain models using biologically-detailed components and open new avenues for investigating the relationship between biophysical mechanisms and functional capabilities.     

A Dynamic Energy Budget (DEB) Model for the Keystone Predator Pisaster ochraceus

We present a Dynamic Energy Budget (DEB) model for the quintessential keystone predator, the rocky-intertidal sea star Pisaster ochraceus. Based on first principles, DEB theory is used to illuminate underlying physiological processes (maintenance, growth, development, and reproduction), thus providing a framework to predict individual-level responses to environmental change. We parameterized the model for P. ochraceus using both data from the literature and experiments conducted specifically for the DEB framework. We devoted special attention to the model’s capacity to (1) describe growth trajectories at different life-stages, including pelagic larval and post-metamorphic phases, (2) simulate shrinkage when prey availability is insufficient to meet maintenance requirements, and (3) deal with the combined effects of changing body temperature and food supply. We further validated the model using an independent growth data set. Using standard statistics to compare model outputs with real data (e.g. Mean Absolute Percent Error, MAPE) we demonstrated that the model is capable of tracking P. ochraceus’ growth in length at different life-stages (larvae: MAPE = 12.27%; post-metamorphic, MAPE = 9.22%), as well as quantifying reproductive output index. However, the model’s skill dropped when trying to predict changes in body mass (MAPE = 24.59%), potentially because of the challenge of precisely anticipating spawning events. Interestingly, the model revealed that P. ochraceus reserves contribute little to total biomass, suggesting that animals draw energy from structure when food is limited. The latter appears to drive indeterminate growth dynamics in P. ochraceus. Individual-based mechanistic models, which can illuminate underlying physiological responses, offer a viable framework for forecasting population dynamics in the keystone predator Pisaster ochraceus. The DEB model herein represents a critical step in that direction, especially in a period of increased anthropogenic pressure on natural systems and an observed recent decline in populations of this keystone species.     

A reexamination of stability in randomly varying versus deterministic environments with comments on the stochastic theory of limiting similarity

The focus of this paper is a general relationship proposed by May (Amer. Natur. 107 (1973)) between the stability properties of stochastic models incorporating environmental variation and the stability properties of the deterministic models from which they are derived. The concepts of stochastic stability underlying this conjectured relationship are discussed and compared to the standard definitions of deterministic stability as well as alternative criteria for stability in stochastic models. It is shown by example that May's qualitative stability criterion does not ensure stability in any sense unless restrictive conditions on the form of the model are satisfied. Even under these conditions, the criterion, which is based on linearization, generally provides information only about the local dynamics of multispecies models. The applicability of such information to stochastic limiting similarity theory is discussed and alternative methods of analysis are proposed.     

Inoculation and recovery rates in the malaria model of Dietz,Molineaux, and Thomas

In the malaria model of Dietz, Molineaux, and Thomas [Bull. WHO 50:347–357 (1974)] the iroculation rate depends on a pseudoequilibrium approximation to a differential equation describing mosquito dynamics. By biasing a key parameter, the approximation can match the predictions of the differential equation; with fixed parameters, the approximation sometimes predicts qualitatively different disease behavior than does its approximand. The model's recovery rate depends on an approximation to a full time-dependent formulation of Macdonald's superinfection hypothesis. Judged by the ability to fit data, the approximation performs better than its approximand. Alternative implementations of the model yield significantly different estimates of scientifically meaningful parameters.     

How do site-specific DNA-binding proteins find their targets?

Essentially all the biological functions of DNA depend on site-specific DNA-binding proteins finding their targets, and therefore ‘searching’ through megabases of non-target DNA. In this article, we review current understanding of how this sequence searching is done. We review how simple diffusion through solution may be unable to account for the rapid rates of association observed in experiments on some model systems, primarily the Lac repressor. We then present a simplified version of the ‘facilitated diffusion’ model of Berg, Winter and von Hippel, showing how non-specific DNA–protein interactions may account for accelerated targeting, by permitting the protein to sample many binding sites per DNA encounter. We discuss the 1-dimensional ‘sliding’ motion of protein along non-specific DNA, often proposed to be the mechanism of this multiple site sampling, and we discuss the role of short-range diffusive ‘hopping’ motions. We then derive the optimal range of sliding for a few physical situations, including simple models of chromosomes in vivo, showing that a sliding range of ~100 bp before dissociation optimizes targeting in vivo. Going beyond first-order binding kinetics, we discuss how processivity, the interaction of a protein with two or more targets on the same DNA, can reveal the extent of sliding and we review recent experiments studying processivity using the restriction enzyme EcoRV. Finally, we discuss how single molecule techniques might be used to study the dynamics of DNA site-specific targeting of proteins.     

Real-Time Imaging of Nuclear Permeation by EGFP in Single Intact Cells

The NPC is the portal for the exchange of proteins, mRNA, and ions between nucleus and cytoplasm. Many small molecules (<10 kDa) permeate the nucleus by simple diffusion through the pore, but molecules larger than 70 kDa require ATP and a nuclear localization sequence for their transport. In isolated Xenopus oocyte nuclei, diffusion of intermediate-sized molecules appears to be regulated by the NPC, dependent upon [Ca²⁺] in the nuclear envelope. We have applied real-time imaging and fluorescence recovery after photobleaching to examine the nuclear pore permeability of 27-kDa EGFP in single intact cells. We found that EGFP diffused bidirectionally via the NPC across the nuclear envelope. Although diffusion is slowed ~100-fold at the nuclear envelope boundary compared to diffusion within the nucleus or cytoplasm, this delay is expected for the reduced cross-sectional area of the NPCs. We found no evidence for significant nuclear pore gating or block of EGFP diffusion by depletion of perinuclear Ca²⁺ stores, as assayed by a nuclear cisterna-targeted Ca²⁺ indicator. We also found that EGFP exchange was not altered significantly during the cell cycle.     

An empirical demonstration of risk-sensitive foraging preferences

We report laboratory experiments with yellow-eyed juncos (Junco phaeonotus) revealing that the birds' foraging preferences for variable rewards respond not only to the mean, but also the variance, of food rewards. The nature of their preferences for variable rewards is related to their expected daily energy budget. We summarize the birds' preferences in utility functions for energetic rewards. Since mean reward size is inadequate to predict their behaviour, we believe that foraging models should consider environmental stochasticity and an animal's response to this variation.     

A Highlights from MBoC Selection: Diffusion of GPI-anchored proteins is influenced by the activity of dynamic cortical actin

Molecular diffusion at the surface of living cells is believed to be predominantly driven by thermal kicks. However, there is growing evidence that certain cell surface molecules are driven by the fluctuating dynamics of cortical cytoskeleton. Using fluorescence correlation spectroscopy, we measure the diffusion coefficient of a variety of cell surface molecules over a temperature range of 24–37°C. Exogenously incorporated fluorescent lipids with short acyl chains exhibit the expected increase of diffusion coefficient over this temperature range. In contrast, we find that GPI-anchored proteins exhibit temperature-independent diffusion over this range and revert to temperature-dependent diffusion on cell membrane blebs, in cells depleted of cholesterol, and upon acute perturbation of actin dynamics and myosin activity. A model transmembrane protein with a cytosolic actin-binding domain also exhibits the temperature-independent behavior, directly implicating the role of cortical actin. We show that diffusion of GPI-anchored proteins also becomes temperature dependent when the filamentous dynamic actin nucleator formin is inhibited. However, changes in cortical actin mesh size or perturbation of branched actin nucleator Arp2/3 do not affect this behavior. Thus cell surface diffusion of GPI-anchored proteins and transmembrane proteins that associate with actin is driven by active fluctuations of dynamic cortical actin filaments in addition to thermal fluctuations, consistent with expectations from an “active actin-membrane composite” cell surface.     

Climate-Based Models for Pulsed Resources Improve Predictability of Consumer Population Dynamics: Outbreaks of House Mice in Forest Ecosystems

Accurate predictions of the timing and magnitude of consumer responses to episodic seeding events (masts) are important for understanding ecosystem dynamics and for managing outbreaks of invasive species generated by masts. While models relating consumer populations to resource fluctuations have been developed successfully for a range of natural and modified ecosystems, a critical gap that needs addressing is better prediction of resource pulses. A recent model used change in summer temperature from one year to the next (ΔT) for predicting masts for forest and grassland plants in New Zealand. We extend this climate-based method in the framework of a model for consumer–resource dynamics to predict invasive house mouse (Mus musculus) outbreaks in forest ecosystems. Compared with previous mast models based on absolute temperature, the ΔT method for predicting masts resulted in an improved model for mouse population dynamics. There was also a threshold effect of ΔT on the likelihood of an outbreak occurring. The improved climate-based method for predicting resource pulses and consumer responses provides a straightforward rule of thumb for determining, with one year’s advance warning, whether management intervention might be required in invaded ecosystems. The approach could be applied to consumer–resource systems worldwide where climatic variables are used to model the size and duration of resource pulses, and may have particular relevance for ecosystems where global change scenarios predict increased variability in climatic events.     

Gillespie eco‐evolutionary models (GEMs) reveal the role of heritable trait variation in eco‐evolutionary dynamics

Heritable trait variation is a central and necessary ingredient of evolution. Trait variation also directly affects ecological processes, generating a clear link between evolutionary and ecological dynamics. Despite the changes in variation that occur through selection, drift, mutation, and recombination, current eco‐evolutionary models usually fail to track how variation changes through time. Moreover, eco‐evolutionary models assume fitness functions for each trait and each ecological context, which often do not have empirical validation. We introduce a new type of model, Gillespie eco‐evolutionary models (GEMs), that resolves these concerns by tracking distributions of traits through time as eco‐evolutionary dynamics progress. This is done by allowing change to be driven by the direct fitness consequences of model parameters within the context of the underlying ecological model, without having to assume a particular fitness function. GEMs work by adding a trait distribution component to the standard Gillespie algorithm – an approach that models stochastic systems in nature that are typically approximated through ordinary differential equations. We illustrate GEMs with the Rosenzweig–MacArthur consumer–resource model. We show not only how heritable trait variation fuels trait evolution and influences eco‐evolutionary dynamics, but also how the erosion of variation through time may hinder eco‐evolutionary dynamics in the long run. GEMs can be developed for any parameter in any ordinary differential equation model and, furthermore, can enable modeling of multiple interacting traits at the same time. We expect GEMs will open the door to a new direction in eco‐evolutionary and evolutionary modeling by removing long‐standing modeling barriers, simplifying the link between traits, fitness, and dynamics, and expanding eco‐evolutionary treatment of a greater diversity of ecological interactions. These factors make GEMs much more than a modeling advance, but an important conceptual advance that bridges ecology and evolution through the central concept of heritable trait variation.     

Quantitative Analysis and Modeling Probe Polarity Establishment in C.?elegans Embryos

Cell polarity underlies many aspects of metazoan development and homeostasis, and relies notably on a set of PAR proteins located at the cell cortex. How these proteins interact in space and time remains incompletely understood. We performed a quantitative assessment of polarity establishment in one-cell stage Caenorhabditis elegans embryos by combining time-lapse microscopy and image analysis. We used our extensive data set to challenge and further specify an extant mathematical model. Using likelihood-based calibration, we uncovered that cooperativity is required for both anterior and posterior PAR complexes. Moreover, we analyzed the dependence of polarity establishment on changes in size or temperature. The observed robustness of PAR domain dimensions in embryos of different sizes is in agreement with a model incorporating fixed protein concentrations and variations in embryo surface/volume ratio. In addition, we quantified the dynamics of polarity establishment over most of the viable temperatures range of C. elegans. Modeling of these data suggests that diffusion of PAR proteins is the process most affected by temperature changes, although cortical flows appear unaffected. Overall, our quantitative analytical framework provides insights into the dynamics of polarity establishment in a developing system.     

The Dynamics of Sex Ratio Evolution: From the Gene Perspective to Multilevel Selection

The new dynamical game theoretic model of sex ratio evolution emphasizes the role of males as passive carriers of sex ratio genes. This shows inconsistency between population genetic models of sex ratio evolution and classical strategic models. In this work a novel technique of change of coordinates will be applied to the new model. This will reveal new aspects of the modelled phenomenon which cannot be shown or proven in the original formulation. The underlying goal is to describe the dynamics of selection of particular genes in the entire population, instead of in the same sex subpopulation, as in the previous paper and earlier population genetics approaches. This allows for analytical derivation of the unbiased strategic model from the model with rigorous non-simplified genetics. In effect, an alternative system of replicator equations is derived. It contains two subsystems: the first describes changes in gene frequencies (this is an alternative unbiased formalization of the Fisher-Dusing argument), whereas the second describes changes in the sex ratios in subpopulations of carriers of genes for each strategy. An intriguing analytical result of this work is that the fitness of a gene depends on the current sex ratio in the subpopulation of its carriers, not on the encoded individual strategy. Thus, the argument of the gene fitness function is not constant but is determined by the trajectory of the sex ratio among carriers of that gene. This aspect of the modelled phenomenon cannot be revealed by the static analysis. Dynamics of the sex ratio among gene carriers is driven by a dynamic “tug of war” between female carriers expressing the encoded strategic trait value and random partners of male carriers expressing the average population strategy (a primary sex ratio). This mechanism can be called “double-level selection”. Therefore, gene interest perspective leads to multi-level selection.     

Hydroxylation of para-chlorotoluene by model complexes of cytochrome P-450

Hydroxylation of p-chlorotoluene with heminthiol complexes, Fenton's system and Udenfriend's system was studied and the complexes assessed as models of cytochrome P-450 monooxygenases. Five species of possible hydroxylation products of p-chlorotoluene, namely, p-chlorobenzyl alcohol, 2-chloro-5-methylphenol, p-chlorobenzaldehyde, 4-chloro-2-methylphenol and 5-chloro-2-methylphenol, were studied using high performance liquid chromatography. The oxidation reactions were characterized by the yields of hydroxylation products and the product ratio. The system consisting of hemin and cysteine ethyl ester as well as Udenfriend's system gave relatively high hydroxylation yields and the former only induced a methyl migration during hydroxylation (methyl NIH shift). However, neither Fenton's nor Udenfriend's systems induced a methyl NIH shift. The hemin-thiol complex is thus concluded to be a good chemical model of cytochrome P-450 monooxygenases.