The Effects of Spatial Legacies following Shifting Management Practices and Fire on Boreal Forest Age Structure

Forest age structure and its spatial arrangement are important elements of sustainable forestry because of their effects on biodiversity and timber availability. Forest management objectives that include specific forest age structure may not be easily attained due to constraints imposed by the legacies of historical management and natural disturbance. We used a spatially explicit stochastic model to explore the synergetic effects of forest management and fire on boreal forest age structure. Specifically, we examined (1) the duration of spatial legacies of different management practices in the boreal forest, (2) how multiple shifts in management practices affect legacy duration and the spatial trajectories of forest age structure, and (3) how fire influences legacy duration and pattern development in combination with harvesting. Results based on 30 replicates of 500 years for each scenario indicate that (1) spatial legacies persist over 200 years and the rate at which legacies are overcome depends on whether new management targets are in synchrony with existing spatial pattern; (2) age specific goals were met faster after multiple management shifts due to the similar spatial scale of the preceding management types; (3) because large fires can erase the spatial pattern created by smaller disturbances, scenarios with fire had shorter lags than scenarios without fire. These results suggest that forest management goals can be accelerated by applying management at a similar spatial scale as existing spatial patterns. Also, management planning should include careful consideration of historical management as well as current and likely future disturbances.     

A Hidden Markov Model for Single Particle Tracks Quantifies Dynamic Interactions between LFA-1 and the Actin Cytoskeleton

The extraction of hidden information from complex trajectories is a continuing problem in single-particle and single-molecule experiments. Particle trajectories are the result of multiple phenomena, and new methods for revealing changes in molecular processes are needed. We have developed a practical technique that is capable of identifying multiple states of diffusion within experimental trajectories. We model single particle tracks for a membrane-associated protein interacting with a homogeneously distributed binding partner and show that, with certain simplifying assumptions, particle trajectories can be regarded as the outcome of a two-state hidden Markov model. Using simulated trajectories, we demonstrate that this model can be used to identify the key biophysical parameters for such a system, namely the diffusion coefficients of the underlying states, and the rates of transition between them. We use a stochastic optimization scheme to compute maximum likelihood estimates of these parameters. We have applied this analysis to single-particle trajectories of the integrin receptor lymphocyte function-associated antigen-1 (LFA-1) on live T cells. Our analysis reveals that the diffusion of LFA-1 is indeed approximately two-state, and is characterized by large changes in cytoskeletal interactions upon cellular activation.     

Spatial Structure and Diffusive Dynamics from Single-Particle Trajectories Using Spline Analysis

Single-particle tracking of biomolecular probes has provided a wealth of information about intracellular trafficking and the dynamics of proteins and lipids in the cell membrane. Conventional mean-square displacement (MSD) analysis of single-particle trajectories often assumes that probes are moving in a uniform environment. However, the observed two-dimensional motion of probe particles is influenced by the local three-dimensional geometry of the cell membrane and intracellular structures, which are rarely flat at the submicron scale. This complex geometry can lead to spatially confined trajectories that are difficult to analyze and interpret using conventional two-dimensional MSD analysis. Here we present two methods to analyze spatially confined trajectories: spline-curve dynamics analysis, which extends conventional MSD analysis to measure diffusive motion in confined trajectories; and spline-curve spatial analysis, which measures spatial structures smaller than the limits of optical resolution. We show, using simulated random walks and experimental trajectories of quantum dot probes, that differences in measured two-dimensional diffusion coefficients do not always reflect differences in underlying diffusive dynamics, but can instead be due to differences in confinement geometries of cellular structures.     

Detection of temporary lateral confinement of membrane proteins using single-particle tracking analysis.

Techniques such as single-particle tracking allow the characterization of the movements of single or very few molecules. Features of the molecular trajectories, such as confined diffusion or directed transport, can reveal interesting biological interactions, but they can also arise from simple Brownian motion. Careful analysis of the data, therefore, is necessary to identify interesting effects from pure random movements. A method was developed to detect temporary confinement in the trajectories of membrane proteins that cannot be accounted for by Brownian motion. This analysis was applied to trajectories of two lipid-linked members of the immunoglobulin superfamily, Thy-1 and a neural cell adhesion molecule (NCAM 125), and the results were compared with those for simulated random walks. Approximately 28% of the trajectories for both proteins exhibited periods of transient confinement, which were < 0.07% likely to arise from random movements. In contrast to these results, only 1.5% of the simulated trajectories showed confined periods. Transient confinement for both proteins lasted on average 8 s in regions that were approximately 280 nm in diameter.     

Diffusion approximation and first passage time problem for a model neuron. II. Outline of a computation method

To account for nonstationary effects the customary diffusion approximations in neuro-biology have to be modified in order to include time-varying terms in the expressions of drift and infinitesimal variance. The evaluation of the statistics of the firing time thus generally requires the use of numerical procedures. In this paper it is shown how one can extend a method due to Durbin (1971) to the case of temporally inhomogeneous diffusion equations by employing a transformation technique. A few simple examples are briefly discussed.     

Projecting the future of an alpine ungulate under climate change scenarios

Climate change represents a primary threat to species persistence and biodiversity at a global scale. Cold adapted alpine species are especially sensitive to climate change and can offer key “early warning signs” about deleterious effects of predicted change. Among mountain ungulates, survival, a key determinant of demographic performance, may be influenced by future climate in complex, and possibly opposing ways. Demographic data collected from 447 mountain goats in 10 coastal Alaska, USA, populations over a 37‐year time span indicated that survival is highest during low snowfall winters and cool summers. However, general circulation models (GCMs) predict future increase in summer temperature and decline in winter snowfall. To disentangle how these opposing climate‐driven effects influence mountain goat populations, we developed an age‐structured population model to project mountain goat population trajectories for 10 different GCM/emissions scenarios relevant for coastal Alaska. Projected increases in summer temperature had stronger negative effects on population trajectories than the positive demographic effects of reduced winter snowfall. In 5 of the 10 GCM/representative concentration pathway (RCP) scenarios, the net effect of projected climate change was extinction over a 70‐year time window (2015–2085); smaller initial populations were more likely to go extinct faster than larger populations. Using a resource selection modeling approach, we determined that distributional shifts to higher elevation (i.e., “thermoneutral”) summer range was unlikely to be a viable behavioral adaptation strategy; due to the conical shape of mountains, summer range was expected to decline by 17%–86% for 7 of the 10 GCM/RCP scenarios. Projected declines of mountain goat populations are driven by climate‐linked bottom‐up mechanisms and may have wide ranging implications for alpine ecosystems. These analyses elucidate how projected climate change can negatively alter population dynamics of a sentinel alpine species and provide insight into how demographic modeling can be used to assess risk to species persistence.     

The frequency of fitness peak shifts is increased at expanding range margins due to mutation surfing

Dynamic species' ranges, those that are either invasive or shifting in response to environmental change, are the focus of much recent interest in ecology, evolution, and genetics. Understanding how range expansions can shape evolutionary trajectories requires the consideration of nonneutral variability and genetic architecture, yet the majority of empirical and theoretical work to date has explored patterns of neutral variability. Here we use forward computer simulations of population growth, dispersal, and mutation to explore how range-shifting dynamics can influence evolution on rugged fitness landscapes. We employ a two-locus model, incorporating sign epistasis, and find that there is an increased likelihood of fitness peak shifts during a period of range expansion. Maladapted valley genotypes can accumulate at an expanding range front through a phenomenon called mutation surfing, which increases the likelihood that a mutation leading to a higher peak will occur. Our results indicate that most peak shifts occur close to the expanding front. We also demonstrate that periods of range shifting are especially important for peak shifting in species with narrow geographic distributions. Our results imply that trajectories on rugged fitness landscapes can be modified substantially when ranges are dynamic.     

Analyzing fish movement as a persistent turning walker

The trajectories of Kuhlia mugil fish swimming freely in a tank are analyzed in order to develop a model of spontaneous fish movement. The data show that K. mugil displacement is best described by turning speed and its auto-correlation. The continuous-time process governing this new kind of displacement is modelled by a stochastic differential equation of Ornstein–Uhlenbeck family: the persistent turning walker. The associated diffusive dynamics are compared to the standard persistent random walker model and we show that the resulting diffusion coefficient scales non-linearly with linear swimming speed. In order to illustrate how interactions with other fish or the environment can be added to this spontaneous movement model we quantify the effect of tank walls on the turning speed and adequately reproduce the characteristics of the observed fish trajectories.     

Observing membrane protein diffusion at subnanometer resolution

Single sodium-driven rotors from a bacterial ATP synthase were embedded into a lipid membrane and observed in buffer solution at subnanometer resolution using atomic force microscopy (AFM). Time-lapse AFM topographs show the movement of single proteins within the membrane. Subsequent analysis of their individual trajectories, in consideration of the environment surrounding the moving protein, allow principal modes of the protein motion to be distinguished. Within one trajectory, individual proteins can undergo movements assigned to free as well as to obstacled diffusion. The diffusion constants of these two modes of motion are considerably different. Without the structural information about the membrane environment restricting the moving proteins, it would not be possible to reveal insight into these mechanisms. The high-resolution AFM topographs suggest that, in future studies, such data revealed under various physiological conditions will provide novel insights into molecular mechanisms that drive membrane protein assembly and supply excellent boundary conditions to model protein-protein arrangements.     

Classification of Dynamical Diffusion States in Single Molecule Tracking Microscopy

Single molecule tracking of membrane proteins by fluorescence microscopy is a promising method to investigate dynamic processes in live cells. Translating the trajectories of proteins to biological implications, such as protein interactions, requires the classification of protein motion within the trajectories. Spatial information of protein motion may reveal where the protein interacts with cellular structures, because binding of proteins to such structures often alters their diffusion speed. For dynamic diffusion systems, we provide an analytical framework to determine in which diffusion state a molecule is residing during the course of its trajectory. We compare different methods for the quantification of motion to utilize this framework for the classification of two diffusion states (two populations with different diffusion speed). We found that a gyration quantification method and a Bayesian statistics-based method are the most accurate in diffusion-state classification for realistic experimentally obtained datasets, of which the gyration method is much less computationally demanding. After classification of the diffusion, the lifetime of the states can be determined, and images of the diffusion states can be reconstructed at high resolution. Simulations validate these applications. We apply the classification and its applications to experimental data to demonstrate the potential of this approach to obtain further insights into the dynamics of cell membrane proteins.     

Classification of Dynamical Diffusion States in Single Molecule Tracking Microscopy

Single molecule tracking of membrane proteins by fluorescence microscopy is a promising method to investigate dynamic processes in live cells. Translating the trajectories of proteins to biological implications, such as protein interactions, requires the classification of protein motion within the trajectories. Spatial information of protein motion may reveal where the protein interacts with cellular structures, because binding of proteins to such structures often alters their diffusion speed. For dynamic diffusion systems, we provide an analytical framework to determine in which diffusion state a molecule is residing during the course of its trajectory. We compare different methods for the quantification of motion to utilize this framework for the classification of two diffusion states (two populations with different diffusion speed). We found that a gyration quantification method and a Bayesian statistics-based method are the most accurate in diffusion-state classification for realistic experimentally obtained datasets, of which the gyration method is much less computationally demanding. After classification of the diffusion, the lifetime of the states can be determined, and images of the diffusion states can be reconstructed at high resolution. Simulations validate these applications. We apply the classification and its applications to experimental data to demonstrate the potential of this approach to obtain further insights into the dynamics of cell membrane proteins.     

Simulation of nitroxide electron paramagnetic resonance spectra from brownian trajectories and molecular dynamics simulations

A simulated continuous wave electron paramagnetic resonance spectrum of a nitroxide spin label can be obtained from the Fourier transform of a free induction decay. It has been previously shown that the free induction decay can be calculated by solving the time-dependent stochastic Liouville equation for a set of Brownian trajectories defining the rotational dynamics of the label. In this work, a quaternion-based Monte Carlo algorithm has been developed to generate Brownian trajectories describing the global rotational diffusion of a spin-labeled protein. Also, molecular dynamics simulations of two spin-labeled mutants of T4 lysozyme, T4L F153R1, and T4L K65R1 have been used to generate trajectories describing the internal dynamics of the protein and the local dynamics of the spin-label side chain. Trajectories from the molecular dynamics simulations combined with trajectories describing the global rotational diffusion of the protein are used to account for all of the dynamics of a spin-labeled protein. Spectra calculated from these combined trajectories correspond well to the experimental spectra for the buried site T4L F153R1 and the helix surface site T4L K65R1. This work provides a framework to further explore the modeling of the dynamics of the spin-label side chain in the wide variety of labeling environments encountered in site-directed spin labeling studies.     

Lateral diffusion in an archipelago. Single-particle diffusion.

Several laboratories have measured lateral diffusion of single particles on the cell surface, and these measurements may reveal an otherwise inaccessible level of submicroscopic organization of cell membranes. Pitfalls in the interpretation of these experiments are analyzed. Random walks in unobstructed systems show structure that could be interpreted as free diffusion, obstructed diffusion, directed motion, or trapping in finite domains. To interpret observed trajectories correctly, one must consider not only the trajectories themselves but also the probabilities of occurrence of various trajectories. Measures of the asymmetry of obstructed and unobstructed random walks are calculated, and probabilities are evaluated for random trajectories that resemble either directed motion or diffusion in a bounded region.     

Constructing cranial ontogenetic trajectories: A comparison of growth,development, and chronological age proxies using a known‐age sample of Macaca mulatta

Recent morphometric research has generated opposing conclusions regarding the ontogenetic trajectories of catarrhine crania, possibly due to the ontogenetic proxies used to calculate them. Therefore, we used three surrogates: size, molar eruption, and chronological age to generate trajectories in a known‐age sample to produce ontogenetic trajectories and determine the similarities and differences between them. Forty‐three landmarks from an ontogenetic series of 160 Macaca mulatta crania, with associated ages at death, were used to produce ontogenetic trajectories of cranial shape change. These were computed by sex through multivariate regression of Procrustes aligned coordinates against three surrogates for ontogeny: natural log of centroid size (growth), molar eruption stage (development), and chronological age. These trajectories were compared by calculating the angles between them. Each trajectory was also used to produce simulated adults from juveniles, which were then compared with each other and actual adults. The different trajectories are nearly parallel as each of the surrogates track similar aspects of ontogenetic cranial shape change, but chronological age was the most divergent. Simulated adults produced using the developmental stage trajectories were most similar to actual adults. When simulated adults were produced from opposite sex trajectories, they resembled the sex from which the trajectory was produced, not the sex of the juvenile specimen. We discuss properties of the trajectories produced from each of the surrogates, the possible reasons for previously opposing conclusions, how these properties can inform future investigations, and how our investigation bears on analyses of heterochrony.     

Substrate steering by electrostatic fields of enzymes: visualization by computer graphics

The rates of reactions catalysed by certain enzymes are increased by electrostatic effects that steer substrate molecules toward the active sites of the enzyme. This phenomenon can be studied by using Brownian dynamics simulation to generate and analyse diffusional trajectories of substrate in the field of an enzyme. The paper demonstrates that computer graphics can be used to show how electrical fields influence the special dependence of substrate flux as calculated by Brownian dynamics. Because of the statistical noise associated with typical simulation runs, filtering techniques are helpful in determining systematic trends in the graphic displays.     

Collective bacterial dynamics revealed using a three-dimensional population-scale defocused particle tracking technique

An ability to monitor bacterial locomotion and collective dynamics is crucial to our understanding of a number of well-characterized phenotypes including biofilm formation, chemotaxis, and virulence. Here, we report the tracking of multiple swimming Escherichia coli cells in three spatial dimensions and at single-cell resolution using a novel three-dimensional (3D) defocused particle tracking (DPT) method. The 3D trajectories were generated for wild-type Escherichia coli strain RP437 as well as for isogenic derivatives that display smooth swimming due to a cheA deletion (strain RP9535) or incessant tumbling behavior due to a cheZ deletion (strain RP1616). The 3D DPT method successfully differentiated these three modes of locomotion and allowed direct calculation of the diffusion coefficient for each strain. As expected, we found that the smooth swimmer diffused more readily than the wild type, and both the smooth swimmer and the wild-type cells exhibited diffusion coefficients that were at least two orders of magnitude larger than that of the tumbler. Finally, we found that the diffusion coefficient increased with increasing cell density, a phenomenon that can be attributed to the hydrodynamic disturbances caused by neighboring bacteria.     

Determining Intrachain Diffusion Coefficients for Biopolymer Dynamics from Single-Molecule Force Spectroscopy Measurements

The conformational diffusion coefficient for intrachain motions in biopolymers, D, sets the timescale for structural dynamics. Recently, force spectroscopy has been applied to determine D both for unfolded proteins and for the folding transitions in proteins and nucleic acids. However, interpretation of the results remains unsettled. We investigated how instrumental effects arising from the force probes used in the measurement can affect the value of D recovered via force spectroscopy. We compared estimates of D for the folding of DNA hairpins found from measurements of rates and energy landscapes made using optical tweezers with estimates obtained from the same single-molecule trajectories via the transition path time. The apparent D obtained from the rates was much lower than the result found from the same data using transition time analysis, reflecting the effects of the mechanical properties of the force probe. Deconvolution of the finite compliance effects on the measurement allowed the intrinsic value to be recovered. These results were supported by Brownian dynamics simulations of the effects of force-probe compliance and bead size.     

Determining Intrachain Diffusion Coefficients for Biopolymer Dynamics from Single-Molecule Force Spectroscopy Measurements

The conformational diffusion coefficient for intrachain motions in biopolymers, D, sets the timescale for structural dynamics. Recently, force spectroscopy has been applied to determine D both for unfolded proteins and for the folding transitions in proteins and nucleic acids. However, interpretation of the results remains unsettled. We investigated how instrumental effects arising from the force probes used in the measurement can affect the value of D recovered via force spectroscopy. We compared estimates of D for the folding of DNA hairpins found from measurements of rates and energy landscapes made using optical tweezers with estimates obtained from the same single-molecule trajectories via the transition path time. The apparent D obtained from the rates was much lower than the result found from the same data using transition time analysis, reflecting the effects of the mechanical properties of the force probe. Deconvolution of the finite compliance effects on the measurement allowed the intrinsic value to be recovered. These results were supported by Brownian dynamics simulations of the effects of force-probe compliance and bead size.