Species may be driven extinct by climate change, unless their populations are able to shift fast enough to track regions of suitable climate. Shifting will be faster as the proportion of suitable habitat in the landscape increases. However, it is not known how the spatial arrangement of habitat will affect the speed of range advance, especially when habitat is scarce, as is the case for many specialist species. We develop methods for calculating the speed of advance that are appropriate for highly fragmented, stochastic systems. We reveal that spatial aggregation of habitat tends to reduce the speed of advance throughout a wide range of species parameters: different dispersal distances and dispersal kernel shapes, and high and low extinction probabilities. In contrast, aggregation increases the steady-state proportion of habitat that is occupied (without climate change). Nonetheless, we find that it is possible to achieve both rapid advance and relatively high patch occupancy when the habitat has a “channeled” pattern, resembling corridors or chains of stepping stones. We adapt techniques from electrical circuit theory to predict the rate of advance efficiently for complex, realistic landscape patterns, whereas the rate cannot be predicted by any simple statistic of aggregation or fragmentation. Conservationists are already advocating corridors and stepping stones as important conservation tools under climate change, but they are vaguely defined and have so far lacked a convincing basis in fundamental population biology. Our work shows how to discriminate properties of a landscape's spatial pattern that affect the speed of colonization , and properties that affect a species' probability of persistence once established. We can therefore point the way to better land use planning approaches, which will provide functional habitat linkages and also maintain local population viability. There is a major concern that climate change and land use change could interact to cause the extinction of many species N states. We assume an “initial state” with one patch occupied (henceforth referred to as the origin patch) and all others unoccupied. We define the speed of advance as the probability that a distant, “target patch” is ever colonized divided by the mean time taken until it is colonized (if it is colonized). The successful colonization of the target may happen via any number of intermediate states, as long as the “absorbing state” of extinction is not reached first. The calculation is made simpler by the fact that we can lump together all states where the target patch is occupied, and treat them as an alternative absorbing state, henceforth referred to as the “target state”.We first developed an exact solution for the speed of advance, which can be computed for small numbers of patches. We use the fact that a metapopulation with N patches behaves as a Markov system with 2μ (we do not include a rescue effect) and each occupied patch can colonize an empty patch so that the colonization probability of an empty patch is j indexes the occupied patches and ijd is the distance between patches i and j. The parameter g changes the shape of the dispersal kernel, with g<1 making it “fat-tailed”, with a higher proportion of long distance dispersal. The parameter α changes the mean dispersal distance for a given g. The parameter R changes the rate of production of propagules by an occupied patch.The probabilities of transition between states are determined by combining the probabilities of individual patch transitions: each patch can go extinct with probability I is the identity matrix, W is a matrix of transition probabilities between all non-absorbing states and q is a vector of probabilities of reaching the target state directly from each non-absorbing state. Conditional on not going extinct, the mean time to reach the target state is given byX is W normalized to exclude the probability of extinction, and ω is a vector of ones. We combine these two quantities to give “speed” as P/T.The probability of ending up in the target state rather than extinction, given that the system starts in the initial state, that is with the origin patch occupied and all other patches unoccupied, is given byWe evaluated speed for systems of between 1 and 8 patches between the source and the target, varying the colonization and extinction parameters and the spatial locations of patches. For more than one “stepping stone” patch, we did not investigate all possible spatial arrangements, but added stepping stones iteratively, each time choosing the location that gave the greatest increase in speed. For comparison we then calculated speeds given a number of idealized patterns, including a regularly-spaced chain of patches.We extended our analysis to consider a more realistic scenario where a species is found in one large landscape (of hundreds of patches), and has to extend its range into an adjacent, newly climatically suitable landscape. We found the speed of advance by simulating the metapopulation dynamics, because the exact calculation of speed was not feasible for such large numbers of patches (see “simulation” section below). We wanted to investigate the interplay between the probability of the chains of colonization events which are needed for the species' range advance, and the spatial clustering of habitat which is known to increase metapopulation viability. To this end, we investigated three families of spatial habitat pattern (see “landscape generation” section below), while keeping the total amount of habitat constant. We also aimed to test the performance of several summary metrics that might be efficient approximations to the speed of advance, and that could be used in conservation planning (see “landscape summary metrics” section below).We generated a wide variety of habitat spatial patterns using fractals as a basis, as well as two idealized patterns, a strictly regular spacing of suitable cells, and a giant cross with continuous east-west and north-south corridors  for a negative exponential dispersal kernel with a mean distance of 10. We added 64 habitat cells as stepping stones along these paths so as to minimize the maximum step distance, and bring the amount of habitat up to 1%.The giant cross landscape consisted of one completely full row and one completely full column of cells  and the adjacent row and column evenly populated with the remaining cells to make up 655. The regular landscape was generated by overlaying a 10×10 lattice over the landscape, at an angle (11°) to avoid extra space at the edge because 256 does not divide by 10.i is j indexes the occupied patches and ijd is the distance between patches i and j. R can be interpreted as the number of emigrants leaving each occupied patch per time step (referred to as ‘fecundity’ for short), and 2/α as the mean dispersal distance. The extinction probability of each patch is μ(1−c), where the (1−c) term constitutes a rescue effect. We investigated mean dispersal distances between 1 and 10 cell units, R values of 10, 45 and 100 and μ values of 0.05, 0.2 and 0.4.We implemented a simple stochastic patch occupancy model. The colonization probability of each patch R = 100 and μ = 0.2 .Each simulation started with a 200 time step “burn in” using one landscape tile , we hypothesized that a simpler metric based only on the expected time to colonization between pairs of patches would be an efficient predictor of speed. This makes it possible to analyze systems with many more patches.We represent the landscape as a weighted, undirected graph We decided to use the edges of the landscape (north and south or east and west) as nominal start and end points for our speed of advance metrics. The edges are represented in our network as special nodes which have a link to all the habitat cells but not to each other. The cost of links to the edges is based on the shortest straight-line distance between a cell and the edge. This is more-or-less equivalent to assuming that colonists are equally likely to come from anywhere along the edge. This assumption could be modified to suit particular conservation applications, using sources and targets relevant to species' predicted range shifts.M is a N×N matrix with elementsC is the single step conductivity  in the absence of any other patches . If a stepping stone is placed at a fraction f of the distance between the source and the target, the time to colonize the target via the stepping stone can be approximated as f = 0.5, except in cases  when it is faster to colonize directly from the source to the target than to use the stepping stone at all. The benefit of having a centrally placed stepping stone as opposed to colonizing the target directly from the source . This explains the major patterns observed in The benefit of regular spacing can be appreciated through the following heuristic argument. The mean time to colonize one patch from another is proportional to exp for 9 different colonization kernels illustrated in (b). In both panels g =  is shown with  lines and α = ×4/4g shown with  (the correction 4/4g is to make the kernels more comparable in average height even though they have different shapes). In panel (a) two different extinction rates are also shown . The arrangement “found by iteration” is found by testing all locations on a 3 by 81 lattice between the source and the target and choosing the location that gives the highest speed for each new stepping stone in turn. This iterative patch addition is not a reliable way of finding the best arrangement for several patches, because early choices severely limit the options for subsequent arrangements.(PDF)Click here for additional data file.Figure S2Speed of expansion in simulations versus four putative summary metrics of the landscape. In all panels (a–d) the y axis is the rate of advance (cells/time step) of a simulated metapopulation across one of 332 landscapes (see apes see  in one o(PDF)Click here for additional data file.Figure S3Speed of expansion in simulations for species with different dispersal distances. In all panels the y axis is the rate of advance (cells/time step) of a simulated metapopulation across one of 332 landscapes , and numbers denote the count of runs falling above or below these lines. Points with conductivity less than 10−4, all with speeds indistinguishable from zero, are not shown on the graph but are included in the counts.apes see  in one o(PDF)Click here for additional data file.Figure S4Speed of expansion in simulations for species with different fecundity and extinction rates (R and μ). In all panels the y axis is the rate of advance (cells/time step) of a simulated metapopulation across one of 332 landscapes . Red points are from runs where the starting patch occupancy (at the end of the 200 time-step burn-in) was less than 2/3, to show that these landscapes tended to have a lower speed than predicted from their conductivity.apes see  in one o(PDF)Click here for additional data file.Data S1Raw data for this study is made available here.(ZIP)Click here for additional data file.


A modeling framework was developed to simulate large and gradual conformational changes within a macromolecule (protein) when its low amplitude high frequency vibrations are not concerned. Governing equations were derived as alternative to Langevin and Smoluchowski equations and used to simulate gating conformational changes of the Kv7.1 ion-channel over the time scale of its gating process (tens of milliseconds). The alternative equations predict the statistical properties of the motion trajectories with good accuracy and do not require the force field to be constant over the diffusion length, as assumed in Langevin equation. The open probability of the ion–channel was determined considering cooperativity of four subunits and solving their concerted transition to the open state analytically. The simulated open probabilities for a series of voltage clamp tests produced current traces that were similar to experimentally recorded currents. Molecular dynamics (MD) simulations is a method used to model the molecular motion of proteins jx. These degrees of freedom are usually translation and rotation of protein segments with reinforced secondary structure (e.g. helices). The proposed modeling framework of this paper simulates the protein dynamics within the entire configuration space of large conformational changes , without explicitly simulating high frequency vibrations of single atoms and thermodynamic properties. It simulates the average conformational changes over Δt, a time window sufficiently larger than the time between collisions. In this study we analyzed the molecular motion of particles under the influence of a conservative force field using a kinematic theory approach and derived the governing equations of the motion. These equations are more accurate alternatives to Langevin equation and Smoluchowski equation which have been used to model the gradual motion of proteins in a reduced configuration space, neglecting its atomic vibrations Large conformational changes usually involve gradual dislocations of protein segments which can be modeled with a limited set of degrees of freedom, m is the mass of the target particle, iv is its velocity along coordinate ix, Φ is the potential of any existing conservative force field, and the last term on the right represents the stochastic force of thermodynamic collisions on the particle. kt is a time when another particle (referred to as a colliding particle in this paper) hits the target particle and According to Newton's second law of motion, velocity of a particle (referred to as a target particle in this paper) on a molecular scale is determined by:However, modeling the non–zero mean of the stochastic force by the term In Langevin model of motion, the stochastic force is assumed to be a zero mean Gaussian process with a Dirac delta autocorrelation function. Based on these assumptions, the variance of the stochastic force can be determined in terms of the variance of the velocity distribution using the fluctuation-dissipation theorem Δt.Motivated by the need for a model that can accurately replicate the velocity distribution and does not constraint the conservative force field as in the Langevin equation, we introduce here a new modeling framework that is applicable to large, gradual conformational changes of a protein. We analyze the actual motion of a particle (protein segment) that undergoes multiple collisions in a probabilistic domain (a kinetic theory approach) and derive a governing equation for the average velocity. Unlike the Langevin equation, the governing equation of the average velocity does not have an inertial term. It should be emphasized that the inertial term is not neglected assuming a large friction coefficient (as assumed in the high friction limit of Langevin equation), it simply does not appear in the governing equation of the average motion after applying Newton's law of motion for a system with multiple collisions. Because during the conformational changes of a protein segment the conservative force may vary significantly over the diffusion length, we do not consider it constant in our analysis and derive a more general equation for the effect of a conservative force on the motion trajectory and the probability distribution. The stochastic term in our equation appears as stochastic velocity (rather than force) and does not have a delta autocorrelation function. Note that the autocorrelation of the stochastic force is assumed to be a Dirac delta function in deriving the statistical properties of motion from Langevin equation and in deriving the Einstein–Smoluchowski relation between the friction coefficient and the diffusion constant. In reality, the autocorrelation decreases gradually over The newly developed modeling framework is used to simulate the conformational changes of the voltage sensor region (S1 to S4) of the Kv7.1 ion–channel during gating and the resultant open probability is compared to experimentally recorded macroscopic currents.We develop a model for the gradual motion of the helical transmembrane segments of an ion–channel protein. We use this model to simulate the motion trajectories and the transient probability distribution in the configuration space. Further, we use the results of these simulations to compute the single channel and macroscopic currents carried by the ion–channel. The model developed in this paper represents the stochastic motion of a particle (here a protein segment) on a molecular scale. It consists of two key sets of equations: equations (38)–(40) and equations (43), (47)–(49). The first set governs the average motion (over a time window) of the particle and can be used to simulate the motion trajectories. Compared to Langevin equation, a key feature of this equation set is that it accounts for potential fields that are not constant over the diffusion length by including the higher derivatives of the potential function. Additionally, it provides more accurate estimates for the stochastic properties of the motion. The second set of equations (derived based on the first set) governs the transient and steady state distribution of the particle in its configuration space. These equations also account for potentials that are not constant over the diffusion length.For simulating the structural dynamics and electrophysiological function of the ion–channel, the first equation set is used to generate motion trajectories and from those the single channel current traces. The second equation set is used to compute the channel open probability and the macroscopic current through a large ensemble of ion channels.0v in the space shown in z direction is chosen along the velocity. Assume an arbitrary location on the surface of the target particle that may be hit by a colliding particle. Panel A shows the impact point and the relative position of the target particle and the colliding particle. An impact occurs if the velocity of the colliding particle along the impact direction (dashed line) is larger than the velocity of the target particle along this direction: tv, can have any value. For a collision at an opposite location (panel B), the condition of impact is: θ (between 0 and π/2) and ϕ (between 0 and 2π) in a spherical coordinate system. Particles with equal mass exchange their component of velocity along the impact direction. Therefore, θ and ϕ determine the direction of the impact line. Because the probability of occupying any location in space is uniform, the probability distribution of the impact line is uniform. The probability distribution of x, y or z), which is known to be a Gaussian distribution. The expectation value of the target particle velocity after the impact is:z direction is non–zero, meaning that the stochastic velocity of a particle is not memory–less and therefore its autocorrelation function is not a delta function. However, this value vanishes after about ten impacts, implying that the autocorrelation approaches zero after a time required for about 10 impacts.The stochastic motion of a target particle (representing a protein segment) and colliding particles are considered in 3D Cartesian space. Because of symmetry, the velocity of the colliding particles is uniformly distributed in all directions. Also, the probability that a point in space is occupied by a colliding particle is equal everywhere. Consider a target particle moving with a velocity thk impact, thk impact and thk and (k+1)th impacts. Between the thk and (k+1)th impact, the particle accelerates under the influence of the conservative force field and as a result, its deterministic velocity increases by k+1)th impact is:k+1)th impact, kγ is the fraction of particle velocity that is preserved during the impact. If there is no conservative force field, there would be no deterministic velocity and the velocity after the (k+1)th impact is:j refers to all collisions prior to the thk collision. The right side of equation (14) is a convergent series. Note that although we used the notion of infinity, in practice the series approaches a constant value if we calculate it up to a few preceding collisions .If there is no conservative force field acting on the particle, it will have a zero mean normally distributed stochastic velocity, Δt. The time interval is chosen sufficiently long, such that the average of a stochastic parameter over Δt closely approximates its expectation value, and sufficiently short such that gradual conformational changes are small during Δt. Δt. Autocorrelation of Δt is orders of magnitude longer than the time of several collisions.To find the global velocity of the particle we average the velocity over a time interval N is the total number of collisions during Δt. Between the thk and (k+1)th impact, the particle travels between ix(k) and ix(k+1) and is subjected to a conservative force if(jx) (per unit mass). The global motion of the protein segment (target particle) during Δt is associated with the deterministic velocity. Δt is assumed sufficiently small such that the conservative force field can be considered constant over the associated global displacement. However, during Δt the range of motion (diffusion length) depends on the stochastic velocity that is much larger than the deterministic velocity and consequently the particle travels much farther  in both the positive and negative directions. The conservative force may or may not be constant over this range. Note that Δt needs to be sufficiently large, such that the time average and the ensemble average are the same (e.g. Δt includes at least 100 collisions). Therefore, we assume that if(jx) is not constant over the diffusion length, thk and (k+1)th impacts we may write:t is between kt and kt+1. Substituting for The average of the deterministic component of velocity is:ix direction, but the location of the particle (jx) varies in all directions. Because i and j refer to any two arbitrary degrees of freedom and if with its Taylor expansion in equation (20) and substituting for t as a result of friction. Note that the motion is a multidimensional motion. Equation (20) integrates over the k refers to a collision incident in the middle of a time window if and its derivatives should be calculated at jx(t), the expectation location of the particle at time t. γ and τ are uncorrelated stochastic processes and are independent from each other and from γ and τ. Considering that the motion is symmetric in the positive and negative directions we may write:if will be eliminated. Combining equations (30), (31) and (32), the deterministic component of the velocity is:We define the global deterministic velocity as the expectation value of equation (25) :iD. At steady state, the time derivative of the probability distribution is zero:ssP to exist, its partial differentials in equation (45) must present a total differential, meaning that for any i and k:C1 and C2j are constants that vary with temperature. Therefore, the principle of detailed balance requires that the terms i. Under this condition the steady state distribution of probability has the following closed form:Time variation of the probability distribution is equal to the negative divergence of the probability flux. If the velocity field is a time invariant process, then the probability flux is the expectation value of the multiplication of the velocity field and the probability distribution. Therefore:z direction); rotation is about an axis parallel to the first principal direction of the S4-S3 complex, located 5 Angstrom away from its center. The axes of rotation and translation were chosen to comply with the geometrical constraints imposed by the S4-S5 linker and to reduce the probability of a steric overlap of the S4-S3 complex with neighboring segments. Based on our recent loop closure technique A 3D structure of the Kv7.1 channel was derived previously Energy landscapes were constructed at multiple membrane potentials by computing the total electrostatic potential of a voltage sensor at all the conformations it assumes in the configuration space described above. The transmembrane potential adds a uniform electric field to the dielectric region (protein and membrane); it approaches zero outside this region. The potential energy of steric interactions was modeled with the Lennard–Jones potential z direction (outward S4–S3 conformations). As a consequence, at higher membrane potentials the local minima in the energy landscape shift in the positive z direction.The electrostatic interactions between the four voltage sensors of the tetrameric channel are negligible as they are located far from each other. Therefore, we only consider the internal interactions within one voltage sensor plus its interaction with the S5–S6 complex of the neighboring subunit. C1 = kT and neglecting the second order term. Depolarizing the membrane potential causes the S4–S3 complex to transition to more outward locations. It moves outward up to 12 Angstrom and rotates up to 0.5 radians during this transition.If the membrane has been kept at a constant potential long enough, the distribution of voltage sensors in the configuration space is stationary (steady state condition). zζ = 0.5*10−3 kg/s in z direction and ϕζ = 12.5*10−3 kg.Ang2/s in ϕ directions. The higher order frictions were neglected. Comparison of these friction coefficients requires transformation to a common dimension. An equivalent translational friction coefficient for rotation about an axis can be derived as:a is the root mean square distance of the moving residues with respect to the axis of rotation. a is about 5 Angstrom in our simulation and we choose ϕζ = a2zζ = 25zζ so that resistance to motion is similar for the two degrees of freedom.Transition of the probability distribution from one equilibrated distribution to another in response to a sudden change in membrane potential was computed using equation (43) taking into account the cooperativity of the four subunits during channel opening Movies S1 and Movies S2). By increasing the membrane potential from the resting potential of −100 mV to the test potential of +60 mV, the transition starts with a rapid outward translation and counter clockwise rotation of the voltage sensors to an intermediate conformation, from which they gradually diffuse to a final conformation associated with an additional energy minimum at +60 mV (Movies S1).Equation (43) was solved using the Finite Difference (FD) method to simulate the transition of probability distribution during step depolarization from resting potential of −100 mV to a test potential of +60 mV (activation test), and also during step repolarization from the depolarized potential of +60 mV to the test potential of −100 mV (deactivation test). These transitions and the resultant open probability were visualized for the activation and the deactivation tests in two supplement movies (e1(t) was less than 10−7 (<0.00001% error). Additionally, we computed the difference between the probability distribution when the FD simulation approaches steady state (t = 1000 ms) and the probability distribution at steady state calculated analytically using equation (48):e2 for the voltage clamp test from −100 mV to +60 mV was less than 10−6 (less than 0.0001%). To ensure the accuracy of the FD solver, we refined the integration intervals. The FD solver maintained a relative error less that 10−7 throughout the gating duration  during the transition:z direction and less than 1% in θ direction . The channel can transition to the open state only if the S4-S3 complexes of all four subunits are in the activated state. The permissive state is the channel state when all four subunits are in their activated states. The cooperative transitions of the channel from the permissive state to the open state and from the open state to the permissive state are modeled by Markovian transitions with transition rates α and β, respectively.At the open state the S4–S3 complex is in an outward position, and during channel closing it moves inward between 7 and 13 Angstroms Transitions to the open state reduce the concentration of voltage sensors in activated states (and consequently in resting states) at depolarized membrane potential due to flux of channels from the permissive state to the open state. Below we explain how to incorporate the cooperativity between the four subunits during transition to the open state in order to calculate the channel open probability. This extends the application of our previous formulation AssC is the steady state probability of voltage sensors in activated conformations with no transitions to the open state:At steady state the net transition to and from the open state is zero and the probability distributions of the voltage sensors in configuration space are independent of each other. The open probability can be calculated as F(t), from or to the open state. Therefore, the transient probability distribution in subunit configuration space, CP, is the sum of the probability distribution of the subunits that are initially in the subunit configuration space, and the probability distribution of the subunits that enter or exit the configuration space from or to the open state, EP:O0 is the initial open probability. RP shows the distribution of the subunits that are initially in the configuration space . Therefore, probability distribution of the subunits that are initially in the configuration space among all the subunits (taking into account the subunits that are in open state as well) would be (1−0O)RP, because only (1−0O) of the subunits are initially in the subunit configuration space.We showed previously how to include the cooperativity of channel opening in computing channel open probability from the probability distribution of its subunits in a discrete Markov model α and β:A(t) is the probability of the channel occupying the permissive state. As a subunit can transition to the open conformation from any activated conformation, we express the channel open probability, O(t), in terms of two activated-state probabilities, AU and AR, defined as follows UP is the solution of equation (43) (where there is no net flux to or from the open state) when all subunits are initially in activated conformations distributed proportionally to their final steady state distribution , and AU is probability of activated state under this condition. AR is the probability of activated state associated with the redistribution of the subunits that are initially in the subunit configuration space. Once the net flux, the open probability and the permissive state probability were computed using the above equations, flux from the open to the permissive state, βO(t), and from the permissive to the open state, αA(t) can be determined.The transient flux can be determined using the transition rates α and β were calibrated to 0.2 /ms and 0.004 /ms. The current traces are similar to those calculated using Monte Carlo simulations and an intermediate Markov model in our previous study Open probabilities for a series of voltage clamp tests from resting potential of −100 mV to test potentials of −40, −20, 0, 20, 40 and 60 mV were simulated and compared with experimentally recorded currents for the same protocol in Movie S3.The trajectory of conformational changes within the configuration space can be estimated using equation (38). the effect of the Brownian forces on the velocity of the particle (friction term) is much larger than that of the external (conservative) force” Δt. As a consequence, the prediction of equation (61) for the stochastic component of the motion is inaccurate. Finally, unlike equation (61), application of equation (38) is not limited to problems where the conservative force is approximated to be constant over the diffusion length  was derived for simulating the average motion trajectory. At the high friction limit, where “iζ and ijω, are considered as the calibrating parameters. The temperature dependent constant, C1, is also a calibrating parameter of the model. However, we chose C1 = kT based on the Einstein–Smoluchowski relation, derived for ideal gases. We also assumed that the friction coefficient is the same along all degrees of freedom, as we had no reason to include anisotropy. This reduces the calibrating parameters of the modeling framework to a single friction coefficient, iζ .In our modeling approach, the friction coefficients, ϕ = 0 in configuration space is associated with the activated state  to provide the best fit between the model prediction and the experimental steady state activation curve  under a variety of different conditions. Therefore, a close match between simulated and recorded open probability may be used as verification of appropriate simulations of conformational changes. Consistent with our previous studies The electrostatic energy of the protein was computed at different points of its configuration space using MATLAB codes. Protein and membrane were modeled as continuum dielectrics with dielectric constant of 6, consistent with the range of 2 to 25 determined for lipid bilayers The governing partial differential equation of the motion (equation (43)) was solved using a central difference FD method and imposing natural boundary conditions. The FD was implemented using MATLAB programming environment and integrated over 50 ns time steps.Figure S1Velocity trace of a particle computed using the Langevin equation during 10,000 impacts. Panel A) shows the entire trace and panel B) enlarges the small region marked by red ribbon in panel A). Dashed lines in panel B) mark the impact incidents.(TIF)Click here for additional data file.Figure S2Probability density function of the velocity computed using Langevin model (blue curve) compared with its equivalent Maxwell-Boltzmann distribution (red curve). The expectation value of the velocity square is the same for both distributions.(TIF)Click here for additional data file.Figure S3Velocity trace of a particle computed using the model developed in this paper during 10,000 impacts. Panel A) shows the entire trace and panel B) enlarges the small region marked by red ribbon in panel A). Dashed lines in panel B) mark the impact incidents.(TIF)Click here for additional data file.Figure S4Probability density function of the velocity computed using the model developed in this paper (blue curve) compared with its equivalent Maxwell-Boltzmann distribution (red curve). The expectation value of the velocity square is the same for both distributions.(TIF)Click here for additional data file.Movie S1Probability distribution of subunits in configuration space during gating, when the membrane potential increased to +60 mV from a resting potential of −100 mV. Vertical dashed line is the assumed border between resting and activated conformations. The white vertical curve shows the flux within configuration space across this border. The net flux from the permissive state to the open state is shown by the arrow exiting the subunit configuration space at the top right corner. Open state probability is shown  in the square on the upper right and plotted as a function of time in the right bottom panel.(WMV)Click here for additional data file.Movie S2Probability distribution of subunits in configuration space during gating, when the membrane potential decreased to −100 mV from an activated potential of +60 mV. Vertical dashed line is the assumed border between resting and activated conformations. The white vertical curve shows the flux within configuration space across this border. The net flux from the open state to the permissive state is shown by the arrow exiting the open state toward the subunit configuration space at the top right corner. Open state probability is shown  in the square on the upper right and plotted as a function of time in the right bottom panel.(WMV)Click here for additional data file.Movie S3A trajectory of conformational changes during gating, when the membrane potential is increased from a resting potential of −100 mV to a depolarized potential of +60 mV. The two orange helical segments are the S4–S3 complex. At −100 mV they are at their most downward location (deep resting). In response to increase of membrane potential to +60 mV they move upward (with slight rotation) to the intermediate resting state. They stay at that state for a while and then move upward along with a noticeable rotation to the activated state. Legend in upper right corner shows the voltage sensor state and in upper left corner the membrane potential.(WMV)Click here for additional data file.Text S1Langevin equation prediction for the velocity distribution. The prediction of Langevin equation for the velocity, in response to a stochastic force with a Gaussian distribution, contradicts the Boltzmann–Maxwell distribution (a Gaussian distribution) for the velocity. Compared to a Gaussian distribution with the same variance, the velocity distribution in Langevin equation has higher densities for velocity magnitudes in close vicinity of zero and for large velocity magnitudes.(DOC)Click here for additional data file.


It was evaluated by plotting ln LT50 against LC50 with data on organic toxicants obtained from the scientific literature. Linear relationships between LC50 and ln LT50 were obtained and a Calculated NLT was derived from the plots. The Calculated NLT obtained was in good agreement with the Reported NLT obtained from the literature. Estimation of toxicity at any exposure time and concentration is possible using the model. The use of NLT as a reference point is important since it provides a data point independent of the toxicity data set and limits the data to the range where toxicity occurs. This novel approach, which represents a departure from Haber's rule, can be used to estimate long term toxicity from limited available acute toxicity data for fish exposed to organic biocides.A model based on the concept of reduction in life expectancy (RLE model) as a result of long term exposure to toxicant has been developed which has normal life expectancy (NLT) as a fixed limiting point for a species. The model is based on the equation (LC Toxicity is a function of both exposure time period and concentration or dose –4. NeverHowever there have been studies where exposure time has been evaluated as a quantifiable variable of toxicity , 10–12 aC − C0) × t = k) between exposure time (t) and exposure concentration as the lethal dose to 50% of organisms (C). In this equation C0 is a threshold concentration below which no apparent toxic effects are observed and k is a constant. But Haber [C × t = k). This relationship was later described as Haber's rule of inhalation toxicology. Conventionally Haber's rule, or its modified forms, is used for empirical evaluation of the effect of exposure time on toxicity. To expand the relationship and its application, different variants have been proposed by various researchers [The significance of exposure time in toxicological evaluations was first recognised by Warren  describiut Haber  describeearchers –22.C × t) is not always constant [However some researchers have noted that the product of  with the natural log of the exposure time with the normal life expectancy (NLT) as a limiting point. Unlike Haber's rule when the exposure concentration is zero, the exposure time is not infinite but the normal life expectancy of a particular organism.A reduced life expectancy (RLE) model has been proposed to study the effects of long term exposure , 27, 28.50  against LC50  in ambient water  [The RLE model has been evaluated with zooplanktons using data from the scientific literature by plotting ln LT., 2012) . The RLE., 2012) . The con., 2012)  who stud., 2012)  observedThe objective of this paper was to use life expectancy in the evaluation of the relationship between exposure time and exposure concentration utilising the fish toxicity data available in the scientific literature especially in the situation when exposure time is relatively long.50) and the natural logarithm of the corresponding exposure time (LT50). The ILC50 is preferred as it is the concentration of toxicant in the organism body at the target site [50 and LT50 for toxicants can be described by the following equation:50 is the internal lethal concentration resulting in the death of 50% of the organisms exposed for the time LT50, LT50 is the exposure time, NLT is the time until 50% of the organisms die without exposure, and d is a constant. Constant d is a measure of toxicity and represents the reduction in life expectancy of the organism per unit concentration of the toxicant.The RLE model proposed by Yu et al.  is basedget site –34. Also50 represents a reduced life expectancy from the normal life expectancy (NLT) of the organism. When ILC50 is plotted against the LT50, the regression line can be extended up to the point where ILC50 becomes zero which corresponds to a toxicant free medium at which the organism would be expected to live its normal life expectancy. The model can be used to predict the reduction in life expectancy at different concentrations of toxicant in the environment. This model has already been tested using the data obtained from earlier work [50 and LT50 was observed [The value of LTier work  and a hiobserved .50 data in the scientific literature [50 to the LC50 with the proposal that the relationship between LC50 and exposure time period (LT50) can be used to estimate the reduction in life expectancy of organisms [However the limited availability of ILCterature  restrictrganisms .KB) for aquatic organisms is the ratio between the concentration of toxicant in the organism (CB) and the concentration in water (CW) at equilibrium [CW is the lethal concentration (LC50) in water and CB the corresponding Internal Lethal Concentration (ILC50). Thus,The bioconcentration factor (ilibrium . Thus(250 with LC50 in (a is −1/d and b is ln(NLT)/d.The relationship obtained after replacing ILC LC50 in  is given50 is zero, the organism would have a normal life expectancy; thusWhen LC50 zero (toxicant free environment) organisms would be expected to live to their normal life expectancy. Therefore the model can be used to predict the reduction in the life expectancy at different concentrations of a toxicant in the external environment.According to this model, at LCFish were selected as study organisms since a large volume of toxicity data related to fish are available in the literature. The routes of toxicant uptake common to all fish are through gills, outer body surface, and food. The organic toxicants used in this study were those which had significantly different modes of toxic action and included various organic compounds including organophosphates, organochlorines, pyrethroids, and antiparasites.50 at various exposure durations. Most of the data sets had 4 points where the LC50 had been recorded at 24, 48, 72, and 96 hrs , so all were converted into day. The Reported NLT data for each organism was also obtained from the literature. When ranges of NLT values were given, the average was calculated to obtain the Reported NLT. The temperature of ambient water used in the experiments ranged between 14°C and 30°C.Toxicity data related to fish and organic toxicants were obtained from an extensive search of the literature. The data sets were used which included records of LCd 96 hrs  and only50 and ln(LT50) with the RLE Model as expressed in (50 (μg/L) was plotted against ln(LT50) and linear regression analysis was used to obtain the regression equation and correlation coefficient (R2) using Excel. The regression line obtained was extended to the point where the LC50 became zero which corresponded to a toxicant free medium at which the organism would be expected to live its full NLT. The values of the slope (a) and intercept (b) were obtained from the regression equations. These values were then used to obtain the Calculated NLT of each species using  with different mechanisms of action but the relationship of LC50 and ln LT50 was linear with all toxicants. All plots irrespective of toxicant type had negative slopes indicating that lethal toxicity is related to exposure time and LC50 required to cause toxicity decreases consistently in a systematic pattern.The relationship of toxicity with exposure time was linear and had a negative slope in all cases; examples of the plots are shown in C × t = k) when the toxicant concentration is zero, the exposure time is infinity. But according to the RLE model, the NLT is a limiting reference point and when LC50 approximates zero, the exposure time (ln LT50) should be the organism's normal life expectancy (NLT). Thus maximum possible exposure time of any toxicant to any fish would be the NLT of that particular fish species. The use of NLT as a reference point is important since it limits the maximum exposure time from being infinity at zero exposure concentration (Haber's rule) to the NLT of the test organism 8). The RO. niloticus as 1 to 1.5 yrs, while Ofori-Danson and Kwarfo-Apegyah [There are several sources of error in carrying out a comparison of the Reported NLT with that calculated from the RLE Model. An important factor is the decision on death during the toxicity test with fish considered dead becoming mobile when transferred to toxicant free water . In addi-Apegyah  have rep-Apegyah . In warm-Apegyah . Thus mo-Apegyah . It can -Apegyah –75.Even though there are possible reasons for the Calculated NLT to differ from the Reported NLT as outlined above but overall the Reported NLT and Calculated NLT are in reasonably good agreement.50 at other exposure times and also chronic toxicity. The equations obtained from regression analysis of LC50 versus ln LT50  is −5800 and the intercept (b) is 36000, the LC50 of dimethoate with fish O. niloticus at any exposure time (ln LT50) can be calculated. Similarly using (50 of m-cresol to G. affinis at any exposure time (ln LT50) can be obtained.Using , where tly using , the tox50 versus lnLT50) obtained from the literature for a particular toxicant and fish species, application of  available the lethal toxicity estimation at any exposure is possible.


The concentrations of NaDCC should appear as 0.30–0.50% (and The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


The intake of food and the expenditure of calories is modelled by a system of differential equations. The state variables are the amount of calories stored in adipose tissue and the level of plasma leptin. The model has as input a drive that controls the intake of food. This drive consists of a collective of physiological and psychological incentives to eat or to stop eating. An individual based approach is presented by which the parameters of the system can be set using data of a subject. The method of analysis is fully worked out using weight data of two persons. The model is prone to extensions by transferring incentives being part of the input to the collection of state variables. The change in C follows directly from the difference between the intake and the expenditure of calories. The goal is to fit the model to available data on food intake. For that purpose two long-term time series on body weight are taken in consideration. This form of data-directed modelling limits the use of highly complex model systems.In psychology the notion of D defined in this way this study concretizes the hypothesis underlying a general model of intake regulation as formulated by Speakman et al. D is reconstructed from the model by posing a number of assumptions. Most importantly it is assumed that weight increase (or decrease) is due to decreased (increased) calorie expenditure. Furthermore, rate coefficients in the two differential equations are estimated by using quantitative results from the literature. This differential equation approach of describing food intake and the corresponding energy flow was also followed by Chow and Hall As remarked the intake of calories requires attention in order to stay healthy. This was also the case in prehistoric times, but for a different reason than in present days: surviving was the main issue. Just eating for killing the hunger of the moment was not wise. It was better to eat some more to overcome possible hardships in later days. Maybe we still carry this trait in our genes which makes us consciously or unconsciously decide to store a sufficient amount of energy in our body It is understood that the pattern of food intake can be quite diverse C [cal] stored in adipose tissue and the concentration L [ng ml−1] of plasma leptin. The assumption that the dynamics is linear leads for the change in time t [day] of the state variables to the following differential equations I [cal day−1] denotes the intake rate of calories and Q [cal day−1] the expenditure rate due to body maintenance and physical action. The intake rate I follows from the drive D damped by the effect of leptin.The dynamical system under consideration consists of two state variables: the calories D, being composed out of a number of incentives to eat and not to eat, facilitates in a simple manner the inclusion of the feedback role of the hormone leptin in the process of taking up calories.If one rejects the inclusion of the drive component, then formula (1b) is void, so that Eq.(1a) becomes merely a matter of book keeping of calories while Eq.(1c) only gives an account of the presence of leptin in the blood circulation. However, the aim of this study is to analyse the effect of leptin upon the intake of food The first task in dealing with the model (1abc) is to find values for the parameters. Since the level of leptin accommodates more quickly than the weight, see L = 20.3 [ng ml−1], see −1]. Together with Eqs.(2)–(3) it yields The mean concentration of plasma leptin in women is estimated www.livestrong.com). It is also supposed that the gain in weight through the years is not caused by an increase of the intake of calories but merely by a decrease of physical exercise D is also set at a constant value. The parameter ρ is kept at the value given above in agreement with the literature; the value is independent of the subject under consideration. In the quasi-steady state with condition (2) the differential equation (1a) becomesThe expenditure of calories may vary in time. It is in the order of 2000 calories per day  readsK can be set zero provided that η>>ε; in that case the influence of the starting value is not felt anymore (quasi-steady state). In order to use the weight data we replace the variable stored calories C by the weight W:The integration constant D, Q(0), δ, η and ε are estimated by minimizing the residual sum of squares of the difference between observation iW and model value W(it). From their values the parameters α and β can be computed.The parameters D is let free to change in time. Furthermore, the time scale of months is chosen and the transformation (8) is used. In this way the evolution of the drive D is reconstructed by using the weight data and the estimated parameters, see L(τ) depending on Wobs(τ) as given by (2) and (8).In a next step Eq.(5a) is considered again, but now 1. While estimating the parameter (ε2) for that interval one has to impose a continuity condition upon Q at the point of transition. In Application of nonlinear regression based on least squares (Matlab: lsqnonlin) leads to the estimates given in D(τ) fluctuates around a constant level . For subject A (B) the regular D-values, outside the dietary periods, have a mean −1] with a standard deviation of 131 (183). Also for each of the dietary periods the mean differenceE-values a significant positive effect of dieting was found with p = 0.000 (0.009). In L).It is found that the reconstructed drive D is brought down. In X(s) with the block function B(τ) with B(τ) = 1 during the dietary periods and 0 outside these periods. The variable s denotes a time shift with B(τ) shifted in time over an interval s (backwards for s<0). The function X(s) is normalized such that X(0) = 1, see Thus, during dieting the drive Many studies have appeared on obesity and deal with the question why do people eat too much, or in the case of anorexia why are other people fasting in such an extreme way drive was introduced as a collective noun for all incentives that apply to the intake of food that are not explicitly contained in the model. One of them was following a diet, while the effect of leptin also being a stimulus not to eat was contained in the model. As a result of this investigation one may quantify and thus also compare the way both leptin and dieting diminish the intended intake of food, see I = D−E−αI, where E denotes the amount by which the intended intake rate is reduced from dieting. This process of subsequently extending the model can be continued by including a next incentive in the model e.g. the hormone ghrelin which stimulates eating. In this way step by step the drive D can be peeled off.The notion of The dynamics of the differential equation model (1abc) has two time scales: one in the order of minutes in which the system may reach the quasi-steady state. In this state the leptin concentration follows the calorie content of the adipose tissue. In a time scale of the order of days the system adapts to changes in the food intake. For the problem of structural weight change there is still a scale of months to years in which some trend is expressed. The system, as we have formulated, does still produce a meaningful result at this scale because of the condition that has been imposed upon the long term expenditure of calories. Ravussin and Bogardus Weight fluctuations from dieting represent a type of relaxation oscillation K = 0. Then parameter estimate methods for differential equations have to be used. The Matlab package Grind (www.sparcs-center.org/grind) provides the procedure “optimpars” being a suitable tool for such an investigation in a more extended version of the present model. Another limitation of the present study is the number of only two subjects. From the results presented in The linear model (1abc) does not suffice to analyse the dynamics of subjects with an eating disorder and/or deficiencies in their metabolism. In order to account for the effect of such a malfunctioning, nonlinear elements should be included. An example is the lower sensitivity of the brain for high leptin concentrations caused by an extended period of overweight Appendix S1Reconstructing the drive D.(DOCX)Click here for additional data file.Program S1Parameter estimates.(XLSX)Click here for additional data file.


Whispering gallery mode resonances in liquid droplets and microspheres have attracted considerable attention due to their potential uses in a range of sensing and technological applications. We describe a whispering gallery mode sensor in which standard optical fibre is used as the whispering gallery mode resonator. The sensor is characterised in terms of the response of the whispering gallery mode spectrum to changes in resonator size, refractive index of the surrounding medium, and temperature, and its measurement capabilities are demonstrated through application to high-precision fibre geometry profiling and the detection of unlabelled biochemical species. The prototype sensor is capable of detecting unlabelled biomolecular species in attomole quantities. The exquisite sensitivity of such resonances to the shape and size of the resonator and also to the refractive indices of the resonator and the surrounding medium  have ledet al. [Recently, there has been increasing interest in the potential applications of whispering gallery modes in the field of chemical and biochemical sensing. The field has recently been reviewed by Vollmer and Arnold . Sensorset al. , the excHere we describe a whispering gallery mode sensor based on excitation of whispering gallery modes in a standard communications-grade optical fibre. We provide a brief introduction to the physical principles behind whispering gallery mode spectroscopy, before describing our experimental setup and a range of experiments carried out to characterise the technique and explore its potential applications. We demonstrate the sensitivity of the spectrometer to fibre diameter and to surface refractive index, characterise the response to temperature, and present proof-of-concept data on using fibre-based WGM spectroscopy for targeted sensing of an unlabelled protein. To our knowledge, while cavity enhanced absorption spectroscopy has recently been demonstrated using a fibre-based microcavity , chemica2.i.e.,m is an integer , λ is the free space wavelength and ν the frequency of the light used to excite the resonator, n is the refractive index of the resonator, and d is the resonator diameter. This treatment becomes more accurate as m increases and the path taken by the light within the resonator becomes more closely approximated by a circle. A key result is that both the WGM frequencies and the frequency spacing between adjacent modes are inversely proportional to the resonator diameter. However, in reality the true path is ‘polygonal’ in nature, such that the path length is shorter, and therefore the resonant frequency higher, than predicted by this simple model.WGM frequencies in an optical fibre, or indeed any cylindrical resonator, are reasonably well approximated by matching an integral number of wavelengths to the resonator circumference, l; the azimuthal mode number, m; and the slab mode number, p [m, l and p has 2m field maxima around the cylinder azimuth , l − 1 nodes along the radial coordinate of the cylinder, and p nodes along the cylinder axis. Outside the resonator, the field decays exponentially with a decay constant d of approximately [λ is the wavelength, an is the refractive index of the resonator, bn is the refractive index of the surroundings, and θ is the angle at which total internal reflection takes place at the internal surface of the resonator. We see that the evanescent field from the resonator decays over a distance scale on the order of a wavelength. This has the important consequence that WGM sensing techniques are sensitive only to chemical species in close proximity to the sensor surface.The true mode frequencies within a cylindrical resonator are eigenvalues of the appropriate wave equation. The solutions to such an equation take the form of Bessel functions  and the whispering gallery modes are defined by three mode numbers: the radial mode number, umber, p . A WGM dximately (2)d=λ2πν0 is usually expressed in terms of a ‘quality factor’ or ‘Q-factor’, Q.Pstored and Plost are the power stored within the cavity and lost from the cavity, respectively, and δν is the spectral linewidth of the cavity mode at frequency ν0. Gorodetsky et al. [Q of a perfect microsphere resonator is around 1010. Experimentally measured Q factors are generally much lower than this due to scattering from surface roughness and impurities, absorption by the material, and simultaneous excitation of multiple closely spaced modes. Cylindrical resonators have been characterised with Q factors of the order of 105 or higher [The measure of an optical resonator or cavity’s ability to confine light of a given frequency y et al.  have calr higher .et al. [δν associated with binding of N randomly located molecules to the resonator surface is given byαex is the excess polarisability, σp is the surface density of adsorbed molecules, E0(r) is the electric field at position r, ɛ0 is the permittivity of free space, and ɛrs is the relative permittivity of the resonator. The integrals are over the surface area and volume of the whispering gallery mode, with the volume integral yielding the total energy of the mode within the resonator. Of key importance is that the fractional shift on binding depends on both the fraction of the mode energy found at the positions of the bound molecules, and on the excess polarisability of the molecules. For proteins, αex is approximately proportional to the molecular mass, such that larger shifts are predicted for heavier molecules.The response of the whispering gallery mode spectrum to molecular binding at the resonator surface has been explained in terms of a change in optical path length caused by direct interactions between the resonant photons and the bound molecules. When a molecule binds to the resonator surface, it displaces water and creates an excess dipole moment at the binding point. Arnold et al.  considerS = δλ/δn to changes in the refractive index of the bulk, where δλ is the observed wavelength shift accompanying a change δn in refractive index, the fractional wavelength shift on molecular binding to the sensor surface isnm is the refractive index of the sensor material and ns is the refractive index of the solvent. This expression allows the surface density of detected molecules to be determined from the measured wavelength shift.White and Fan  have genQ factor of the cavity, since a higher Q leads to sharper cavity mode spectra and detection of smaller shifts in resonance frequency. A consequence of this is that the detection limit of a fibre-based whispering gallery mode sensor is likely to be somewhat lower than for microsphere-based sensors, due to the lower Q factor of cylindrical relative to spherical cavities. However, there are a number of potential advantages that make fibre-based sensors worth exploring. Some of these have been pointed out previously by Farca et al. [The detection limit is highly sensitive to the a et al. . Optical3.μm diameter single mode optical fibre with its plastic buffer coating removed. Buffer removal was achieved either using a standard fibre stripper or by placing the fibre in the flame of a butane torch. The small core diameter of a single mode fibre (∼8 μm), together with the fact that the WGMs excited are predominantly confined near to the surface, means that the sensor fibre essentially acts as a uniform cylindrical dielectric with the refractive index characteristics of the fibre cladding. Light is coupled into WGMs of the sensor fibre via the evanescent field generated in a tapered region of a second fibre, labelled as the ‘delivery fibre’ in μm fibre. Transmission losses through the fibre on tapering varied depending on the exact tapering conditions and final taper diameter, but were generally of the order of a few tens of percent.The basic experimental setup for recording the WGM spectrum of an optical fibre is shown schematically in The light source used to excite whispering gallery modes in the sensor fibre is a 20 mW fibre-coupled Santur TL2020-C DFB communications laser, tuneable over the range from 191.7 to 196.1 THz in 0.05 THz steps (1564 – 1529 nm in ∼0.4 nm steps), and the detector is a DET10C high speed InGaAs photodiode sensor (Thorlabs). Light from the laser source is coupled into the input end of the delivery fibre, and the signal at the output end is detected by the photodiode. The measured signal as a function of wavelength is recorded on a PC via a USB analogue to digital converter (LabJack U12). In the absence of a sensor fibre, the measured signal is uniform with wavelength and simply reflects the emission spectrum of the laser, modified somewhat by the transmission characteristics of the delivery fibre. To measure the WGM spectrum of a sensor fibre, the fibre is crossed at right angles with the tapered region of the delivery fibre and the two fibres are brought into contact. Each time the laser wavelength comes into resonance with a WGM, light is coupled efficiently from the delivery fibre into the sensor fibre. This reduces the amount of light transmitted through the delivery fibre, resulting in a dip in the measured signal intensity at each resonant wavelength.The detailed mode structure of the WGM spectrum depends on the particular combination of delivery and sensor fibre used, since small variations in taper geometry can have quite marked effects on the relative coupling efficiencies into different modes. This may be seen in the data presented in Details of individual experiments will be described along with the results in Section 4.3.1.i.e., the simulations are limited to modelling p = 0 modes), so the simulated spectra are somewhat simpler in form than many of our measured spectra. However, with the exception of experiments carried out specifically to investigate coupling into axial modes, described in the next section and illustrated in Efficient coupling between the evanescent field generated at the surface of the tapered delivery fibre and a whispering gallery mode in the sensor fibre requires optimum overlap between the two mode fields. This is strongly dependent on the diameters of the two fibres, and in particular on the diameter of the tapered delivery fibre. To determine the optimum taper diameter, a two-dimensional model of the crossing region was simulated using a commercial finite difference time domain (FDTD) software package . Two-dimμm fibre, there was considerable overlap between the range of fibre sizes for which we could perform simulations and the range of sizes used experimentally in the fibre profiling experiments described below. Simulations run with a range of resonator diameters (with a background refractive index corresponding to that of air) indicate that coupling is inefficient until the taper diameter falls to around one micron or less. This is in line with our experimental observations and with the taper dimensions used by other authors to excite WGMs in cylindrical cavities [μm diameter sensor fibre excited by the evanescent field from a 1 μm diameter taper. The simulation software also allows the electric field distribution within the two fibres to be plotted, which provides a useful visualisation of the mode field overlap between the delivery and sensor fibres. Build-up of the electric field inside the resonator with time during excitation at a frequency corresponding to a WGM is shown in The frequency resolution of the simulations is controlled via a Fourier transform relationship by the total time period simulated, and was chosen to match the frequency resolution of our experiments. The computational requirements of the FDTD code limited simulations to fairly small resonator diameters, even for two-dimensional simulations. However, while we were not able to carry out simulations matching our experimental frequency resolution for a 125 cavities . While ocavities , 33, in The simulations also allow us to examine the coupling between the delivery and sensor fibres when the fibres are in contact and when they are separated. Since the evanescent field from the delivery taper decays exponentially away from the surface, coupling is unsurprisingly stronger at very small separations and when the fibres are in physical contact. Another important observation is that there is very little perturbation of the electric field of the WGM inside the sensor fibre as the delivery fibre is brought into close proximity or even into contact.Currently, the resolution of the instrument is limited by the frequency resolution of the communications laser used to make the measurements. In the next phase of the project, the tuneable laser/photodiode setup will be replaced with a broadband supercontinuum source and high resolution spectrometer. This will not only improve the instrument resolution, but will also greatly increase the speed of data acquisition, since wavelength scans will no longer be necessary. However, here we will demonstrate that even with the relatively low resolution currently attainable, the instrument provides a powerful tool for the characterisation of optical fibres, and we will also provide what we believe is the first proof of concept for targeted chemical sensing using whispering gallery modes in optical fibres.4.4.1.μm diameter single mode fibre in air. The spectrum is seen to consist of a series of equally spaced dips, corresponding to WGMs of successively increasing azimuthal mode number m (mode numbers are labelled on the top axis). A lower limit for the Q factor of our fibre resonators may be determined from the linewidth of the measured whispering gallery modes to be around 1.5 × 103. Given the limited resolution of the laser currently being used, and our present inability to resolve individual sub-modes as a result (as discussed in the following paragraph), the true Q factor is almost certainly considerably higher than this. An indication of the potentially realisable Q factors in our configuration may be obtained from the FDTD simulations. The simulations described above were carried out at a resolution approximately matching that of the experiments. However, simulations carried out at higher resolution predict a Q factor of 1.73 × 105 for a 50 μm resonator at an excitation wavelength of 1558 nm. Since 2D FDTD simulations are unable to account for longitudinal dispersion of the WGM along the fibre axis, it is likely that the true value lies somewhere in between our experimentally measured value and that predicted by the simulations. For comparison, Q factors reported for WGMs excited in thin-walled capillaries by Zamora et al. [et al. [5, while planar disk resonators[Q factors of around 104.a et al.  and Whit [et al.  range fresonators–37 have l = 1, p = 0 , though the visible shoulder to the high frequency side of each peak may almost certainly be attributed to unresolved higher order modes in l and p. This may be explored further by examining the effect of crossing angle  on the spectra. Reasonably accurate values for the coupling angle in our experiments may be determined from video microscope images of the crossed fibres. A number of spectra recorded at different crossing angles by mounting the resonator on a rotating stage are shown in ϕ moves away from 90 degrees, and the peaks broaden.The modes that are most efficiently coupled into using the excitation geometry employed have l = 1, p = 0 mode, the overlap is at a maximum when the two fibres are crossed at right angles. The projection of the delivery fibre field onto the WGM field reduces as the angle is moved away from 90 degrees, explaining the observed reduction in signal from this mode as the crossing angle is reduced. At such geometries, overlap with other helical WGMs of the resonator is improved, and we see increased signal from coupling into these modes. This manifests mainly as a broadening in the spectral peaks. However, structure is clearly visible within the broadened signal, corresponding to the various modes excited.The coupling efficiency into a given whispering gallery mode depends on the extent of overlap between the evanescent field of the delivery fibre taper and the mode of interest inside the resonator. When considering the l = 1, p = 0 mode on a more quantitative level. The electric field of this mode lies in the plane containing the fibre axis, and the projection of the delivery fibre field onto this axis varies as the cosine of the crossing angle ϕ between the two fibres. We may therefore expect the intensity of the l = 1, p = 0 peak  to vary in proportion to cos ϕ. As shown in We can consider coupling into the Taking these results into account, care was taken in all subsequent experiments to ensure that the delivery and sensor fibres were crossed at right angles.δν, i.e., the spacing between adjacent peaks, is given approximately byn in the above expression could more rigorously be replaced with the ‘effective refractive index’ en seen by the propagating WGMs in the resonator. The effective index will have some dependence on the size of the resonator due to the differing evanescent field fractions. However, other authors have shown [ν and d. The increase in scatter of the data points at large values of 1/d is largely due to the difficulty in obtaining an accurate microscopic measurement of the fibre diameter as the diameter reduces. The slope of the plot is 6.31 × 107 m s−1, in fairly good accord with the model, which predicts a proportionality constant of From ve shown  that for4.2.et al. [The extreme sensitivity of WGM spectra to changes in the resonator shape and size make WGM spectroscopy a promising tool for topological measurements at the sub-wavelength scale. The use of WGMspectroscopy in the characterisation of fibre diameter variations has been demonstrated previously by Birks et al.  and by Wet al. . In Figuet al. [The results of this approach are validated by comparing the taper geometry determined from these spectroscopic measurements with a microscopic image of the taper; the two sets of data are overlaid in et al. , improvi4.3.Using the WGM spectroscopy technique for chemical sensing requires the ability to make measurements in aqueous solution. A small sample cell, shown in S = 42 ± 3 nm/RIU (RIU = refractive index units). A typical high resolution tuneable laser system has a bandwidth Δλ of the order of 0.001 nm or better, which would yield a value of Δn = Δλ/S =∼ 2.4 × 10−5 for the detectable change in refractive index using our current setup.In order to determine the sensitivity of our fibre resonator to changes in the bulk refractive index, measurements were made in which ultrapure water (prepared using a Milli-Q Synthesis system) and ethanol were alternately flushed through the sample reservoir, with spectra recorded on each change of solvent. The associated wavelength shift averaged over all of the peaks in the spectrum, together with a one-standard-deviation error estimate, was 1.3 ± 0.1 nm. The refractive indices of water and ethanol at 1,550 nm and 293 K are  1.323 anet al. [et al. showed that higher sensitivities could be achieved using capillaries with sub-micron wall thicknesses, and achieved sensitivities of 50–70 nm/RIU for TM modes, and 130–170 nm/RIU for TE modes in their resonators at refractive indices near that of water. In a related approach, Tan et al. [Since in our experiments the electric field of the WGMs is relatively localised near the fibre surface, we can compare our calculated sensitivity with similar measurements made for capillary-based sensors. Our sensitivity compares favourably with the 16.1 nm/RIU reported by White et al. . Zamora n et al.  coated gChanges in temperature have a small effect on both the diameter of the resonator  and the refractive indices of the resonator and the surrounding medium. For this reason, the temperature dependence of the WGM frequencies was evaluated prior to attempting any targeted binding studies in order to determine whether temperature-induced shifts in WGM frequencies should be considered as a source of error in our measurements.Differences in the thermal expansion coefficients of the PMMA sample cell and the silica fibre meant that the cell could not be used reliably for these measurements. Instead, a simple alternative arrangement was used, in which the delivery and sensor fibres were fixed onto a glass substrate immersed in a water-filled reservoir resting on a hot plate. For each measurement, the hot plate was set to the required temperature, the system allowed to come to thermal equilibrium, and a spectrum recorded. The temperature was recorded using a digital thermometer.Over the range from 20 to 40 °C, any shifts in WGM frequencies due to changes in temperature were essentially within our measurement uncertainty, and were at least a factor of ten smaller than the shifts observed in the analyte binding experiments described in the next section. This was in accord with the results of FDTD simulations in which the effects of the expected thermal variation of resonator diameter and refractive indices were explored. Temperature effects are therefore unlikely to have a significant impact on targeted sensing experiments carried out at a nominally constant temperature.unlabelled molecular species.Targeted sensing involves chemically modifying the surface of the fibre to enable specific binding of the analyte of interest. If the analyte is present in the sample solution then the change in refractive index at the fibre surface as the analyte binds to the fibre will lead to a shift in the WGM frequencies, providing a mechanism for detection. For example, using this technique, a sensor for a particular protein could in principle be fabricated by coating the fibre surface with the appropriate antibody. As outlined in the Introduction, a distinct advantage of WGM-based techniques over many existing biosensors is that they enable the detection of Our proof-of-concept measurements utilised the biotin-streptavidin interaction. Biotin is a small B-complex vitamin molecule which binds through simple physisorption to bare silica surfaces, or may be bound covalently to suitably functionalised surfaces , while sμm sensor fibre (fabricated from standard 125 μm fibre using the same tapering rig employed to make the tapered delivery fibres) was employed in order to give well spaced WGMs within the frequency range of the laser. The sample cell containing the crossed delivery and sensor fibres was flushed first with ultrapure water and then with buffer solution. No shift in WGM frequencies was observed on switching between water and buffer, indicating that within our frequency resolution, the refractive index of the buffer may be assumed to be the same as that of water. After flushing with buffer solution, the fibres were treated with a 0.1 μM solution of biotin (Sigma-Aldrich) in buffer and flushed again with pure buffer solution to ensure that only chemisorbed biotin remained attached to the fibre surface. In principle, we might expect to see a shift in WGM frequencies on binding of biotin to the fibre surface. However, no such shift was observed, indicating that our current frequency resolution is insufficient to detect the small refractive index changes associated with binding of a small molecule to the surface. After rinsing, a 1 μM solution of streptavidin (Sigma-Aldrich) was admitted to the sample cell. Soon after injection of streptavidin into the sample cell, a shift of approximately 0.9 ± 0.2 nm in the WGM wavelengths was observed (see A 45 rved see . The shiσp of detected streptavidin molecules. The excess polarisability of streptavidin is αex = 4πɛ0(3.2 × 10−21) cm3 [δλ/λ ∼ (6±2) × 10−4, this yields a surface density of ∼ (2.7 ×0.9) × 1017 molecules m−2. We note that the evanescent field fraction in the 45 μm fibre used to make the streptavidin measurements will be somewhat larger than for the 125 μm fibre used to make our bulk sensitivity measurements, with FDTD simulations indicating a ratio of around 1.4 between the evanescent field fractions in the two fibres. A sensitivity increase of this order would yield a surface coverage of 1.9 × 1017 molecules m−2, still within our experimental errors.−21) cm3 , and theμm resonator fibre excited with a 1 μm taper corresponds to a surface area of approximately 1.4 × 10−10 m2 , so the frequency shift shown in Figure 9 is caused by adsorption of around 3.8 × 107 molecules (6.3 × 10−17 moles).A WGM in a 45 16 m−2, somewhat lower than the result obtained above. This may simply be a reflection of the uncertainties in the values used for αex and the various refractive indices in the above calculation, or may indicate multilayer coverage of the fibre surface at the rather high concentrations used in our experiments.Based on the somewhat limited information available on the dimensions of streptavidin, a monolayer would correspond to a surface density of ∼ 4 × 10i.e., around 7 × 1016 molecules m−2.Based on analysis of all of our existing experimental data on the biotin-streptavidin system, and taking into account the frequency resolution of the laser source, we estimate the detection limit for streptavidin using the current experimental setup to be around four times lower than the amount detected in the results reported above, 2O rather than H2O as the solvent could mitigate this issue, but since ideally a biological sensor will be optimised for operation in a range of aqueous environments, employing a tuneable laser source covering a different wavelength range is a more realistic long-term solution.While the streptavidin concentrations used in these experiments are higher than is likely to be found in biologically relevant systems, these experiments provide a first demonstration that targeted chemical sensing is possible using whispering gallery modes in optical fibres. Our detection sensitivity is currently limited primarily by the spectral resolution achievable with the fairly large minimum frequency step size of the communications laser employed to excite the WGMs. A second consideration is the reduction in Q factor resulting from the significant optical absorption of water at the wavelength of our laser system. Performing the experiments using DQ factor of the optical fibre resonator, is at least two to three orders of magnitude better than we are currently able to achieve. Alternative laser sources will be investigated in future work and should lead to a considerable improvement in our ability to detect small shifts in WGM frequencies. In addition, the observed shifts on binding of streptavidin vary according to the quality of the tapered delivery and sensor fibres used . A more robust process for creating reproducible tapers is needed in order to optimise this technique for chemical sensing. However, these early proof-of-concept results provide an encouraging indication that the fibre WGM technique could be developed into a robust and low-cost method for targeted sensing.The FDTD simulations discussed earlier indicate that the ultimate resolution of the technique, dictated by the 5.We have presented a whispering gallery mode sensor based on a a simple optical arrangement consisting of a pair of crossed optical fibres, and demonstrated potential applications in fibre size profiling and sensing of unlabelled chemical and biochemical species. A light source upgrade should allow us to improve the frequency resolution of the instrument to the point where detection of species at typical biological concentrations will be possible.