Capturing nature’s statistical structure in behavioral responses is at the core of the ability to function adaptively in the environment. Bayesian statistical inference describes how sensory and prior information can be combined optimally to guide behavior. An outstanding open question of how neural coding supports Bayesian inference includes how sensory cues are optimally integrated over time. Here we address what neural response properties allow a neural system to perform Bayesian prediction, i.e., predicting where a source will be in the near future given sensory information and prior assumptions. The work here shows that the population vector decoder will perform Bayesian prediction when the receptive fields of the neurons encode the target dynamics with shifting receptive fields. We test the model using the system that underlies sound localization in barn owls. Neurons in the owl’s midbrain show shifting receptive fields for moving sources that are consistent with the predictions of the model. We predict that neural populations can be specialized to represent the statistics of dynamic stimuli to allow for a vector read-out of Bayes-optimal predictions. Many behaviors require predictive movements. Predictive movements are especially important in prey capture where a predator must predict the future location of moving prey. How sensory information is transformed to motor commands for predictive behaviors is an important open question. Bayesian statistical inference provides a framework to define optimal prediction and Bayesian models of the brain have received experimental support. However, it remains unclear how neural systems can perform optimal prediction in time. Here we use a theoretical approach to specify how a population of neurons should respond to a moving stimulus to allow for a Bayesian prediction to be decoded from the neural responses. This provides a novel theoretical framework that predicts properties of neural responses that are observed in auditory and visual systems of multiple species. Predicting the future position of an object in the environment is a common and critical component of many tasks that involve reaching or orienting toward moving targets –4. To exPrey capture for moving targets has been studied at the behavioral and neural levels for animals that rely on visual –9 and auThese studies open several questions about the neural basis of predictive behaviors. What information is represented in these populations of neurons? Is the observed neural representation an optimal solution to the prey capture problem faced by each species? An optimal solution to the prediction problem would take into account the source dynamics, sensory statistics, and prior information to guide the solution. This approach to an optimal solution can be formulated as Bayesian prediction . There iHere we specify how a population of neurons should respond to a moving stimulus to allow for a Bayesian prediction to be decoded from the neural responses. We approach this question in the context of auditory-based prey capture by the barn owl. The Bayesian prediction problem we consider is that of predicting a sound source’s future direction, given a sequence of sensory observations and a prior distribution for direction and angular velocity. It has been shown that the owl’s sound localization for brief sounds is consistent with a Bayesian model . Here, eThe success of the PV in decoding predictive movements of visual targets in the salamander  and dragWe consider the problem of predicting the future direction of a moving source from a temporal sequence of auditory observations. Specifically, the prey capture problem is that of predicting the direction of a moving sound source a short time in the future based on the sequence of interaural time difference (ITD) measurements from the sounds reaching the left and right ears . ITD is k the direction at a point n time steps in the future θk+n, given the sequence of observations up to time k, ITDk1: = , is to compute an estimate from the posterior distribution pk+n. The form of the posterior distribution is determined by a model for the dynamics of the moving target and the statistical relationship between the state of the target and the ITD observations.The Bayesian filtering approach to predicting at time θk is the target direction, ωk is the angular velocity, ηk is a zero-mean circular Gaussian noise process, νk is a zero-mean Gaussian noise process, k is the current time step, and Δt is the time step duration. The sensory information ITD is modeled as a sinusoidal function of direction that is corrupted by noise:ξk is a zero-mean Gaussian noise process with standard deviation 12.5 μs and the amplitude A and frequency f are determined by the shape of the owl’s head and facial ruff  and ξk is a Gaussian noise process with zero mean and variance R that is referred to as the observation error.The output of the system is a linear approximation to the mapping from direction to ITD plus noise:i|j to be the mean and covariance, respectively, of the posterior at time i given observations up to time j. The mean of the posterior distribution is computed recursively through a process of prediction and updating. The prediction one step ahead in time is computed asThe Kalman filter is used to compute the mean and covariance of the posterior at each time. Define Updating the estimate with a new observation is computed ask+n. This requires the estimate at time k to be multiplied by the state transition matrix n times:When an estimate has been made for the state k+n is computed asThe covariance of the posterior at time We used a particle filter to compute the Bayesian prediction for simulations where the linear approximation to the relationship between direction and ITD was not valid. Particle filtering algorithms are sampling-based approaches to approximating the posterior distribution that are valid for nonlinear and non-Gaussian models . The parm particles for direction j = 1, 2, …, m. Here we used m = 10,000. Set the time index k = 1. For each time k iterate the following steps.Initialize by selecting k:Compute weights for the particles using the likelihood of the ITD measurement at time n particles with replacement from the previous set of particles, where the probability of selecting particles k.Normalize the weights k + n is approximated by propagating the particles according to the source dynamics over n stepsi = 0, 1, …, n– 1, where The posterior at time pk+n is found as the circular mean of the particles The estimate of the source direction under the predictive posterior k + 1 and repeat from step 2 using the particles Update the time step to pk+n as described above. We then used our main result specifying that the activities are proportional to the ratio of the posterior and prior to generate the spiking probabilities for the population of neurons. We scaled the ratio of the posterior to the prior so that firing rates would be approximately 10 spikes/s for neurons with peak responses. Spike counts were generated for the population at each time step using independent Poisson neurons with the specified rate. The direction of the PV was used to estimate the predicted source direction at each time.The neural population model consists of 5000 Poisson neurons with receptive fields that shift according to the prediction given in the proposition proved in the results. The preferred directions of the neurons were drawn from the prior Gaussian distribution with mean zero and standard deviation 23.3 deg. These preferred directions match the model of Fischer and Peña (2011). To generate the neural responses to a sequence of ITD inputs we first computed the predictive posterior θPVA(t) and the Bayesian prediction θBayes(t) to quantify the approximation error whereThe PV was tested for counterclockwise source trajectories with initial directions covering -180 deg to 180 deg in 10 deg steps and angular velocities ranging from 0 deg/s to 150 deg/s in 25 deg/s steps. We calculated the RMS error between the PV estimate


Breakage–fusion–bridge cycles in cancer arise when a broken segment of DNA is duplicated and an end from each copy joined together. This structure then ‘unfolds’ into a new piece of palindromic DNA. This is one mechanism responsible for the localised amplicons observed in cancer genome data. Here we study the evolution space of breakage–fusion–bridge structures in detail. We firstly consider discrete representations of this space with 2-d trees to demonstrate that there are  Breakage–fusion–bridge (BFB) cycles potentially arise whenever a stretch of DNA is broken and a cell division cycle takes place. The first stage in this division process is DNA replication, where duplication will take place up to the DNA break, leaving two exposed ends. Prior to cell division, the cells checkpoint machinery will look for mistakes and the two exposed ends may be erroneously joined together by double stranded break repair. This results in a palindromic sequence of DNA, often containing a duplicated centromere see Fig. a. SpindlNote that the first break in the cascade occurs on the longer arm of the chromosome to the right of the centromere, as presented in Fig. The process is unlikely to continue indefinitely because eventually repair machinery will attach exposed ends to other portions of the genome to produce a translocation, for example, or telomerase may cap the end to produce a somatic telomere. However, this process of repeatedly duplicating, repairing and unfolding is known to produce complex rearrangements of the original genomic assembly, and are frequently observed in cancer genomes . We can thus label each point along the original DNA sequence with its genomic position and consider how these labels are duplicated in the BFB process. By comparing the positions on the BFB product with the original reference sequence, we can fold the BFB product so the same labels (i.e. reference positions) are vertically aligned, such as in Fig. fold and the folded structure relative to the reference will be used in the majority of the work. We will also use breakpoint to refer to the reference position of the fold. The stretch of DNA between two consecutive folds will be referred to as a segment.There are some key differences to note however. Whilst paper is intrinsically the same material at all positions, DNA is composed of a variable sequence of nucleotides and subsequently is identifiably unique along its length This representation mirrors that observed experimentally. In Fig. mentclass2pt{minimcopy number profile, summarized in Fig. This inference relies in part on the linear relationship between the experimental signal and the number of copies of DNA ‘folded’ across a given reference region, the These data constitute an example of an ‘amplicon’, which are frequently observed in cancer genome data. These are clusters of rearrangements with a high signal in the reference genome, indicating an abnormally large number of copies are present in the cancer genome. These ‘amplified’ regions are usually restricted to a few megabases of DNA, a small portion of a typical human chromosome. The BFB process is one mechanism by which these events can arise   positions of folds. This representation is different to that in Kinsella and Bafna , where aConsider the example in Fig. one-fold word We now implement the BFB process. Firstly, we have a break in our segment at the first fold, labeled two-fold word We continue the process, the next fold occurring at position Note also that the word is palindromic, if we reverse the order of the symbols we obtain the same word. This reflects the chromosome symmetry; if we turn the chromosome upside down, we end up with an identical structure.Note that we had two choices to construct the fold at reference position If a BFB fold is positioned a length The palindromic nature of BFB products then means we always have two choices to place the fold position for any given product. In what follows, we always assume that we are duplicating DNA above the position of the fold, with respect to the representations drawn in Fig. three-fold word four-fold word We now continue the process, iteratively building up the word. The next new fold is at position Note also that in Fig. We summarize the word representations of a BFB process as follows.Word constructionWe initialize with five-fold word We now consider the opposite problem; given a BFB word, we need to reverse the process and identify the events that have taken place. This represents the typical inference problem in genomics, where we have the final structure of a genome and wish to reverse engineer the process to identify the evolutionary history. This can be achieved by identifying the unique BFB fold that demarcates the centre of the palindromic structure and undoing the duplication. For example, the BFB sequence of Fig. In general we have the following result.STEP 1:Take palindromic word STEP 2:Identify the rightmost uniquely occurring symbol STEP 3:STEP 2.If  For a viable BFB fold word A fold word is a viable representation of a BFB process if and only if it can be reversed with the following algorithm. This identifies the unique minimal sequence of BFB cycles responsible for the word. This is largely the same as the process described in Kinsella and Bafna , which rpartially order set (poset). This is derived from a 2-d tree, which is required for the more general problem of estimating the total number of evolutions in Sect. We now have a word representation of the BFB process. This representation is not unique and there may be several qualitatively distinct structures for each BFB word. For example, the two structures in Fig. fold sequence representation for any given n-fold word. These sequences each contain To do this, we first obtain a Consider the evolution portrayed in Fig. s Remark  means weThus for the example of Fig. reduced fold sequenceWe noted previously that because the 3rd BFB is deleted by the 4th, the end product can be obtained by simply implementing the undeleted BFBs. Note that the 4th BFB fold can be positioned on the third segment after the midpoint of the structure arising from the 2nd BFB cycle. We can thus equally represent the final structure with cumulative sequencedirection sequence, Consider the cumulative sum of the fold sequence, the s}=1,2,4,,6$$\end{Representing the process this way thus enables us to capture some nice properties. We summarize this as follows.fold sequence of cumulative sequence. The number of segments of the structure after reduced fold sequence contains strictly positive values and is obtained by combining direction sequenceAny fold word partially ordered set (poset).We now have an algebraic representation of BFBs. We will see that each representation can account for many different BFB structures. The two structures given in Fig. The general situation is described in the following result.If For example, from fold sequence It is natural to attempt to count and construct the different orders we get for a single fold sequence. We do this with the aid of a 2-d tree construct; a special kind of directed graph that generalizes the notion of a tree, as exemplified in Fig. parent nodes already present. Two edges are then constructed from these two nodes to the new node , extends from the node with greater value minor edge (dashed), extends from the other node labeled Each pair of edges introduced during the 2d-tree construction consists of a plain or flipped. If the word plain (black). If they change from flipped (orange).Each pair of major/minor edges and corresponding daughter node are either For example, in Fig. The following observation is important when we later consider the number of possible evolutions from a fixed number of BFB cycles.(i)If node (ii)If node Consider a node with value Thus consider Fig. 2-d Poset Tree, This construction is termed the We now have the 2-d tree poset to encapsulate the order relations between the fold reference positions, where a daughter node labeled Order tree, Now if we just consider the major edges of the 2-d tree we get the tree structure of Fig. The number of possible orders for Fig. Let We can now count the number of distinct evolutions for a single BFB word or fold sequence. Consider next two problems, firstly, how to count the number of different words given a fixed number of BFB cycles, and secondly how to then utilize Theorem We then first consider the question of how many different fold words can form from Let The resulting counts can be seen in Table If We can also see these counts in Table first fold. In Fig. five-fold word The proof of this theorem relies on an appropriate induction. This involves the reintroduction of a We constructed an order tree from the 2d-tree by removing all minor edges. We require the capacity to modify the shape of an order tree with the following operation.ES: edge switchingRemove a major edge and insert the corresponding minor edge.This move can be seen in Fig. This is a specific form of the Subtree Prune and Regraft (SPR) operation that has seen application to many other problems in evolution (Semple and Steel We require a specific set of ES operations. Let SS: subtree switchingIPerform an ES operation on any flipped (orange) edge contained in subtree IIPerform an ES operation on any plain (black) edges adjacent to (so not contained in) subtree IIIForm a new root node for I). Edge II). We also have a new root node (SS move III).Examples can be seen in Fig. We find there is a unique correspondence between the possible introductions of a first fold Let This correspondence between the introduction of a new first fold and tree operations allows us to count a subset of order trees, as described below.Let Examples of this can be seen in Fig. Summing over the possible values of For any order tree five-fold word An example of this can be seen in the last column of Fig. We are finally in a position to prove our main result and count evolutions.n-fold word with a corresponding order tree We have seen that any The formula counting evolutions that retain at least one copy of all folds is obtained as follows. The first BFB cycle results in a single fold retained in the structure. Consider any evolution on The total number of distinct copy number profiles were determined for a fixed number of BFB cycles, as summarised in Table So far we have treated BFB cycles as a discrete process, treating the folded structures as functions of fold sequences, a space we have now explored in some detail. However, the BFB process relies on the fold occurring somewhere along the length of the structure. We can thus consider the fold positions in a sequence of BFB structures as a stochastic process, and investigate the implications of this on the BFB structure.We first consider the stochastic nature of the structures length under the simplest assumption that the fold position is uniformly distributed along the structure. This is likely to be an approximation as rearrangements are known to be linked to fragility The probability that the (ii)The probability density function of the minimum BFB length in a sequence of (iii)The distribution of the number (iv)The amplicon size, (v)If we observe two folds in a final BFB structure with consecutive lengths Some of these distributions are plotted in Fig. This can be intuited as follows. If a BFB product has current minimum length We also know from Theorem We have seen different BFB structures arising due to different fold sequences. It is thus natural to investigate the likelihood of a particular fold sequence occurring. We extend the stochastic process approach above to elucidate this problem. We again make the simplifying assumption that BFB folds occur uniformly along the length of the structure.Suppose we are interested in the likelihood of observing fold sequence such as The likelihood of seeing reference positions The conditional probability of next fold sequence element The probabilities for the first few BFB sequences can be seen in Fig. Note that these probabilities are not for reduced sequences. For example, although fold sequences We now put the framework we have described into context with some real data.read depth measurement; a count of illumina reads lying within a Consider the amplicon of Fig. We thus have six regions, each with a fixed (unknown) copy number, and five folds. We would like to identify the underlying fold sequence and provide an explanatory evolution of events. This would indicate the order that the folds occurred in this process, the resultant structure, and subsequent copy numbers across the amplicon.A natural approach would be to first determine integer copy numbers within each segment through a method of segmentation, and then explore the evolutionary space of BFBs to find a structure that fits the estimated copy number. This is an approach utilized in Greenman et al.  for moreThe calculation of integer copy number remains challenging. There are several factors that can make this difficult, including variable mapability across the genome, the presence of subclonality, the presence of normal contamination, dispersion in read depth (Greenman et al. An illumina sequencing library will produce a large number nd Tammi , where and Tammi . For examentclass2pt{minimWe would now like to use the BFB machinery developed previously.In order to analyse the amplicon, we first need to know whether the two ends of the BFB either both face left or both face right. The rightmost region, For any proposed fold sequence The amplicon in Fig. The fold likelihood mentclass2pt{minimThese unfortunate limitations will apply in general because evolutions do not always have unique copy number profiles Table , and it The portrait of BFB cycles sketched in Fig. Our models have so far assumed that the BFB mutation process is randomly sampled and under no forces of selection. This is unlikely to be the case in general and the selection of any cells that have growth advantage are likely to emerge in cancer samples. We have seen in Fig. If this process proceeds over We can also argue this from a different perspective. Note that the distribution of the amplicon size in Fig. Selection thus plays a fundamental role in the evolution of these structures and a fuller investigation of selection acting across a growing set of cells undergoing a BFB process is warranted.We have highlighted some of the genomic complexities that arise from the BFB process that underlies the copy number profile of many amplicons observed in cancer. Although not every copy number profile can arise from a BFB process, the number of different BFB evolutions rises spectacularly quickly with the number of BFB cycles. Furthermore, a single copy number profile may be possible from more than one BFB evolution, making inference of the correct evolution difficult. For such degenerate cases, use of additional in-silico methods such as Greenman et al. , or expeThis work provides some understanding to the evolution of amplicons. However, amplicons can arise from other processes such as tandem duplication (McBride et al. This analysis also assumes that the data arise from a single dominant clone, which is not always the case (Nik-Zainal et al.


The finite elements method (FEM) is a useful tool for the analysis of the strain state of semiconductor heterostructures. It has been used for the prediction of the nucleation sites of stacked quantum dots (QDs), but often using either simulated data of the atom positions or two-dimensional experimental data, in such a way that it is difficult to assess the validity of the predictions. In this work, we assess the validity of the FEM method for the prediction of stacked QD nucleation sites using three-dimensional experimental data obtained by atom probe tomography (APT). This also allows us to compare the simulation results with the one obtained experimentally. Our analysis demonstrates that FEM and APT constitute a good combination to resolve strain–stress problems of epitaxial semiconductor structures. In the last decades, semiconductor quantum dots (QDs) have been extensively investigated because they are attractive structures for electronic and optoelectronic advanced devices-3. The cThe simulation of the strain–stress fields in a semiconductor material in order to predict the location of stacked QDs lead to a better understanding of the behaviour of these complex nanostructures. The finite elements method (FEM) is a widespread tool to calculate the strain and stress fields in semiconductor nanostructures, and it has been used in the study of QDs,14,15, QIn this paper, we have performed a strain analysis using FEM based on APT experimental data of a sample of InAs-stacked QDs. We have used the 3D compositional data obtained by APT from a layer of QDs to predict the nucleation site of the next layer of QDs, and we have compared the predictions obtained by FEM with the experimental observations by APT. Our results show that the combination of FEM with APT constitutes a powerful methodology for the analysis of the nucleation sites in stacked semiconductor QDs.0.2Al0.2Ga0.6As layer grown by molecular beam epitaxy. A specimen with the needle-shaped geometry required for APT has been milled using a dual-beam FEI Quanta200 3D focused ion beam (FIB) instrument  equipped with an in situ OMNIPROBE micromanipulator , and following the procedure described in Hernández-Saz et al.[.[xAlyGax-y1-As elastic constants and the lattice parameters; it is based on the atomic concentration obtained from the APT data . Initial strain was assumed to be ϵ0 = (axAlyGa1-x-yAsIn - aGaAs)/aGaAs in all subdomains except in the base, where ai denotes the lattice parameter of i. The elastic properties have been taken from[σijϵij/2, where σij (ϵij) with i,j = x,y,z are the components of the stress (strain) matrix (the Einstein summation convention is assumed). The normalized SED is expressed as SED/SEDmax, where SEDmax is the maximum value of SED at the top layer surface.The sample used to exemplify the study consists of InAs/GaAs-stacked QDs covered by a 2-nm Inaz et al.. The nee et al.[. criteriaken from. The elaFigure Figure In order to predict the nucleation site of the QD in the second layer, the chemical potential of the material during growth should be considered. In this case, the chemical potential has two major contributions: the one related to the surface energy and the one corresponding to the elastic strain. With regard to the first one, a previous analysis of the structure by transmission electron microscopy has shown that the structure grows with a flat surface, as no undulations have been observed in the wetting layers or in the surface of the structure. Because of this, the surface energy is not expected to have a major effect in the chemical potential of the structure in the prediction of the nucleation sites because prior to the formation of the second layer of QDs, the wetting layer is flat, therefore this term is neglected. As a result, the elastic strain is expected to be the determining factor for the growth process. This parameter will be calculated in this work using FEM based in the APT data.zz) calculated by FEM corresponding to the area of the APT data in the model of Figure ϵzz but with a different scale in order to appreciate variations in strain. As it can be observed, an area of the material strained in compression appears in the region above the QD, which means that the lattice is expanded in the growth plane. Because of the higher size of In atoms, they will be attached preferably to these areas with higher lattice parameter; therefore, it is expected that the next QD will grow in this position. In Figure Figure x and y directions (ϵxx and ϵyy), which are two perpendicular axes contained in the growth plane, (c) shows ϵzz, and (d) shows the normalized SED. In order to compare the predictions calculated by FEM with the experimental results obtained by APT, superimposed to these strain maps, we have included the APT data corresponding to the upper layer of QDs in the form of In concentration isolines, ranging from 25% In (dark blue) to 45% In (red), in steps of 5%. Also, in (d), we have included an inset showing a complete map of the APT data for clarity. As it can be observed in Figure ij as explained earlier, and which maximum value determines the most favoured localization of the QD[Figure f the QD. In thisf the QD,19. Thisf the QD) is therf the QD or kinetf the QD. These pf the QD. In the On the other hand, our results have shown that the upper QD does not grow vertically aligned with the lower QD, but there is some deviation. Previous theoretical analyses have shown that this misalignment is, in part, related to the elastic anisotropy in the material, where tIn conclusion, we have evaluated the accuracy of strain calculations by FEM using 3D atomic scale data obtained by APT for the prediction of the preferential nucleation sites of InAs stacked QDs. Our results by FEM have shown a very good agreement with our experimental observations, showing that this is a very useful tool for the analysis of the strain distribution in semiconductor systems. The combination of APT with FEM opens up the possibility of understanding the behaviour of complex semiconductor systems where strain plays a major role.APT: Atom probe tomography; FEM: Finite elements method; FIB: Focused ion beam; GaAs: Gallium arsenide; InAs: Indium arsenide; QDs: Quantum dots; SED: Strain energy density.The authors declare that they have no competing interests.JHS has participated in the design of the study, prepared the experimental specimens and carried out the APT analysis with SD, performed the FEM study, taken part in discussions and in the interpretation of the results, and written the manuscript. MH has participated in the FEM data analysis; she has supervised the research and revised the manuscript, and has taken part in discussions and in the interpretation of the results. SD has taken part in discussions and in the interpretation of the results. SIM has conceived the study; he has coordinated the work and the collaboration between groups, and he has participated in its design and supervised the manuscript. All the authors have read and approved the final manuscript.JHS is a PhD student at the Universidad de Cádiz. MH is an Associate Professor at the Departamento de Ciencia de los Materiales e Ingeniería Metalúrgica y Química Inorgánica, Universidad de Cádiz. SD holds an Associate Professor at Université et INSA de ROUEN and he is the responsible of the Matériaux de la Microélectronique et de la Photonique (ER2MP) group. SIM is a full professor at the Departamento de Ciencia de los Materiales e Ingeniería Metalúrgica y Química Inorgánica, Universidad de Cádiz and the head of the Materials and Nanotechnology for Innovation group (INNANOMAT). This group belongs to the Institute of Electron Microscopy and Materials (interim stage) of the University of Cádiz.


The paper presents a new model for kinetically controlled adsorption at the fluid/fluid interface. The main purpose of the presented approach is to relate easy to estimate bulk surfactant concentration with Gibbs surface excess. Two adsorption isotherms are involved in the new model development: Frumkin and Szyszkowski isotherms. Additionally the Johannsen time profile of concentration in the adsorption layer is assumed and estimated in the model derivation. The proposed approach assumes the near interface, adsorptive layer which is described based on Fick’s transient diffusion law. The solution to the model contains the estimation of effective diffusivities with adsorptive layer thickness as well. The experimental results of toluene/water + sodium dodecyl sulfate are presented and used for model verification. The work presents a new kinetic approach to describe the adsorptive layer. The fundamental concept of the proposed mathematical model is to distinguish and analyze the space, located in the vicinity of the interface, where mass transfer is dominated by adsorptive forces. The description proposed is based on the Fick’s second diffusion law of molecular mass transport where the typical molecular diffusion coefficients are expressed as local effective diffusivities. Such an approach is typical for many systems in which the underlying transport mechanism is not yet clearly explained as for mass transfer in liquids or solids, in porous media or during some processes like extraction, drying, and many others.A typical process in which the key phenomena takes place at the fluid interface is solvent extraction of metals using hydrophobic extractants. In order to estimate the rate of the process, dynamic surface tension experiments must be performed. This is achieved by forming a fresh interface, typically during the droplet creation, and analyzing the transient states of the surface tension. The kinetic interpretation divides the adsorption process into two parts. The first part concerns molecular diffusion from the bulk phase towards the interface. This process can be described by Fick’s laws or Maxwell–Stefan theory –5.The second part is the mass transfer through the adsorptive sublayer located near the interface. In this space the main driving force is adsorption. Due to the nature of mass transport near the interface, the process must be described by a different mathematical approach. In this case the typical description includes the Fick’s law of mass transfer with the assumption that the diffusion coefficient is not molecular but attains some specific effective value. The values of the effective diffusion coefficients can differ greatly from the molecular ones.Finally the adsorption itself is the mass exchange between interface and the sublayer space. The rate of the process can be thus limited by both phenomena, diffusion from the bulk to the sublayer and adsorption at the interface, or in the case of comparable rates, a diffusion-adsorptive process. In general, the adsorption step is faster than the diffusion so the kinetic description is typically limited to the diffusion process. But in the case of surfactants with a complicated structure, the diffusion process is not the only transport mechanism . The limJads is the adsorptive flux, Jdes is the desorptive flux and Jint is the source flux. The internal source flux results from the reorganization of adsorbed molecules at the interface. In the case where the total surface excess does not uniquely explain the state of the interface, two characteristic states are introduced into the process description [ω1 and ω2. The alteration and transition from one state to the other state is described by the so called two-state model. The resultant internal flux is proportional to the rate of reorientation described by the rate constant k12 and is given by Eq. The dynamics of the adsorption is described by three component fluxes given in Eq. cription , 8. Thesentclass1pt{minimaJdif given by Eq. The diffusion limited adsorption processes are described by several mathematical models, based on the Ward and Tordai approach . They asDef takes into account the fact that the concentration gradient is not the primary driving force in the sublayer space. Consequently it is valid only in the vicinity of the interface and attains a different value than that for the bulk phase. Applying Fick’s second law to Eq. cs(τ) and the initial concentration c0 in the sublayer, gives Eq. The description of the mass flux in the sublayer is based on Fick’s law where the molecular diffusion coefficient becomes effective. The effective diffusion coefficient cs(τ) function in order to solve it analytically. There exist several simplified approaches, e.g., Fainermann [kads/kdes dependent on the temperature according to the Arrhenius–Eyring law [∞ and the bulk concentration c given by Eq. The choice of isotherm that relates the surface excess with the bulk phase concentration is important. Equation inermann  proposedinermann . On the ring law . The kinSeveral other mixed-kinetic models are proposed where the process is limited by both the kinetics of adsorption and surfactant diffusion –23.The space where the adsorptive forces tend to dominate is considered as a layer located near the interface and is assumed to play a fundamental role in the adsorption process. It is typical during fluid flow to consider two separate regions which differs by a specified property or set of properties. The Prandtl concept  of boundDads. Because the layer is dominated by adsorptive forces, the classical correlations for bulk liquid phase diffusion coefficients are not suitable. In this case the diffusion coefficient becomes an effective diffusion coefficient for adsorption in the sublayer which holds the nature of the media, namely molecular diffusion with adsorptive driving forces. The concentration c is the surfactant concentration in the sublayer at position x in process time t. Figure Fick’s second law given by Eq. 6mentclass2pt{minimThe origin of the coordination system is fixed at the interface. The following boundary conditions are applied to the equation:x = L) it is assumed constant bulk concentration cbulk, by which the total bulk concentration is introduced for use in further modeling derivations.At the boundary of the adsorptive layer (x = 0) it is assumed that the total mass flux influences the Gibbs surface excess. Because Gibbs surface excess Γ is defined only at the interface location it is represented by an ordinary differential equation dependent only on t.At the interface  equal to cbulk which is the situation in the moment of contact of the phases.The initial condition for the boundary layer can be proposed in several ways. One possibility is to set the adsorptive layer specie concentration To find the general solution to Eq. x, the constant is chosen to give an exponential form.The left and right-hand sides of the equation are set equal to a specified parameter, which must be arbitrarily selected according to the required form of solution. Because diffusive mass transport of surfactant in the adsorption layer produces an exponential rise of concentration along its width ω is the reciprocal of area m−2. It should be mentioned that the possibility exists to choose other constants which gives a trigonometric form of integration result. Equation A and B that appear in the solution   8) to obsolution  gives:14A,By isolating constant solution  one of tx = 0 to  represents the general dependence of concentration on time t and  on distance x at the surface. To solve Eq. a and b are constants specific to the kinetics of the analyzed system. The Johannsen equation is applied for x = 0. Reformulated this way Eq. Equation  represeninit(cbulk). It refers to the typical situation when some initial state exists in the system which is at equilibrium with the bulk concentration and at specified initial moment when this bulk concentration is altered. This is mathematically more complex then applying trivial zero initial Gibbs surface excess, but allows us to track the dynamics of the surface excess in a more general way. Applying initial condition Γinit(cbulk) at t = 0 gives:At this point the initial state of the surface excess is assumed to be described by the function depending on bulk concentration Γx which in fact has no physical meaning except at x = 0 (interface location). Applying the interface condition of x = 0 simplifies  and refer to the adsorptive and desorptive fluxes when the interface condition shifts towards the equilibrium state.Equation sion Eq. , is consinit(cbulk) at t = 0 to Eq. cbulk at t = 0) used for sample preparation, is then disturbed by the change of the bulk surfactant concentration. There also exists the possibility to apply non-equilibrium initial condition, e.g., zero value of Gibbs surface excess which leads to a simpler derivation. At this point it is assumed some local equilibrium state exists and is described by typical isotherms used to describe liquid/liquid adsorption.The description of initial condition Γ∞ and adsorption equilibrium constant KL from the Langmuir isotherm reads:The well-known Szyszkowski isotherm which was derived from the Gibbs and Langmuir isotherms, assumes ideal behavior of the bulk surfactant component. That leads to a condition of dilute solution of the surface active specie in the bulk phase. The classical formulation  of the Sγ to the bulk concentration cbulk by the use of ASz and BSz Szyszkowski constants:The empirical formulation of the Szyszkowski equation , 30 relaThe Gibbs surface excess in terms of Szyszkowski constants is then presented by the relation :28\documinit gives the formulation for the time evolution of surface excess based on the Szyszkowski isotherm:Applying  as the iA′ attains negative values for repulsion and positive for attraction forces. The formulation of the Szyszkowski–Langmuir equation [The Langmuir isotherm is transformed to the Frumkin isotherm by considering the lateral interactions between adsorbed molecules. The additional interaction parameter equation  in the sAFr, BFr and A′.φ is the energy of interaction between one pair of adsorbed molecules, and the term Nφ is the interaction of one molecule with its N nearest neighbors in the totally covered surface [The Frumkin empirical equation relates surface tension to bulk concentration incorporating three constants init to /toluene system was selected as a test of the proposed model. Sodium dodecyl sulfate was obtained from Sigma-Aldrich with a purity of 99 % and used without further purification. SDS is water soluble and does not transfer to organic phase. The additional reason to use SDS is fact that its surface properties are very well known thus it is a good test system for the presented model. All the surfactants solutions were prepared by using water from the PURELAB Classic, Elga with a resistivity of 18.2 MΩ cm. Toluene  was distilled before use. Samples were shaken for 4 h to assure the equilibrium state at the interface. The measurements were carried out at 25 °C.The calculation of isotherms parameters is straightforward and is done using a least squares algorithm. The calculated isotherms’ parameters are needed for further calculation and for the comparison with the proposed model. The calculated data sets Table  are in gDads in adsorption layer, its size L, and parameters a and b from the Johannsen Eq. 34) for tnsen Eq. . Additioe) dynamic Gibbs surface excess values Γe and equivalent bulk phase concentrations ce. The procedure consists of several steps which are presented in Fig. ASz, BSz or AFr, A′, BFr) are estimated because they are needed for further calculations. In the sense of the proposed method, this is an initial approximation of their values that will be improved in the next steps. The calculation is based on the equilibrium surface tension data γ(cbulk). Estimation of isotherms parameters is easily done by the typical least squares method which is used to adjust the fitting line to the experimental data. For that purpose several numerical tools can be used. The author used genfit from Mathcad [Modeling of the experimental measurements requires estimation of the experimental  values give important information on the correctness of estimated isotherms and the trend of changes of surface excess due to surfactant bulk concentration.In the second step, which is in fact optional, when the isotherms parameters are given, the surface excess is estimated by Eqs. ce and corresponding equivalent surface excess Γe. Those parameters are calculated on the basis of dynamic experimental measurements of surface tension γdyn(t) by the use of the proper equations. In the case of modeling with Szyszkowski isotherm the equation reads:The third step is based on the assumption that for the transient state, when the adsorption process has not reached a steady state condition, the isotherm parameters can be used for further calculation. In fact these parameters describe the equilibrium state of the adsorption process and it is assumed that for specified transient state there is some equivalent equilibrium state. In the view of this statement, new equivalent parameters are introduced: the equivalent concentration of surfactant in the bulk phase ce calculation can only be done by the use of a numerical method because Eq. c. This issue is in fact straightforward and typical numerical methods for nonlinear problems can be used. The tool from Mathcad (root function or given/find block), Maple (fsolve procedure) or numerical libraries package IMSL (subroutine ZBREN [In the case of Frumkin isotherm the ne ZBREN ) can be e. For every time measurement the calculation is as follows:for Szyszkowski isothermfor Frumkin isothermThe estimated concentrations from the dynamic measurements are used in the fourth step to calculate the equivalent dynamic surface excess Γe versus time t and concentration ce are obtained to be used in Eqs. In this way, based on experimental results, the calculated values of ΓThe experimental verification was done using the liquid/liquid system water and toluene. The experimental setup is shown in Fig. The software managing the measurement sequence controls the flow rate of liquid from the syringe 4 with a motor driven piston 5, recording at the same time the image obtained from the camera 6. Proper recording is possible by using a uniform light source 2. Removable basis 1 allows precise setting of the field of view and focus at the area of the recorded image of the drop shape. During the measurement, software records the volume of the liquid droplet and its shape. On the basis of the droplet size, the software calculates the surface tension, contact angle, the droplet radius, the droplet surface area and its volume. The camera performs measurements for a drop hanging from normal and inverted vertical capillary or for a droplet lying on a selected surface.−4 N/m the system reached the equilibrium state [In order to prepare two phase mixtures, aqueous solutions of appropriate SDS concentration were shaken with toluene for 4 h to achieve a state of saturation while maintaining equal volumes of the phases. The flasks were allowed to stand for phase separation for 24 h. Several equilibrium measurements with altered surfactant concentration were done which are necessary to determine the adsorption isotherms. Also the dynamic measurements of surface tension changes in time were performed. Injecting the SDS solution droplet from the capillary creates a fresh interface between the saturated phases. The subsequent observation of transient changes in the shape of a droplet gives a picture of changes in interfacial tension, and conducting the measurement until the changes are very small, determines the equilibrium of adsorption. It is assumed that when the changes are less than 10um state . The chaγ∞ and parameters of chosen isotherms using Eqs. γ∞ and paa, b and c parameters, the equilibrium surface tension is calculated in the limit when time t approaches infinity using Eq. Using the values of a, b and c parameters are only of statistical significance and do not represent any physical quantity. To calculate their values two different tools were chosen: Mathcad using the genfit procedure and Maple using the NonlinearFit procedure. Both gave different result but the difference is not relevant. The residual values (estimated fit error) for both tools are less than 10−6 (8.71 × 10−7 and 6.72 × 10−7 respectively). Maple software was used for the model calculation due to its analytical capabilities. The parameter values are presented in Table R2 determination parameter.Equation The parameter values of the best fit isotherm model are given in Table ce is obtained using Eq. ce as a function of interfacial tension for a given sample set is presented on Fig. Having estimated the isotherms parameters, the equivalent surfactant concentration e for the equilibrium state. The model assumes that the isotherms are used to estimate the transient states of surface excess approaching equilibrium. Determination of the parameters for the proposed model including diffusion coefficient in the adsorption sublayer, the thickness of this layer, and numerical fine-tuning of the isotherm parameters is realized by fitting the model function to the experimental data. The model parameters were calculated using the minimization procedure LSSolve contained in the NAG [e for a given time t and equivalent bulk concentration ce.The next step is to estimate the equivalent dynamic surface excess Γ the NAG  numericae is the experimental equivalent surface excess and Γ is the model function given by Eq. ω > 0Dads > 0L > 0The minimization problem is stated as follows:Dads and size L are reasonable physiochemically. The ω parameter bound results from the mathematical structure of the model and is applied to obtain only real value solutions.The bounds on the diffusion coefficient e corresponding to the experimental values of surface excess. They are calculated for Szyszkowski isotherm using Eq.  system. In the initial stage of the process, for a very short time, the sudden change in the character of the profile is revealed. The typical, exponentially decaying diffusion profile changes into a profile dominated by the adsorption at the very initial stage of the process. The character of the concentration profile becomes linear for a longer time of the adsorption. The flux of the surfactant to the interface in terms of diffusion law is estimated by the Eq. . In order to obtain such a result the relative difference Δ was calculated:The concentration profile shown presents the dynamics of evolution of the surfactant in the adsorptive sublayer in the  and initial time t = 0. The difference is not significant due to the very short time and is presented to accent its existence. For longer time values the difference approaches zero.The differences between both approaches are small and reach their maximum at about six percent of the relative value for i is defined by the Eq. (The last result that is useful for computation in different fields of simulation is the surfactant interface mass flux. During the process of developing the model one of the steps is the formulation of the mass flux of the surfactant to the interface. The formulation is given by Eq. . The mas the Eq. :48\documx = 0 presents the interface. Consequently the flux is formulated as a mass diffusive flux from the bulk phase to the interface. The flux reaches a steady value for long times at about −2.24 × 10−8 mol/m2 s as shown in Fig. The obtained time profile of adsorptive mass flux gives negative values due to the formulation of the coordinate system in which the location at The work presented includes the mathematical model of the adsorption process at the interface. The model was developed in two distinct variants using Szyszkowski and Frumkin isotherms. The model was developed based on the Fick’s second law for transient diffusion. To apply this approach to the adsorptive sublayer where the concentration gradient is not the primary driving force for mass transfer, the effective diffusion coefficient was introduced. Such an approach is widely used in the literature –5 and alThe specific subspace, called the adsorptive sublayer, was distinguished and characterized by the model equation with boundary and initial conditions applied. It is assumed that surface excess is altered by the mass transfer of the surfactant from the sublayer to the interface. The mass transfer itself in the sublayer is dependent also on the bulk surfactant concentration. The model was solved by an analytical method to formulate the solution in the most general way.−9 m2/s and correspond to the literature data. The calculated size of the sublayer is of the order 10−7 m. From the mathematical assumption applied and analogy to the boundary layer theory such size is considered to be reasonable.The model was verified using experimental results for the water solution of SDS with the organic phase as toluene. The estimated diffusion coefficients are of the order 10


FST adjustment. To investigate the degree of conservativeness of this approach, we performed extensive simulations of one- and two-contributor crime scene profiles, in the latter case with, and without, the profile of the second contributor available for the analysis. The genotypes were simulated using five population databases, which were also available for the analysis, and evaluations of WoE using our heuristic rule were compared with several alternative calculations using different databases. Using FST = 0.03, we found that our heuristic gave WoE more favourable to the defence than alternative calculations in well over 99% of the comparisons we considered; on average the difference in WoE was just under 0.2 bans (orders of magnitude) per locus. The degree of conservativeness of the heuristic rule can be adjusted through the FST value. We propose the use of this heuristic for DNA profile WoE calculations, due to its ease of implementation, and efficient use of the evidence while allowing a flexible degree of conservativeness.When evaluating the weight of evidence (WoE) for an individual to be a contributor to a DNA sample, an allele frequency database is required. The allele frequencies are needed to inform about genotype probabilities for unknown contributors of DNA to the sample. Typically databases are available from several populations, and a common practice is to evaluate the WoE using each available database for each unknown contributor. Often the most conservative WoE (most favourable to the defence) is the one reported to the court. However the number of human populations that could be considered is essentially unlimited and the number of contributors to a sample can be large, making it impractical to perform every possible WoE calculation, particularly for complex crime scene profiles. We propose instead the use of only the database that best matches the ancestry of the queried contributor, together with a substantial  •A heuristic rule of assuming the database of Q for all unprofiled individuals in a CSP is proposed.•We simulate a total of 105 000 one- and two-contributor CSPs with no dropin or dropout.•The heuristic rule is conservative compared to an alternative for the majority of simulated CSPs.•We suggest that the use of this heuristic will allow for evaluation of complex cases with many possible databases. The choice of database can have an important impact on weight of evidence (WoE): the rarer an allele the stronger the evidence implicating a queried contributor (Q) if he has that allele and it is observed in the crime scene profile (CSP). The most appropriate population is the one that best matches the ancestry of X, the true source of the DNA. Under the prosecution case X is assumed to be Q, but under the defence case there is often little or no information about the ancestry of X. Many authors have noted that the database most appropriate for Q is not necessarily most appropriate for X In current forensic practice, when the ancestry of X is unknown, it is common to consider multiple population databases and choose the one that generates the lowest WoE. There should be no requirement to favour defendants in this way. Suppose for example that Q is Caucasian but it is discovered that the lowest WoE is obtained using a database of Vietnamese individuals. If the population local to the crime includes few Vietnamese and there is no evidence to suggest that a Vietnamese person was the source of the DNA, it may not be helpful to the court to report the WoE arising from the Vietnamese database. Similarly, the world's population can be categorised in a vast number of different ways, and it is not possible to investigate them all in order to report the smallest WoE. However, a forensic expert should make reasonable allowance for the different possible ancestries of X, given the available knowledge about the location and nature of the crime. It can be expedient to make approximations that favour the defence in order to permit simplified analyses while avoiding courtroom challenges. Here we propose a heuristic for WoE analysis that involves only one calculation, using the database most appropriate for Q. We show that our heuristic tends to strongly favour defences compared with a range of alternative calculations.FST adjustment may be required FST estimates appropriate for forensic use FST = 0.03 is sufficiently large to be almost always conservative. The effect of the FST adjustment is to increase the probability assigned to the alleles of Q, and consequently decrease the probability for other alleles. Although the rationale for an FST adjustment is to allow for the possibility that X has ancestry similar to that of Q, we illustrate below that for FST = 0.03 our heuristic calculation is conservative even if X could have come from one of several different populations. It is for this reason that our heuristic uses the same value of FST whatever the population of Q, even though within-population FST values differ across populations.For a one-contributor CSP when there are only, say, five population databases, it is usually easy to compute the WoE for each database and choose the one most favourable to the defence. However, for mixed profiles, the computational effort to consider multiple databases for each unprofiled contributor can be substantial. Thus our heuristic that computes the WoE only using the database of Q would be attractive, provided that it can be established to be conservative (favourable to the defence). If X is from the same population as Q then it becomes relevant to consider that they may also come from the same subpopulation, in which case an FST = 0.03. The FST adjustment under our heuristic only increases the population allele fraction for the alleles of Q, which is helpful to defences because it increases the probability that X or U share alleles with Q, thus increasing the support for the defence explanation of the observed CSP.A similar argument applies to other contributors to a mixed CSP. Consider a two-contributor profile, one of the contributors being X, who is alleged to be Q. If the reference profile of the other contributor is known, as is often the case for a victim or bystander, there are no probabilities to assess for the alleles of that individual and so the question of the appropriate population database is essentially the same as for the one-contributor case. When the reference profile of the other contributor, say U, is unavailable, then we show that it is conservative to use for both X and U the database best matching the ancestry of Q, again with FST, and show that our heuristic remains conservative compared to the WoE from the four alternative databases.It is not feasible or desirable to guarantee that a proposed WoE calculation is more favourable to the defence than any conceivable alternative calculation. We perform simulation experiments which show that for UK population databases our heuristic WoE calculation is, with probability ≫ 0.99, more favourable to defendants than a range of reasonable alternative calculations. We first simulate single-contributor CSPs matching the reference profile of the alleged contributor Q. Then the WoE for Q to be a contributor is calculated using the correct database (that used for the simulation) and is compared with the smallest WoE calculated using in turn four other databases. We repeated this exercise for one database using allele fractions that differ from the database values according to each of three values of FST = 0.03) with (a) the correct assignment of databases, (b) the minimum WoE using each of the four alternative databases for both X and U, and (c) the minimum WoE over the four databases for X, always using the correct database for U. In all our calculations, an adjustment using FST = 0.03 is applied to the alleles of Q when the database of Q is used for X.We then simulate two-contributor CSPs using all possible choices of two databases from the five available, and compare the WoE computed using the database of Q for both contributors , African and African Caribbean (IC3), South Asian (IC4), East Asian (IC5) and Middle Eastern (IC6) . For furlog10(likelihood ratio)) comparing a hypothesis that includes Q as a contributor with an alternative in which Q is replaced by X, assumed to be unrelated to Q. We implement FST adjustment FST = 0.03, and otherwise FST = 0. In all calculations one was added to the database count for each allele of Q, introducing a bias against understating the frequencies of rare alleles The WoE is computed using the likelihood ratio framework 2.2FST = 0.03), so that a negative result indicates that it is favourable to the defence to report our heuristic WoE irrespective of the ancestry of X.Initially a series of 10 000 one-contributor CSPs were simulated, using in turn allele fractions from each of the five population databases . The WoE for each simulated CSP was calculated five times, each time comparing hypotheses of the form:FST values . Ten thousand CSPs were simulated for each FST value . The hypotheses compared were the same as above, and our heuristic was again applied (using the IC1 database) from which was subtracted the minimum WoE using each of the four other databases.A second set of one-contributor analyses was conducted to investigate the effect of Q having an ancestry that differs from all of the available databases. Simulations were based on the IC1 database but with allele fractions differing from the IC1 values according to three Next, 25 sets of 1000 two-contributor profiles were created, one for each choice of databases for the two contributors. The hypotheses compared were of the form:We then performed a series of analyses based on the same simulations but now assuming that the uncontested contributor to the two-contributor profiles was unknown, and so the hypotheses compared were of the form:For each dataset we computed the WoE using our heuristic with three alternative WoE calculations.FST adjustment increases the probability for U to share alleles with Q. The second alternative, applicable in all 25 datasets, uses the lowest WoE obtained over all possible databases for X, using the correct database for U. The third alternative WoE calculation, also applied in all 25 datasets, uses the lowest WoE over the four alternative databases, the same for both X and U.The first alternative WoE calculation used the correct database for each of X and U, which differs from our heuristic in the 20 datasets with Q and U simulated from different databases. This alternative may be regarded as the most appropriate WoE, while our heuristic WoE is biased in favour of the defence because the 3FST = 0.01, 0.02 and 0.03, we found that the number of comparisons that are not conservative is at most 3 out of 10 000, which is as expected higher than when Q is simulated directly from the IC1 database (0 non-conservative out of 10 000) but the difference is small and not significant.In one-contributor tests, our heuristic gives, with probability > 0.999, a lower WoE than any of the four alternative calculations . There wIncluding a known contributor reduces the WoE for both our heuristic and the minimum of the four alternatives by about 3 bans  = 0.903, and so the LR is usually but not always in excess of one billion.When the additional contributor is unknown (U rather than K), the fraction of simulations in which our heuristic was conservative compared with using the correct databases for each of X and U was on average 0.997, and at least 0.993 over the 20 choices of databases for X and U . The reaP > 0.995) less than the minimum value over the four alternative choices of database for X, with U always assigned the correct database. Finally, P > 0.996) to use our heuristic.4FST = 0.03 when the database of Q was used, and FST = 0 otherwise. The FST adjustment increases the population probabilities for the alleles of Q, but not other alleles observed in the CSP nor any other available reference profiles.We have shown that for a one-contributor setting, our heuristic WoE calculation that uses only the database of the queried contributor Q is almost always conservative (favours defences) compared with choosing the lowest WoE among four other databases for the alternative contributor X . SimilarIn all our simulations, our heuristic is conservative compared with the alternative calculations considered in at least 99.3% of the simulations, and in the few instances that it was not conservative the difference was always < 1.5 bans. The world's population can be divided into an unlimited number of different subpopulations; therefore there can be no precisely correct choice of alternative subpopulations to consider. What is required is an average WoE over each possibility for the ancestry of the alternative contributor X, weighted by its plausibility given the known circumstances of the crime. Our heuristic will almost certainly give a result that is more favourable to defendants. We have verified that our good results are not favourably biased because Q is sampled from the same database used in the analysis.FST from 0.03 used here. FST = 0.03 exceeds almost all median FST values from world-wide comparisons of subpopulations with continental-scale populations, and we have shown here that it also suffices to ensure that using the database of Q with this FST value almost always returns a lower WoE than a range of alternative calculations.The degree to which our heuristic favours defences can be controlled by changing the value of We have not performed simulations for three or more unknown contributors because of the prohibitive time required for a simulation study, but the same principles apply to ensure that our heuristic will be similarly conservative. The large average reduction in WoE compared with the two-contributor case suggests that the difference in WoE will also be reduced, although it is expected to increase as a fraction of the overall WoE see .ST is the main factor in ensuring that our heuristic tends strongly to favour defendants, and so while misassigning the database of Q will have some detrimental impact, it will usually be small and outweighed by the impact of the FST value.We believe that our heuristic offers a good policy for WoE calculation based on DNA evidence that is easy to implement, and almost always favourable to defendants relative to reasonable alternative policies. Because it is favourable to defendants to use the database most appropriate for Q, it will therefore generally be unfavourable to defendants if the wrong database is used because the ancestry of Q is misassigned, or because there is no appropriate database. However the relatively large value of F


Bombus terrestris): The system was used both for training of the bees and for the experimental data collection. The robotic system was reliable, with no flight in our studies failing due to a technical malfunction. The data recorded were easy to apply for further analysis. The software and the hardware design are open source. The development of cheap open‐source prototyping platforms during the recent years has opened up many possibilities in designing of experiments. Automatization not only reduces workload, but also potentially allows experimental designs never done before, such as dynamic experiments, where the system responds to, for example, learning of the animal. We present a complete system with hardware and software, and it can be used as such in various experiments requiring feeders and collection of visitation data. Use of the system is not limited to any particular experimental setup or even species.Human observations during behavioral studies are expensive, time‐consuming, and error prone. For this reason, automatization of experiments is highly desirable, as it reduces the risk of human errors and workload. The robotic system we developed is simple and cheap to build and handles feeding and data collection automatically. The system was built using mostly off‐the‐shelf components and has a novel feeding mechanism that uses servos to perform refill operations. We used the robotic system in two separate behavioral studies with bumblebees ( Use of artificial flowers in pollination experiments has many advantages compared with field studies, including a comprehensive control of the environment. The easiest way of conducting such a laboratory experiment is to establish a group of artificial flowers in an experiment arena and refilling their nectar reservoirs by hand, and then making observations visually , a common model species in behavioral and pollination experiments, its hardware parts can be scaled up or down to suit other species.Here, we describe an artificial flower system that is fully automatic in terms of its refill operation and data collection and that uses a novel feeding mechanism to deliver discrete doses of sugar solution. The system we designed is highly reliable, relatively easy and cheap to construct, and uses mostly off‐the‐shelf components. Although the original system was developed for bumblebees  low‐level handling of the electronics of each flower, (2) collection of visitation data from the sensors, and (3) interpretation of the commands from the computer.http://www.arduino.cc), which is based on ATmega2560 processor chip by Atmel . The microcontroller is connected by ribbon cables to two separate connector cards which contain components needed to connect the flowers. The microcontroller is powered by USB connection to the computer, whereas flowers receive electricity from a separate power supply via connector card. Arduino Mega 2560 can simultaneously use up to 32 flowers.The heart of the control unit is Arduino Mega 2560 single‐board microcontroller platform  or even more, depending upon the type of servos used. As the microcontroller platform is powered by computer's USB port, its maximum current output is limited to 500 mA, which is not enough for all flowers, and therefore, a separate power supply unit is needed.The program running on the microcontroller (hereafter referred to as the “firmware”) handles the electronics in the system: It reads infrared sensors at specific intervals and generates correct electrical pulses to drive the servomotors. The firmware is operated by sending specific commands to it via the USB cable, which describe some simple operations, such as “refill flower #8”. The control unit sends back data collected from the sensors.Infrared sensors are constantly read at 10 Hz frequency . The control unit uses the following algorithm to determine the time and length of a visit: When an animal enters a flower, and sensor detects it, system time (seconds elapsed since the start of the experiment) is recorded. Simultaneously, a timer is started; this timer value is incremented at 100 ms intervals and zeroed after it reaches the value 10, that is, timer resets once per second. Positive readings  from IR sensor are counted during this 1‐sec time period, and if detection count is greater than a certain preset value , a variable containing the visit length is incremented by one. Otherwise, if detection count fails to reach this preset value, visit to flower is determined to be ended. The minimum length of a visit is 1‐sec; in reality, the actual visit may be shorter than this , but the algorithm rounds value up to the next full second.This algorithm was designed because bumblebee behavior may not be consistent during every visit; instead of going straight into the flower, for example, the animal may go in and back out quickly, and then go into the flower again. Without our algorithm, the system would detect each such erratic movement as a separate visit and interpreting the data would be much more difficult.The control unit requires a USB connection to a computer, as its behavior is determined by the commands it receives from the computer. This greatly simplifies the design of the hardware and firmware, because the user interface for the device can be a standard desktop software instead using a “stand‐alone” device with its own LCD display, command buttons, and data storage .μL. The cup is fixed to an end of a steel wire, which in turn is attached to servomotor's shaft via a servo horn . The empty weight of a single flower is 50 g without a cable.Each flower in the system is a self‐contained unit and operates under the control unit's command, independently of the other flowers. Power and data connection to a flower is provided via a four‐wire cable, which was 5–10 m long in our system. The flower itself has a nectar reservoir (approx. 300 mL volume), from which the nectar cup is refilled. In our system, the nectar cup was a small Phillips screw head, which had a volume of 1.7 When in its normal position, the arm is held horizontal to position the nectar cup directly below the feeding hole. During a refill operation, the servo arm turns 60° downwards and submerges the cup into the nectar reservoir to fill it; after this, the servo is returned to the normal position. The exact position of the nectar cup can be fine‐tuned simply by bending the steel wire arm by hand. The refill cycle takes 14 sec; while a faster rate of refill is technically possible, we found during testing that this led to a periodic formation of a small air bubble in the nectar cup we used, thereby preventing it from filling up completely.A lid covers the reservoir, and mounted on top of the lid is the servo and a small circuit board. A rectangular hole is cut to the lid, through which the servo arm operates. Nectar is available to the animal via a 3‐mm feeding hole directly above the nectar cup. A short plastic tube is vertically attached to the lid, so that the feeding hole is at the bottom of a “pit” where animal has to crawl. Mounted on the sides of the tube are an infrared LED and a phototransistor, which detect the feeding visit. Peak spectral wavelength of the IR LED is 940 nm, and the phototransistor was selected to spectrally match the LED. When the animal crawls into the sensor tube, it blocks the passage of the IR light beam, causing the phototransistor to shut off; this in turn is sensed by the control unit.On top of the lid is the top cover, which protects the circuit board and servo. Different kinds of symbols can be attached to the top cover to symbolize, for example, flower corolla. Another function of the top cover is to hide the moving parts of the refill mechanism (servo and servo arm) so that the animals cannot visually detect when the flower is refilling the nectar cup.The control software running on the computer is responsible for sending commands to the control unit and receiving flower visitation data from it. The software was written in Java programming language and works stably at least on Windows XP and Windows 7 operating systems.Before starting an experiment, the system is configured using the control software. Each flower's refill rate can be individually configured, and if needed, the refill mechanism and/or sensor of specific flowers can be disabled. The refilling rate can be fixed  or it can include random variance (within an adjustable range). Once an experiment is started, the software controls the experimental setup automatically, refilling flowers according to the predefined configuration, and gathering sensor data from them. When the user decides to stop the experiment (via the control software), the sensor data are formatted and saved automatically as a Microsoft Excel file.When an experiment is not running, the flowers can be commanded to refill themselves manually. The servo arms can also be stopped and held in downward position, a feature that proved useful in some maintenance operations (such as cleaning of the nectar cups).We also included a “deception mode” feature in the control software and firmware: In this mode, the servo arm is not brought to a fully upward position, so that the nectar cup is below the feeding hole but still inaccessible to the animal. The purpose of this feature is to simulate deceptive flowers, such as some orchids, which do not produce nectar. Independent flowers can be configured to be in deception mode; they respond to refill commands such as flowers in “normal” mode, but keep the nectar cup out of the reach of the animal.See Table We used this robotic system in two separate behavioral studies during 2014 and 2015 . A training phase was arranged in a separate nest cage with five flowers and an experiment phase conducted in a flight arena that was connected to the nest cage via a gated tube. The flight arena had 27 flowers, and the same system operated flowers in both cages simultaneously. Bumblebees readily visited the flowers. Based on our observations, the bumblebees were unable to detect the refill operations before actually probing the feeding hole.System maintenance consisted of periodic cleaning of the flowers and exchanging of old nectar with fresh. Some of the servos in nest cage flowers had occasional malfunctions after long continuous use  and were replaced. Servos failed because their internal mechanical parts wore out, which probably could be avoided by purchasing higher quality servos. However, not a single experiment flight in either of our studies failed because of these malfunctions, so overall reliability of the system can be considered good. The servo replacement was also relatively quick and easy, requiring about ten minutes of work. This delay was avoided by building spare flowers.The servos used in this system were “micro” size class analog servos commonly used in remote‐controlled model aircrafts. These are nowadays widely available and cheap . With larger structural changes to the flowers, the system could possibly be used to feed butterflies and maybe even with hummingbirds and flower bats. With major hardware changes, the system could be used as a general feeder to many other types of animals, such as mice, dogs, and pigeons, as the only critical mechanical parts are the standardized servos, which are widely available in different sizes. The flowers in the current design are lightweight and compact (90 mm diameter), offering flexibility in experimental designs.Flowers can be scaled down by placing the parts  more tightly on the lid, and using a smaller diameter nectar container. We estimate that this can reduce the flower diameter by 2–3 cm. We used “micro” class servos, but there are smaller ones available (“submicro” and “nano” class), which could be used to reduce the diameter even further (about 0.5–1 cm). The height of the flower  is mostly limited by the length of the servo arm, which has to have enough vertical room to turn to the downward refill position. If the diameter is made smaller, the height can also be lower, as the servo arm is shorter in that case. The circuit boards of the flowers can be made more compact by fabricating them instead of using stripboard. The circuit board can actually be omitted altogether, by soldering components directly to the wires  if a very small flower is desired. Also note that the flower does not have to be round as in our case, as the only factors limiting the geometry of the nectar container are the height and width in the servo arm's direction, which has to have enough space to allow the refill movements. Scaling up is more straightforward, simply using larger parts and if necessary, more powerful servos.Both the firmware and control software are released under open‐source license and can be freely modified. The control software we provide here is flexible enough to be used in various experiments without modifications. However, modification of the source code allows the control program to be tailored to run any specific experiment.One interesting possibility for modification is to change the behavior of the program from predetermined to dynamic. In the current version, the program runs according to preset parameters throughout the experiment. However, the rate of refill operations of independent flowers could be adjusted dynamically in response to learning of the animal, based on visitation data from sensors. For example, the refill rate of a specific (type of) flower could be increased or decreased when it is visited more often than other (types of) flowers.We have developed a complete, affordable, and automatic robotic flower (feeding) system for pollination and behavioral experiments. Our system not only handles flower refill operations but also data collection and storage, in a user‐friendly way. All software and hardware design are open source and freely modifiable.The development of low‐cost open‐source microcontroller prototyping platforms in recent years made this system possible. We encourage other researchers to explore the possibilities of this approach.None declared.Figure S1. An example of visitation data produced by our software, viewed in Microsoft Excel.Click here for additional data file.Figure S2. Circuit diagram of a single flower.Click here for additional data file.Figure S3. Circuit diagram of the connector cards of the control unit.Click here for additional data file.Figure S4. Use of the system.Click here for additional data file.Figure S5. Movement range of the servo arm: in refill position (left) and in normal feeding position (right).Click here for additional data file.Figure S6. Flower with the top cover removed.Click here for additional data file.Figure S7. Control unit.Click here for additional data file.Figure S8. Examples of individual flowers (with top cover in place) we used .Click here for additional data file.Appendix S1. ZIP package file containing control software, its source code, firmware's sources and user instructions.Click here for additional data file.


The blood flow through an overlapping clogged tapered artery in the presence of catheter is discussed. Since cholesterol deposition is resulting in the stenosis formation, velocity slip at the arterial wall is considered. The equations governing the fluid flow have been solved analytically under the assumption of the mild stenosis. The analysis with respect to various parameters arising out of fluid and geometry considered, on physiological parameters such as impedance and wall shear stress at the maximum height of the stenosis as well as across the entire length of the stenosis has been reported. A table summarizing the locations of extreme heights and the corresponding annular radii is provided. It is observed that the wall shear stress is the same at both the locations corresponding to the maximum height of the stenosis in case of nontapered artery while it varies in case of tapered artery. It is also observed that slip velocity and diverging tapered artery facilitate the fluid flow. Shear stress at the wall is increasing as micropolar parameter is decreasing and the trend is reversed in case of coupling number. The results obtained are validated by comparing them with the experimental and theoretical results. The cardiovascular system permits blood and bodily fluid to transport nutrients, gas, and hormones to any or all elements of the body and collects the waste product, deoxygenated blood from all elements of the body, thus enabling all organs of the body to work efficiently. In this world majority of the deaths are associated with vessel diseases. Among them maximum deaths are related to the characteristics of blood flow and vessel moments. In particular the circulatory connected issues are the major causes of health problems and deaths in the present world. This is supported by the report of the World Health Organization (WHO), according to which, seventeen million deaths in 2008 are associated with the heart. The basic cause of heart related disorders is due to occlusion, which compromise the functioning of the vital organs. Most notably, the heart and brain are often affected, as happens in MI (destruction of heart tissue resulting from obstruction of the blood supply to the heart muscle) and stroke.The deposition of excessive fatty components and abnormal intravascular growth within the lumen of the artery results in the formation of the stenosis. This results in the narrowing of the artery and this condition is known as atherosclerosis. Formation of the stenosis results in the obstruction of the flow of blood thus resulting in abnormal variations of the blood flow characteristics. Catheterization is a method in which a long skinny versatile tube is inserted into the stenotic artery to treat atherosclerosis. Insertion of such a tube  into the lumen of the artery results in significant changes in the blood flow, even in massive arteries. Therefore, the study of catheter effects in physiological fluid flow through a stenotic artery is vital. Hence the mathematical understanding of such a scenario was taken up by many assorted researchers. In particular understanding of biological development of blood flow through stenosed arteries has become prominent and has played a leading role in the diagnosis and the treatment of heart related diseases.Huo and Fu  reportedAll the above mentioned studies modelled blood as a classical Newtonian fluid. Because of the suspension nature of the blood cells, human blood exhibits non-Newtonian behavior at low shear rates. Dash et al.  discusseAmong the several non-Newtonian fluids present, the micropolar fluid encompasses special importance as it exhibits microrotational inertia and microrotational effects which arise from the naive structure of the fluid. This micropolar fluid model can adequately represent fluids consisting of dipole elements and can support couple stress and body couple effects as well. The classical Navier-Stokes theory is incapable of predicting these new physical phenomena. Since Blood consists of dipole elements, it seems very appropriate to model blood using micropolar fluid. The angular velocity at any point in the extended Navier-Stokes equation as described by Hansen et al.  is givenaszewicz  and Erinaszewicz . Srinivaaszewicz  studied aszewicz  considerA lot of work has been done in relation to the development of sclerosis, wherein the majority of the studies, shape of the stenosis was considered to be either symmetric or asymmetric. But it is understood that the stenosis may be multiple in nature or develop in an irregular manner. Further, stenosis may be overlapping or composite in nature. Haldar  investigIn all the above studies usual no-slip condition at the arterial wall has been considered. However variety of studies of suspensions of blood flow, have established the likely presence of slip at the boundary. In particular Brunn  theoretiHence in this paper we intend to explore the effects of overlapping stenosis on the physiological parameters of the blood flow that is modelled using micropolar fluid through a catheterized tapered artery. Velocity slip at the wall is additionally thought of. The analysis is done analytically. This configuration and the related results could be very useful in the development of various clinical related technologies and equipments. It is also useful to the scientists engaged in the design and development of artificial organs.rc is fixed. r0 is the annular radius in the nonstenotic portion for nontapered artery. ζ( = tan(ϕ)) is the tapered parameter with ϕ being the tapering angle. ϕ < 0, ϕ > 0, and ϕ = 0 correspond to converging, diverging and nontapered artery. Here zL and zR correspond to the locations where maximum heights of the stenosis is occurring. zC indicates the location on z-axis corresponding to the critical height of the overlapping stenosis.The schematic diagram of the overlapping stenotic tapered artery in the presence of catheter is as shown in L1 is the length of the stenosis. ϵ is the maximum height of the stenosis and the locations of the maximum heights of the stenosis are dependent on ζ. The locations of the extreme heights of the stenosis are obtained as the roots of the cubic equationThe mathematical form of the geometry of the overlapping stenotic tapered artery is as described by , 19, whiP is the fluid pressure. j, ρ are the microinertia moment and density parameters, respectively, while f and l are body force and body moment, respectively. The material constants α, β, γ, μ, and κ satisfy the following inequalities:r, θ, z) with r = 0 as the axis of symmetry. Since the flow is assumed to be axisymmetric, all the flow variables are independent of θ. The velocity vector vr, νθ, and vz are functions of r and z.The balance equations which govern the flow of micropolar fluid areDρDt+ρ∇·Vu0 is the typical axial velocity.Under the above assumptions,  will getN = κ/(μ + κ) is the coupling number (0 ≤ N < 1), m2 = [r02(2μ + κ)κ]/[γ(μ + κ)] is the micropolar parameter, Re = (ρu0r0)/μ is the Reynolds number, and δ = r0/L1 and ξ = ϵ/r0.Introducing the above nondimensional variables into  and dropξ( = ϵ/r0) ≪ 1 and subject to the additional condition δ( = r0/L1) ≈ O(1), we find that ∂p/∂r ≪ ∂p/∂z and ∂νθ/∂z ≪ ∂νθ/∂r.Under the assumptions of mild stenosis; that is, γ = L0/L1.Hence  get redu0∂vz∂z=0,zL, zC, and zR for various values of tapered parameter are calculated, is given byzL, zC, and zR) and the corresponding radii of the annular region for various values of the tapered parameter ζ are numerically calculated with fixed values of L = 2,  L1 = L/2,  L0 = (L − L1)/2,  r0 = 1. The values are summarized in the form of The nondimensional form of , from whu is the velocity slip at the wall of the artery.The dimensionless boundary conditions are given byI1(mr), K1(mr), I0(mr), and K0(mr) are the modified Bessels functions of the first order and zeroth order of first and second kind, respectively. ci, i = 1,2, 3,4 are the constants which are evaluated by using the boundary conditions 2rvzdr in the following simplified formdi = ci/(∂p/∂z) and i = 1,2, 3 and 4.The nondimensional volumetric flow is obtained using p across the artery is obtained from  is computed asOne of the basic objectives of the problems related to physiological fluid dynamics is to predict shear stress at the wall in clogged arteries. This significantly influences the rate of mass transport across the artery walls and the formation of atheroma on the walls of the arteries as in atherosclerosis. The expression for shear stress is obtained fromnditions , the nonm (micropolar fluid parameter), N (coupling number), rc (radius of the catheter), ϵ (maximum height of the stenosis), and Γ, ϕ (tapering angle), and u (slip velocity). The results obtained here have been validated and the trend is found to be agreeing very well with the experimental and theoretical results reported in the literature.The objective of this analysis is to study the change in flow pattern and estimate the variation in flow resistance and wall shear stress in a narrow overlapping stenosed tapered artery when a catheter is inserted into it. In addition to the above, velocity slip at the boundary of the artery has been considered. Blood is modelled as steady incompressible micropolar fluid. The impact on the physiological parameters is obtained for different parameters arising out of the geometry and fluid considered, such as λ) per unit length of the artery is calculated using  of the fluid elements, couple stresses (γ) and the direct coupling of the microstructure to the velocity field (κ). All these fluid parameters can have any value greater than or equal to zero. Hence, the ratio of the couple stress to viscous effects, that is, γ/μ and microstrure coupling to the viscous effects, that is, κ/μ, can have any value greater than or equal to zero. This knowledge will help us to understand the physics behind the variation in the physiological parameters with respect to the associated fluid parameters. The influence on impedance for various values of micropolar parameter is shown in m is increasing. This is because, as m increases the coupling effects of the fluid decrease, thus resulting in less impedance. The effect of coupling number is depicted in N → 0 when the ratio of microstructure coupling to the viscous effect κ/μ → 0. Hence it can be technically understood that as the microstructure coupling effects corresponding to a fluid are increasing impedance is increasing. The effect of velocity slip on resistance to the flow is sketched in u = 0.01 and 0.02 are considered at the arterial wall. They correspond to 1% and 2% of the average velocity of the blood, respectively. Hence it can be concluded that even a small slip in the velocity has significant effect on resistance to the flow.It is to be noted that the motion of micropolar fluid may get affected by the viscous action  and height of the stenosis on wall shear stress in the stenotic portion is depicted in (i)zR. This was ignored by all the earlier authors to the best of our knowledge.The locations of the extremum heights are different for different tapering angles. It is observed that impedance and wall shear stress differ significantly at these locations. In particular, impedance is maximum for converging tapered artery at (ii)ϵ, Γ, N, and rc are increasing and the trend is reversed with respect to micropolar parameter and slip velocity.Impedance is increasing as the parameters (iii)ϕ and m.The variation of shear stress at the wall is directly proportional to the coupling number, radius of the catheter, and slip velocity while it is inversely proportional with respect to (iv)rc, N, and u while it increases exponentialy for higher values of these parameters.It is that shear stress is increasing linearly for lower values of the (v)The variation of shear stress across the entire length of the stenosis with respect to various parameters has also been studied. The results obtained are in predicted lines.(vi)The results obtained have been validated with that of Back  and the (vii)The impedance in case of overlapping stenosis is very high when compared to that of symmetric stenosis.The mathematical treatment of the above phenomena is very realistic and is expected to be very useful in predicting the behavior of physiological parameters in the diagnosis of various arterial diseases and in the development of artificial organs.It is very important to understand the dynamics of blood flow in human physiological system. In view of this, a mathematical model is developed for axisymmetric flow of blood through overlapping stenotic tapered artery with velocity slip at the wall. Under the assumption of mild stenosis, closed form expressions for the axial velocity, microrotation vector, impedance and shear stress at the wall are obtained under appropriate boundary conditions. Via the described procedure flow conditions accounting for the presence of various parameters arising out of the fluid and geometry are discussed mathematically. Impedance and wall shear stress are numerically evaluated using Mathematica and Matlab.