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This article proposes that the affective processes that shape our relationship to the world of digital consumption and communication can be illuminated further when viewed through a lens of object relations psychoanalysis. We focus on the use of the mobile phone as both an object in the world and of the psyche in order to reflect upon its uses as an evocative object that shapes the psychosocial boundaries of experience in everyday life. We argue that in contrast to the concepts of interpersonal communication that can be found in some domains of popular culture and in communication studies, object relations psychoanalysis can be usefully deployed in order to explore the unconscious attachments that develop in relation to consumer objects, allowing for the complexity of feeling and reflection that may emerge in relation to them and the potential spaces of the mind. The mobile phone's routine uses and characteristics are widely understood. At the same time, the mobile phone invites critical reflections that identify a paradoxical object of both creative and pathological use. Such reflexivity includes the mobile's relationship to the complexity of psychosocial experience within the contemporary cultural moment. Applying the ideas of psychoanalysts Donald Winnicott, Thomas Ogden and Christopher Bollas, we argue that one explanation for why the mobile phone continues to attract not only enthusiastic cultural commentary but also a degree of apprehension across academic and popular-discursive settings can be found in its capacity to both disrupt and connect as an object of attachment and as a means of unconscious escape.

Water quality monitoring at high temporal frequency provides a detailed picture of environmental stressors and ecosystem response, which is essential to protect and restore lake and river health. An effective monitoring network requires knowledge on optimal monitoring frequency and data variability. Here, high-frequency hydrochemical datasets (dissolved oxygen, pH, electrical conductivity, turbidity, water temperature, total reactive phosphorus, total phosphorus and nitrate) from six UK catchments were analysed to 1) understand the lowest measurement frequency needed to fully capture the variation in the datasets; and 2) investigate bias caused by sampling at different times of the day. The study found that reducing the measurement frequency increasingly changed the interpretation of the data by altering the calculated median and data range. From 45 individual parameter-catchment combinations (six to eight parameters in six catchments), four-hourly data captured most of the hourly range (>90 %) for 37 combinations, whilst 41 had limited impact on the median (<0.5 % change). Twelve-hourly and daily data captured >90 % of the range with limited impact on the median in approximately half of the combinations, whereas weekly and monthly data captured this in <6 combinations. Generally, reducing sampling frequency had most impact on the median for parameters showing strong diurnal cycles, whilst parameters showing rapid responses to extreme flow conditions had most impact on the range. Diurnal cycles resulted in year-round intra-daily variation in most of the parameters, apart from nutrient concentrations, where daily variation depended on both seasonal flow patterns and anthropogenic influences. To design an optimised monitoring programme, key catchment characteristics and required data resolution for the monitoring purpose should be considered. Ideally a pilot study with high-frequency monitoring, at least four-hourly, should be used to determine the minimum frequency regime needed to capture temporal behaviours in the intended focus water quality parameters by revealing their biogeochemical response patterns., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2024 The Authors. Published by Elsevier B.V. All rights reserved.)

Competing Interests: Competing interests: We have read and understood The BMJ policy on declaration of interests and have the following interests to declare: none.

We link continuum models of reaction-diffusion systems that exhibit diffusion-driven instability to constraints on the particle-scale interactions underpinning this instability. While innumerable biological, chemical and physical patterns have been studied through the lens of Alan Turing's reaction-diffusion pattern-forming mechanism, the connections between models of pattern formation and the nature of the particle interactions generating them have been relatively understudied in comparison with the substantial efforts that have been focused on understanding proposed continuum systems. To derive the necessary reactant combinations for the most parsimonious reaction schemes, we analyse the emergent continuum models in terms of possible generating elementary reaction schemes. This analysis results in the complete list of such schemes containing the fewest reactions; these are the simplest possible hypothetical mass-action models for a pattern-forming system of two interacting species.

Mathematical modellers model infectious disease dynamics at different scales. Within-host models represent the spread of pathogens inside an individual, whilst between-host models track transmission between individuals. However, pathogen dynamics at one scale affect those at another. This has led to the development of multiscale models that connect within-host and between-host dynamics. In this article, we systematically review the literature on multiscale infectious disease modelling according to PRISMA guidelines, dividing previously published models into five categories governing their methodological approaches (Garira (2017)), explaining their benefits and limitations. We provide a primer on developing multiscale models of infectious diseases., Competing Interests: Declaration of competing interest The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Ruth Bowness reports a relationship with Medical Research Council that includes: funding grants (grant number MR/P014704/1). Ruth Bowness reports a relationship with Academy of Medical Sciences (London) that includes: funding grants (grant number SBF003/1052). Ruth Bowness reports a relationship with Wellcome Trust that includes: funding grants (grant number SBF003/1052). Ruth Bowness reports a relationship with UK Government Department of Business, Energy and Industrial Strategy that includes: funding grants (grant number SBF003/1052). Ruth Bowness reports a relationship with British Heart Foundation that includes: funding grants (grant number SBF003/1052). Ruth Bowness reports a relationship with Global Challenges Research Fund that includes: funding grants (grant number SBF003/1052)., (Copyright © 2023 The Authors. Published by Elsevier B.V. All rights reserved.)

Embryonic mesenchymal cells are dispersed within an extracellular matrix but can coalesce to form condensates with key developmental roles. Cells within condensates undergo fate and morphological changes and induce cell fate changes in nearby epithelia to produce structures including hair follicles, feathers, or intestinal villi. Here, by imaging mouse and chicken embryonic skin, we find that mesenchymal cells undergo much of their dispersal in early interphase, in a stereotyped process of displacement driven by 3 hours of rapid and persistent migration followed by a long period of low motility. The cell division plane and the elevated migration speed and persistence of newly born mesenchymal cells are mechanosensitive, aligning with tissue tension, and are reliant on active WNT secretion. This behaviour disperses mesenchymal cells and allows daughters of recent divisions to travel long distances to enter dermal condensates, demonstrating an unanticipated effect of cell cycle subphase on core mesenchymal behaviour., Competing Interests: The authors have declared that no competing interests exist., (Copyright: © 2023 Riddell et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.)

Cell migration is frequently modeled using on-lattice agent-based models (ABMs) that employ the excluded volume interaction. However, cells are also capable of exhibiting more complex cell-cell interactions, such as adhesion, repulsion, pulling, pushing, and swapping. Although the first four of these have already been incorporated into mathematical models for cell migration, swapping has not been well studied in this context. In this paper, we develop an ABM for cell movement in which an active agent can "swap" its position with another agent in its neighborhood with a given swapping probability. We consider a two-species system for which we derive the corresponding macroscopic model and compare it with the average behavior of the ABM. We see good agreement between the ABM and the macroscopic density. We also analyze the movement of agents at an individual level in the single-species as well as two-species scenarios to quantify the effects of swapping on an agent's motility.

In this paper, we investigate a generalized model of N particles undergoing second-order nonlocal interactions on a lattice. Our results have applications across many research areas, including the modeling of migration, information dynamics, and Muller's ratchet-the irreversible accumulation of deleterious mutations in an evolving population. Strikingly, numerical simulations of the model are observed to deviate significantly from its mean-field approximation even for large population sizes. We show that the disagreement between deterministic and stochastic solutions stems from finite-size effects that change the propagation speed and cause the position of the wave to fluctuate. These effects are shown to decay anomalously as (lnN)^{-2} and (lnN)^{-3}, respectively-much slower than the usual N^{-1/2} factor. Our results suggest that the accumulation of deleterious mutations in a Muller's ratchet and the loss of awareness in a population may occur much faster than predicted by the corresponding deterministic models. The general applicability of our model suggests that this unexpected scaling could be important in a wide range of real-world applications.

Competing Interests: Competing interests: We have read and understood BMJ policy on declaration of interests and declare we are both members of Independent SAGE and have been active in helping explain mathematical models and their role in policy to the public during the pandemic.

Recent research has highlighted the importance of dissolved organic matter (DOM) for ecosystem function and because of this paradigm shift, it has become crucial to not only quantify its contribution to river nutrient loads but also to characterise its composition. There has been a significant research effort utilising optical methods, such as fluorescence and UV-Vis spectrophotometry, in order to start exploring DOM character. However, these methods still lack the granularity to understand the chemical composition at the molecular level, which is vital to properly understanding its functional role in freshwater ecosystems. As a direct result, there has been a shift towards including molecular-scale analyses to investigate the in-stream processing of the material. Alongside this, recent methodological advancements, particularly in mass spectrometry are opening new opportunities for probing one of the most complex environmental mixtures. However, in order to fully exploit these opportunities, it is key that the way that samples are collected, processed and stored is considered carefully such that sample integrity is maintained. There are additional challenges when collecting water samples for analysis at molecular scale, for example the ultra-low concentrations of individual compounds within DOM means that the samples are sensitive to contamination. This paper discusses current sample collection, processing and storage protocols for this C, N and P quantification and characterisation in freshwaters, and proposes a new standardised protocol suitable for both nutrient fraction quantification and molecular scale analyses, based on method development and testing undertaken in our UK Natural Environment Research Council large grant programme, characterising the nature, origins and ecological significance of Dissolved Organic Matter IN freshwater Ecosystems (DOMAINE)., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2022 Elsevier B.V. All rights reserved.)

The evolution of cooperation in cellular groups is threatened by lineages of cheaters that proliferate at the expense of the group. These cell lineages occur within microbial communities, and multicellular organisms in the form of tumours and cancer. In contrast to an earlier study, here we show how the evolution of pleiotropic genetic architectures-which link the expression of cooperative and private traits-can protect against cheater lineages and allow cooperation to evolve. We develop an age-structured model of cellular groups and show that cooperation breaks down more slowly within groups that tie expression to a private trait than in groups that do not. We then show that this results in group selection for pleiotropy, which strongly promotes cooperation by limiting the emergence of cheater lineages. These results predict that pleiotropy will rapidly evolve, so long as groups persist long enough for cheater lineages to threaten cooperation. Our results hold when pleiotropic links can be undermined by mutations, when pleiotropy is itself costly, and in mixed-genotype groups such as those that occur in microbes. Finally, we consider features of multicellular organisms-a germ line and delayed reproductive maturity-and show that pleiotropy is again predicted to be important for maintaining cooperation. The study of cancer in multicellular organisms provides the best evidence for pleiotropic constraints, where abberant cell proliferation is linked to apoptosis, senescence, and terminal differentiation. Alongside development from a single cell, we propose that the evolution of pleiotropic constraints has been critical for cooperation in many cellular groups., Competing Interests: The authors have declared that no competing interests exist.

We develop theoretical equivalences between stochastic and deterministic models for populations of individual cells stratified by age. Specifically, we develop a hierarchical system of equations describing the full dynamics of an age-structured multistage Markov process for approximating cell cycle time distributions. We further demonstrate that the resulting mean behavior is equivalent, over large timescales, to the classical McKendrick-von Foerster integropartial differential equation. We conclude by extending this framework to a spatial context, facilitating the modeling of traveling wave phenomena and cell-mediated pattern formation. More generally, this methodology may be extended to myriad reaction-diffusion processes for which the age of individuals is relevant to the dynamics.

The COVID-19 pandemic, caused by the coronavirus SARS-CoV-2, has led to a wide range of non-pharmaceutical interventions being implemented around the world to curb transmission. However, the economic and social costs of some of these measures, especially lockdowns, has been high. An alternative and widely discussed public health strategy for the COVID-19 pandemic would have been to 'shield' those most vulnerable to COVID-19 (minimising their contacts with others), while allowing infection to spread among lower risk individuals with the aim of reaching herd immunity. Here we retrospectively explore the effectiveness of this strategy using a stochastic SEIR framework, showing that even under the unrealistic assumption of perfect shielding, hospitals would have been rapidly overwhelmed with many avoidable deaths among lower risk individuals. Crucially, even a small (20%) reduction in the effectiveness of shielding would have likely led to a large increase (>150%) in the number of deaths compared to perfect shielding. Our findings demonstrate that shielding the vulnerable while allowing infections to spread among the wider population would not have been a viable public health strategy for COVID-19 and is unlikely to be effective for future pandemics., Competing Interests: The authors have no competing interests to declare., (Copyright: © 2022 Smith et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.)

The effectiveness of non-pharmaceutical interventions, such as mask-wearing and social distancing, as control measures for pandemic disease relies upon a conscientious and well-informed public who are aware of and prepared to follow advice. Unfortunately, public health messages can be undermined by competing misinformation and conspiracy theories, spread virally through communities that are already distrustful of expert opinion. In this article, we propose and analyse a simple model of the interaction between disease spread and awareness dynamics in a heterogeneous population composed of both trusting individuals who seek better quality information and will take precautionary measures, and distrusting individuals who reject better quality information and have overall riskier behaviour. We show that, as the density of the distrusting population increases, the model passes through a phase transition to a state in which major outbreaks cannot be suppressed. Our work highlights the urgent need for effective interventions to increase trust and inform the public.

Data are crucial for understanding and addressing the pandemic, but there are pitfalls.

Peatlands play an important role in modulating the climate, mainly through sequestration of carbon dioxide into peat carbon, which depends on the availability of reactive nitrogen (Nr) to mosses. Atmospheric Nr deposition in the UK has been above the critical load for functional and structural changes to peatland mosses, thus threatening to accelerate their succession by vascular plants and increasing the possibility of Nr export to downstream ecosystems. The N balance of peatlands has received comparatively little attention, mainly due to the difficulty in measuring gaseous N losses as well as the Nr inputs due to biological nitrogen fixation (BNF). In this study we have estimated the mean annual N balance of an ombrotrophic bog (Migneint, North Wales) by measuring in situ N 2 + N 2 O gaseous fluxes and also BNF in peat and mosses. Fluvial N export was monitored through a continuous record of DON flux, while atmospheric N deposition was modelled on a 5 × 5 km grid. The mean annual N mass balance was slightly positive (0.7 ± 4.1 kg N ha -1 y -1 ) and varied interannually indicating the fragile status of this bog ecosystem that has reached N saturation and is prone to becoming a net N source. Gaseous N losses were a major N output term accounting for 70% of the N inputs, mainly in the form of the inert N 2 gas, thus providing partial mitigation to the adverse effects of chronic Nr enrichment. BNF was suppressed by 69%, compared to rates in pristine bogs, but was still active, contributing ~2% of the N inputs. The long-term peat N storage rate (8.4 ± 0.8 kg N ha -1 y -1 ) cannot be met by the measured N mass balance, showing that the bog catchment is losing more N than it can store due its saturated status., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2021 The Author(s). Published by Elsevier B.V. All rights reserved.)

Understanding synchrony in growing populations is important for applications as diverse as epidemiology and cancer treatment. Recent experiments employing fluorescent reporters in melanoma cell lines have uncovered growing subpopulations exhibiting sustained oscillations, with nearby cells appearing to synchronize their cycles. In this study, we demonstrate that the behavior observed is consistent with long-lasting transient phenomenon initiated and amplified by the finite-sample effects and demographic noise. We present a novel mathematical analysis of a multistage model of cell growth, which accurately reproduces the synchronized oscillations. As part of the analysis, we elucidate the transient and asymptotic phases of the dynamics and derive an analytical formula to quantify the effect of demographic noise in the appearance of the oscillations. The implications of these findings are broad, such as providing insight into experimental protocols that are used to study the growth of asynchronous populations and, in particular, those investigations relating to anticancer drug discovery., (Copyright © 2021 Biophysical Society. Published by Elsevier Inc. All rights reserved.)

Reaction-diffusion mechanisms are a robust paradigm that can be used to represent many biological and physical phenomena over multiple spatial scales. Applications include intracellular dynamics, the migration of cells and the patterns formed by vegetation in semi-arid landscapes. Moreover, domain growth is an important process for embryonic growth and wound healing. There are many numerical modelling frameworks capable of simulating such systems on growing domains; however, each of these may be well suited to different spatial scales and particle numbers. Recently, spatially extended hybrid methods on static domains have been produced to bridge the gap between these different modelling paradigms in order to represent multi-scale phenomena. However, such methods have not been developed with domain growth in mind. In this paper, we develop three hybrid methods on growing domains, extending three of the prominent static-domain hybrid methods. We also provide detailed algorithms to allow others to employ them. We demonstrate that the methods are able to accurately model three representative reaction-diffusion systems accurately and without bias.

The simulation of stochastic reaction-diffusion systems using fine-grained representations can become computationally prohibitive when particle numbers become large. If particle numbers are sufficiently high then it may be possible to ignore stochastic fluctuations and use a more efficient coarse-grained simulation approach. Nevertheless, for multiscale systems which exhibit significant spatial variation in concentration, a coarse-grained approach may not be appropriate throughout the simulation domain. Such scenarios suggest a hybrid paradigm in which a computationally cheap, coarse-grained model is coupled to a more expensive, but more detailed fine-grained model, enabling the accurate simulation of the fine-scale dynamics at a reasonable computational cost. In this paper, in order to couple two representations of reaction-diffusion at distinct spatial scales, we allow them to overlap in a 'blending region'. Both modelling paradigms provide a valid representation of the particle density in this region. From one end of the blending region to the other, control of the implementation of diffusion is passed from one modelling paradigm to another through the use of complementary 'blending functions' which scale up or down the contribution of each model to the overall diffusion. We establish the reliability of our novel hybrid paradigm by demonstrating its simulation on four exemplar reaction-diffusion scenarios.

Pattern formation is a key aspect of development. Adult zebrafish exhibit a striking striped pattern generated through the self-organisation of three different chromatophores. Numerous investigations have revealed a multitude of individual cell-cell interactions important for this self-organisation, but it has remained unclear whether these known biological rules were sufficient to explain pattern formation. To test this, we present an individual-based mathematical model incorporating all the important cell-types and known interactions. The model qualitatively and quantitatively reproduces wild type and mutant pigment pattern development. We use it to resolve a number of outstanding biological uncertainties, including the roles of domain growth and the initial iridophore stripe, and to generate hypotheses about the functions of leopard . We conclude that our rule-set is sufficient to recapitulate wild-type and mutant patterns. Our work now leads the way for further in silico exploration of the developmental and evolutionary implications of this pigment patterning system., Competing Interests: JO, RK, CY No competing interests declared, (© 2020, Owen et al.)

The concentration of dissolved organic matter (DOM) in freshwaters is increasing in large areas of the world. In addition to carbon, DOM contains nitrogen and phosphorus and there is growing concern that these organic nutrients may be bioavailable and contribute to eutrophication. However, relatively few studies have assessed the potential for dissolved organic nitrogen (DON) or dissolved organic phosphorus (DOP) compounds to be bioavailable to natural river phytoplankton communities at different locations or times. Temporal and spatial variations in uptake, relative to environmental characteristics were examined at six riverine sites in two contrasting catchments in the UK. This study also examined how the uptake by riverine phytoplankton of four DON and four DOP compounds commonly found in rivers, varied with concentration. Total nitrogen (TN) and phosphorus (TP) concentrations, the proportion of inorganic nutrient species, and nutrient limitation varied temporally and spatially, as did the potential for DON and DOP uptake. All eight of the DOM compounds tested were bioavailable, but to different extents. Organic nutrient use depended on the concentration of the organic compound supplied, with simple compounds (urea and glucose-6-phosphate) supporting algal growth even at very low concentrations. DON use was negatively correlated with the TN and ammonia concentration and DOP use was negatively correlated with soluble reactive phosphorus (SRP) and dissolved organic carbon (DOC) concentration. The evidence indicates that DOM in rivers has been overlooked as a potential source of nutrients to phytoplankton and therefore as an agent of eutrophication., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2020 The Authors. Published by Elsevier B.V. All rights reserved.)

Faced with increasing demand for their services, Emergency Medical Services (EMS) agencies must find more efficient ways to use their limited resources. This includes moving beyond the traditional response and transport model. Alternative Response Units (ARUs) are one way to meet the prehospital medical needs of some members of the public while reducing ambulance transports. They are non-transport vehicles tasked with very specific medical missions. These can include acute management of low-acuity complaints, ongoing home care for chronic medical conditions, preventive medicine, and post-hospital discharge follow-up visits. Their focus can be tailored to the individual needs of the agency. The Philadelphia Fire Department (PFD) operates one of the busiest EMS systems in the country. It has added additional staff and ambulances in recent years yet continues to face an increasing call volume. In an effort to reduce ambulance transports, the PFD recently introduced two ARUs. The first unit, AR-1, is deployed on a university campus and responds to students with low acuity medical complaints or mild alcohol intoxication. It provides many of these a courtesy ride to one of two university emergency departments for further evaluation, eliminating the need for ambulance transport. The second unit, AR-2, works in an area heavily impacted by the opioid crisis. It responds to individuals who have overdosed, been revived with naloxone, and refuse ambulance transport but are interested in long-term treatment for their opioid use disorder. The staff of AR-2 has successfully placed some of these individuals in treatment programs the same day. The AR-1 program is financially supported by the university while AR-2 is funded by the PFD and a federal grant. Both have the potential to decrease ambulance transports or reduce 9-1-1 calls. Whether these or other ARU programs can be financially sustained long-term is unclear. It is also unknown if ARUs represent a better investment than using the money to purchase additional transport vehicles. However, as health care evolves, EMS must innovate and adapt so it can continue to meet the prehospital needs of the public in a timely and cost-effective manner.

Multi-scale epidemic forecasting models have been used to inform population-scale predictions with within-host models and/or infection data collected in longitudinal cohort studies. However, most multi-scale models are complex and require significant modelling expertise to run. We formulate an alternative multi-scale modelling framework using a compartmental model with multiple infected stages. In the large-compartment limit, our easy-to-use framework generates identical results compared to previous more complicated approaches. We apply our framework to the case study of influenza A in humans. By using a viral dynamics model to generate synthetic patient-level data, we explore the effects of limited and inaccurate patient data on the accuracy of population-scale forecasts. If infection data are collected daily, we find that a cohort of at least 40 patients is required for a mean population-scale forecasting error below 10%. Forecasting errors may be reduced by including more patients in future cohort studies or by increasing the frequency of observations for each patient. Our work, therefore, provides not only an accessible epidemiological modelling framework but also an insight into the data required for accurate forecasting using multi-scale models.

Domain growth is a key process in many areas of biology, including embryonic development, the growth of tissue, and limb regeneration. As a result, mechanisms for incorporating it into traditional models for cell movement, interaction, and proliferation are of great importance. A previously well-used method to incorporate domain growth into on-lattice reaction-diffusion models causes a buildup of particles on the boundaries of the domain, which is particularly evident when diffusion is low in comparison to the rate of domain growth. Here we present an alternative method which addresses this unphysical buildup of particles at the boundaries and demonstrate that it is accurate for scenarios in which the previous method fails. Further, we discuss for which parameter regimes it is feasible to continue using the original method due to diffusion dominating the domain growth mechanism.

Cell proliferation is typically incorporated into stochastic mathematical models of cell migration by assuming that cell divisions occur after an exponentially distributed waiting time. Experimental observations, however, show that this assumption is often far from the real cell cycle time distribution (CCTD). Recent studies have suggested an alternative approach to modelling cell proliferation based on a multi-stage representation of the CCTD. In this paper we investigate the connection between the CCTD and the speed of the collective invasion. We first state a result for a general CCTD, which allows the computation of the invasion speed using the Laplace transform of the CCTD. We use this to deduce the range of speeds for the general case. We then focus on the more realistic case of multi-stage models, using both a stochastic agent-based model and a set of reaction-diffusion equations for the cells' average density. By studying the corresponding travelling wave solutions, we obtain an analytical expression for the speed of invasion for a general N-stage model with identical transition rates, in which case the resulting cell cycle times are Erlang distributed. We show that, for a general N-stage model, the Erlang distribution and the exponential distribution lead to the minimum and maximum invasion speed, respectively. This result allows us to determine the range of possible invasion speeds in terms of the average proliferation time for any multi-stage model., (Copyright © 2018 Elsevier Ltd. All rights reserved.)

Dissolved organic matter (DOM) represents a key component of carbon (C) cycling in freshwater ecosystems. While the behaviour of bulk dissolved organic carbon (DOC) in aquatic ecosystems is well studied, comparatively little is known about the turnover of specific DOC compounds. The aim of this study was to investigate the persistence of 14 C-labelled low molecular weight (LMW) DOC at a wide range of concentrations (0.1 µM to 10 mM), in sediments and waters from oligotrophic and mesotrophic rivers within the same catchment. Overall, rates of DOC loss varied between compound groups (amino acids > sugars = organic acids > phenolics). Sediment-based microbial communities contributed to higher DOC loss from river waters, which was attributed, in part, to its greater microbial biomass. At higher DOC compound concentrations, DOC loss was greater in mesotrophic rivers in comparison to oligotrophic headwaters. A lag-phase in substrate use within sediments provided evidence of microbial growth and adaptation, ascribed here to the lack of inorganic nutrient limitation on microbial C processing in mesotrophic communities. We conclude that the higher microbial biomass and available inorganic nutrients in sediments enables the rapid processing of LMW DOC, particularly during high C enrichment events and in N and P-rich mesotrophic environments.

There are numerous biological scenarios in which populations of cells migrate in crowded environments. Typical examples include wound healing, cancer growth, and embryo development. In these crowded environments cells are able to interact with each other in a variety of ways. These include excluded-volume interactions, adhesion, repulsion, cell signaling, pushing, and pulling. One popular way to understand the behavior of a group of interacting cells is through an agent-based mathematical model. A typical aim of modellers using such representations is to elucidate how the microscopic interactions at the cell-level impact on the macroscopic behavior of the population. At the very least, such models typically incorporate volume-exclusion. The more complex cell-cell interactions listed above have also been incorporated into such models; all apart from cell-cell pulling. In this paper we consider this under-represented cell-cell interaction, in which an active cell is able to "pull" a nearby neighbor as it moves. We incorporate a variety of potential cell-cell pulling mechanisms into on- and off-lattice agent-based volume exclusion models of cell movement. For each of these agent-based models we derive a continuum partial differential equation which describes the evolution of the cells at a population level. We study the agreement between the agent-based models and the continuum, population-based models and compare and contrast a range of agent-based models (accounting for the different pulling mechanisms) with each other. We find generally good agreement between the agent-based models and the corresponding continuum models that worsens as the agent-based models become more complex. Interestingly, we observe that the partial differential equations that we derive differ significantly, depending on whether they were derived from on- or off-lattice agent-based models of pulling. This hints that it is important to employ the appropriate agent-based model when representing pulling cell-cell interactions.

Sewage treatment systems are a common feature across the landscape of the United Kingdom, serving an estimated 96% of the population and discharging approximately eleven billion litres of treated wastewater daily. While large treatment facilities are ubiquitous across the landscape, they are not the only method employed in domestic wastewater treatment. This study investigates whether differences in nutrient export (carbon, nitrogen and phosphorus) and organic matter composition (determined by optical indices, SUVA 254 , S 350-400 and E 2 :E 3 ) from treated effluent could be detected between four of the most common facilities employed in the treatment of wastewater across the UK. Set in the context of the River Wylye, a small headwater catchment, treatment facilities studied included; a septic tank system, small packet treatment works, and two large sewage treatment works, one of which employed phosphorus stripping for phosphorus removal. Inorganic N and P concentrations ranged between 7.51 and 42.4 mg N l -1 and 0.22 and 8.9 mg P l -1 respectively, with DOC concentrations ranging between 1.63 and 11.8 mg C l -1 . Optical indices were comparable to those observed in catchments where organic matter is dominated by autochthonous production, suggesting the dominance of low molecular weight material when compared to values observed across temperate aquatic systems. Combining data from both the Environment Agency and Ordinance Survey we estimate that only 15% of domestic properties not connected to mains sewerage in the study catchment have an Environment Agency consent/exemption permit. This calculation suggests that the quantity of small point sources are significantly underestimated, undermining efforts under current legislation to improve stream ecosystem health., (Copyright © 2018 Elsevier Ltd. All rights reserved.)

Reaction-diffusion systems are used to represent many biological and physical phenomena. They model the random motion of particles (diffusion) and interactions between them (reactions). Such systems can be modelled at multiple scales with varying degrees of accuracy and computational efficiency. When representing genuinely multiscale phenomena, fine-scale models can be prohibitively expensive, whereas coarser models, although cheaper, often lack sufficient detail to accurately represent the phenomenon at hand. Spatial hybrid methods couple two or more of these representations in order to improve efficiency without compromising accuracy. In this paper, we present a novel spatial hybrid method, which we call the auxiliary region method (ARM), which couples PDE- and Brownian-based representations of reaction-diffusion systems. Numerical PDE solutions on one side of an interface are coupled to Brownian-based dynamics on the other side using compartment-based 'auxiliary regions'. We demonstrate that the hybrid method is able to simulate reaction-diffusion dynamics for a number of different test problems with high accuracy. Furthermore, we undertake error analysis on the ARM which demonstrates that it is robust to changes in the free parameters in the model, where previous coupling algorithms are not. In particular, we envisage that the method will be applicable for a wide range of spatial multi-scales problems including filopodial dynamics, intracellular signalling, embryogenesis and travelling wave phenomena., Competing Interests: We declare we have no competing interests.

Identifying and quantifying spatial correlation are important aspects of studying the collective behavior of multiagent systems. Pair correlation functions (PCFs) are powerful statistical tools that can provide qualitative and quantitative information about correlation between pairs of agents. Despite the numerous PCFs defined for off-lattice domains, only a few recent studies have considered a PCF for discrete domains. Our work extends the study of spatial correlation in discrete domains by defining a new set of PCFs using two natural and intuitive definitions of distance for a square lattice: the taxicab and uniform metric. We show how these PCFs improve upon previous attempts and compare between the quantitative data acquired. We also extend our definitions of the PCF to other types of regular tessellation that have not been studied before, including hexagonal, triangular, and cuboidal. Finally, we provide a comprehensive PCF for any tessellation and metric, allowing investigation of spatial correlation in irregular lattices for which recognizing correlation is less intuitive.

Persistence of motion is the tendency of an object to maintain motion in a direction for short time scales without necessarily being biased in any direction in the long term. One of the most appropriate mathematical tools to study this behavior is an agent-based velocity-jump process. In the absence of agent-agent interaction, the mean-field continuum limit of the agent-based model (ABM) gives rise to the well known hyperbolic telegraph equation. When agent-agent interaction is included in the ABM, a strictly advective system of partial differential equations (PDEs) can be derived at the population level. However, no diffusive limit of the ABM has been obtained from such a model. Connecting the microscopic behavior of the ABM to a diffusive macroscopic description is desirable, since it allows the exploration of a wider range of scenarios and establishes a direct connection with commonly used statistical tools of movement analysis. In order to connect the ABM at the population level to a diffusive PDE at the population level, we consider a generalization of the agent-based velocity-jump process on a two-dimensional lattice with three forms of agent interaction. This generalization allows us to take a diffusive limit and obtain a faithful population-level description. We investigate the properties of the model at both the individual and population levels and we elucidate some of the models' key characteristic features. In particular, we show an intrinsic anisotropy inherent to the models and we find evidence of a spontaneous form of aggregation at both the micro- and macroscales.

Many biological and physical systems exhibit behaviour at multiple spatial, temporal or population scales. Multiscale processes provide challenges when they are to be simulated using numerical techniques. While coarser methods such as partial differential equations are typically fast to simulate, they lack the individual-level detail that may be required in regions of low concentration or small spatial scale. However, to simulate at such an individual level throughout a domain and in regions where concentrations are high can be computationally expensive. Spatially coupled hybrid methods provide a bridge, allowing for multiple representations of the same species in one spatial domain by partitioning space into distinct modelling subdomains. Over the past 20 years, such hybrid methods have risen to prominence, leading to what is now a very active research area across multiple disciplines including chemistry, physics and mathematics. There are three main motivations for undertaking this review. Firstly, we have collated a large number of spatially extended hybrid methods and presented them in a single coherent document, while comparing and contrasting them, so that anyone who requires a multiscale hybrid method will be able to find the most appropriate one for their need. Secondly, we have provided canonical examples with algorithms and accompanying code, serving to demonstrate how these types of methods work in practice. Finally, we have presented papers that employ these methods on real biological and physical problems, demonstrating their utility. We also consider some open research questions in the area of hybrid method development and the future directions for the field., (© 2018 The Authors.)

The stochastic simulation algorithm commonly known as Gillespie's algorithm (originally derived for modelling well-mixed systems of chemical reactions) is now used ubiquitously in the modelling of biological processes in which stochastic effects play an important role. In well-mixed scenarios at the sub-cellular level it is often reasonable to assume that times between successive reaction/interaction events are exponentially distributed and can be appropriately modelled as a Markov process and hence simulated by the Gillespie algorithm. However, Gillespie's algorithm is routinely applied to model biological systems for which it was never intended. In particular, processes in which cell proliferation is important (e.g. embryonic development, cancer formation) should not be simulated naively using the Gillespie algorithm since the history-dependent nature of the cell cycle breaks the Markov process. The variance in experimentally measured cell cycle times is far less than in an exponential cell cycle time distribution with the same mean.Here we suggest a method of modelling the cell cycle that restores the memoryless property to the system and is therefore consistent with simulation via the Gillespie algorithm. By breaking the cell cycle into a number of independent exponentially distributed stages, we can restore the Markov property at the same time as more accurately approximating the appropriate cell cycle time distributions. The consequences of our revised mathematical model are explored analytically as far as possible. We demonstrate the importance of employing the correct cell cycle time distribution by recapitulating the results from two models incorporating cellular proliferation (one spatial and one non-spatial) and demonstrating that changing the cell cycle time distribution makes quantitative and qualitative differences to the outcome of the models. Our adaptation will allow modellers and experimentalists alike to appropriately represent cellular proliferation-vital to the accurate modelling of many biological processes-whilst still being able to take advantage of the power and efficiency of the popular Gillespie algorithm.

Dissolved organic matter (DOM) plays a central role in regulating productivity and nutrient cycling in freshwaters. It is therefore vital that we can representatively sample and preserve DOM in freshwaters for subsequent analysis. Here we investigated the effect of filtration, temperature (5 and 25°C) and acidification (HCl) on the persistence of low molecular weight (MW) dissolved organic carbon (DOC), nitrogen (DON) and orthophosphate in oligotrophic and eutrophic freshwater environments. Our results showed the rapid loss of isotopically-labelled glucose and amino acids from both filtered (0.22 and 0.45μm) and unfiltered waters. We ascribe this substrate depletion in filtered samples to the activity of ultra-small (<0.45μm) microorganisms (bacteria and archaea) present in the water. As expected, the rate of C, N and P loss was much greater at higher temperatures and was repressed by the addition of HCl. Based on our results and an evaluation of the protocols used in recently published studies, we conclude that current techniques used to sample water for low MW DOM characterisation are frequently inadequate and lack proper validation. In contrast to the high degree of analytical precision and rigorous statistical analysis of most studies, we argue that insufficient consideration is still given to the presence of ultra-small microorganisms and potential changes that can occur in the low MW fraction of DOM prior to analysis., (Copyright © 2017 Elsevier B.V. All rights reserved.)

In this work, we implement approximate Bayesian computational methods to improve the design of a wound-healing assay used to quantify cell-cell interactions. This is important as cell-cell interactions, such as adhesion and repulsion, have been shown to play a role in cell migration. Initially, we demonstrate with a model of an unrealistic experiment that we are able to identify model parameters that describe agent motility and adhesion, given we choose appropriate summary statistics for our model data. Following this, we replace our model of an unrealistic experiment with a model representative of a practically realisable experiment. We demonstrate that, given the current (and commonly used) experimental set-up, our model parameters cannot be accurately identified using approximate Bayesian computation methods. We compare new experimental designs through simulation, and show more accurate identification of model parameters is possible by expanding the size of the domain upon which the experiment is performed, as opposed to increasing the number of experimental replicates. The results presented in this work, therefore, describe time and cost-saving alterations for a commonly performed experiment for identifying cell motility parameters. Moreover, this work will be of interest to those concerned with performing experiments that allow for the accurate identification of parameters governing cell migratory processes, especially cell migratory processes in which cell-cell adhesion or repulsion are known to play a significant role.

Skin pigment pattern formation is a paradigmatic example of pattern formation. In zebrafish, the adult body stripes are generated by coordinated rearrangement of three distinct pigment cell-types, black melanocytes, shiny iridophores and yellow xanthophores. A stem cell origin of melanocytes and iridophores has been proposed although the potency of those stem cells has remained unclear. Xanthophores, however, seemed to originate predominantly from proliferation of embryonic xanthophores. Now, data from Singh et al. shows that all three cell-types derive from shared stem cells, and that these cells generate peripheral neural cell-types too. Furthermore, clonal compositions are best explained by a progressive fate restriction model generating the individual cell-types. The numbers of adult pigment stem cells associated with the dorsal root ganglia remain low, but progenitor numbers increase significantly during larval development up to metamorphosis, likely via production of partially restricted progenitors on the spinal nerves., (© 2016 The Authors BioEssays Published by WILEY Periodicals, Inc.)

In this work we study the effect of domain growth on spatial correlations in agent populations containing multiple species. This is important as heterogenous cell populations are ubiquitous during the embryonic development of many species. We have previously shown that the long-term behavior of an agent population depends on the way in which domain growth is implemented. We extend this work to show that, depending on the way in which domain growth is implemented, different species dominate in multispecies simulations. Continuum approximations of the lattice-based model that ignore spatial correlations cannot capture this behavior, while those that explicitly account for spatial correlations can. The results presented here show that the precise mechanism of domain growth can determine the long-term behavior of multispecies populations and, in certain circumstances, establish spatially varying species densities.

Reaction-diffusion models are widely used to study spatially extended chemical reaction systems. In order to understand how the dynamics of a reaction-diffusion model are affected by changes in its input parameters, efficient methods for computing parametric sensitivities are required. In this work, we focus on the stochastic models of spatially extended chemical reaction systems that involve partitioning the computational domain into voxels. Parametric sensitivities are often calculated using Monte Carlo techniques that are typically computationally expensive; however, variance reduction techniques can decrease the number of Monte Carlo simulations required. By exploiting the characteristic dynamics of spatially extended reaction networks, we are able to adapt existing finite difference schemes to robustly estimate parametric sensitivities in a spatially extended network. We show that algorithmic performance depends on the dynamics of the given network and the choice of summary statistics. We then describe a hybrid technique that dynamically chooses the most appropriate simulation method for the network of interest. Our method is tested for functionality and accuracy in a range of different scenarios.

Two lowland catchments in the U.K. were sampled throughout 2010-11 to investigate the dominant controls on dissolved organic matter quantity and composition. The catchments had marked differences in terms of nutrient status, land cover and contrasting lithologies resulting in differences in the dominant flow pathways (groundwater vs. surface water dominated). The Upper Wylye is a chalk stream with a baseflow index of 0.98, draining a catchment dominated by intensive agricultural production. Millersford Brook is a lowland peat catchment with a baseflow index of 0.43, draining a semi-natural catchment with heather moorland and coniferous forest. Samples were collected weekly between October 2010 and September 2011 from eleven sampling locations. Samples were analysed to determine dissolved organic carbon, nitrogen and phosphorus fractions with DOM composition evaluated via the DOC:DON ratio, DOC:DOP ratio, specific UV absorption at 254nm, absorbance ratio (a250:a365) and the spectral slope parameter between 350 and 400nm (S350-400). Significant differences were observed in all determinands between the catchments, over time, and spatially along nutrient enrichment and geoclimatic gradients. Seasonal variation in preferential flow pathways mobilising groundwater-derived DOM were identified as likely controls on the delivery of DOM in the permeable chalk dominated catchment. Steeper S350-400 values and elevated a250:a365 ratios in this catchment suggest material of a lower bulk aromatic C content and molecular weight delivered during the winter months when compared to the summer. DOC:DON ratios were markedly lower in the chalk catchment than the peatland catchment, reflecting the paucity of organic matter within the mineral soils of the chalk landscape, and higher fertiliser application rates. This manuscript highlights that DOM composition varies according to catchment landscape character and hydrological function., (Copyright © 2016 The Authors. Published by Elsevier B.V. All rights reserved.)

Stochastic simulation methods can be applied successfully to model exact spatio-temporally resolved reaction-diffusion systems. However, in many cases, these methods can quickly become extremely computationally intensive with increasing particle numbers. An alternative description of many of these systems can be derived in the diffusive limit as a deterministic, continuum system of partial differential equations (PDEs). Although the numerical solution of such PDEs is, in general, much more efficient than the full stochastic simulation, the deterministic continuum description is generally not valid when copy numbers are low and stochastic effects dominate. Therefore, to take advantage of the benefits of both of these types of models, each of which may be appropriate in different parts of a spatial domain, we have developed an algorithm that can be used to couple these two types of model together. This hybrid coupling algorithm uses an overlap region between the two modelling regimes. By coupling fluxes at one end of the interface and using a concentration-matching condition at the other end, we ensure that mass is appropriately transferred between PDE- and compartment-based regimes. Our methodology gives notable reductions in simulation time in comparison with using a fully stochastic model, while maintaining the important stochastic features of the system and providing detail in appropriate areas of the domain. We test our hybrid methodology robustly by applying it to several biologically motivated problems including diffusion and morphogen gradient formation. Our analysis shows that the resulting error is small, unbiased and does not grow over time., (© 2016 The Authors.)

The multi-level method for discrete-state systems, first introduced by Anderson and Higham (SIAM Multiscale Model Simul 10(1):146-179, 2012), is a highly efficient simulation technique that can be used to elucidate statistical characteristics of biochemical reaction networks. A single point estimator is produced in a cost-effective manner by combining a number of estimators of differing accuracy in a telescoping sum, and, as such, the method has the potential to revolutionise the field of stochastic simulation. In this paper, we present several refinements of the multi-level method which render it easier to understand and implement, and also more efficient. Given the substantial and complex nature of the multi-level method, the first part of this work reviews existing literature, with the aim of providing a practical guide to the use of the multi-level method. The second part provides the means for a deft implementation of the technique and concludes with a discussion of a number of open problems.

Numerous processes across both the physical and biological sciences are driven by diffusion. Partial differential equations are a popular tool for modelling such phenomena deterministically, but it is often necessary to use stochastic models to accurately capture the behaviour of a system, especially when the number of diffusing particles is low. The stochastic models we consider in this paper are 'compartment-based': the domain is discretized into compartments, and particles can jump between these compartments. Volume-excluding effects (crowding) can be incorporated by blocking movement with some probability. Recent work has established the connection between fine- and coarse-grained models incorporating volume exclusion, but only for uniform lattices. In this paper, we consider non-uniform, hybrid lattices that incorporate both fine- and coarse-grained regions, and present two different approaches to describe the interface of the regions. We test both techniques in a range of scenarios to establish their accuracy, benchmarking against fine-grained models, and show that the hybrid models developed in this paper can be significantly faster to simulate than the fine-grained models in certain situations and are at least as fast otherwise., (© 2016 The Author(s).)

Domain growth plays an important role in many biological systems, and so the inclusion of domain growth in models of these biological systems is important to understanding how these systems function. In this work we present methods to include the effects of domain growth on the evolution of spatial correlations in a continuum approximation of a lattice-based model of cell motility and proliferation. We show that, depending on the way in which domain growth is implemented, different steady-state densities are predicted for an agent population. Furthermore, we demonstrate that the way in which domain growth is implemented can result in the evolution of the agent density depending on the size of the domain. Continuum approximations that ignore spatial correlations cannot capture these behaviors, while those that account for spatial correlations do. These results will be of interest to researchers in developmental biology, as they suggest that the nature of domain growth can determine the characteristics of cell populations.