Publications by year
In Press
Beardmore R, Gudelj I, Hewlett M, Pena-Miller R, Meyer J (In Press). Canonical host-pathogen tradeoffs subverted by mutations with dual benefits.
The American NaturalistAbstract:
Canonical host-pathogen tradeoffs subverted by mutations with dual benefits
Host-parasite coevolution is expected to drive the evolution of genetic diversity because the traits used in arms races, namely host range and parasite resistance, are hypothesized to tradeoff with traits used in resource competition. We therefore tested data for several tradeoffs among 93 isolates of bacteriophage λ and 51 Escherichia coli genotypes that coevolved during a labora- tory experiment. Surprisingly, we found multiple tradeups (positive trait correlations) but little evidence of several canonical tradeoffs. For example, some bacterial genotypes evaded a trade- off between phage resistance and absolute fitness, instead evolving simultaneous improvements in both traits. This was surprising because our experimental design was predicted to expose resistance-fitness tradeoffs by culturing E. coli in a medium where the phage receptor, LamB, is also used for nutrient acquisition. On reflection, LamB mediates not one but many tradeoffs, allowing for more complex trait interactions than just pairwise tradeoffs. Here, we report that mathematical reasoning and laboratory data highlight how tradeups should exist whenever an evolutionary system exhibits multiple interacting tradeoffs. Does this mean that coevolution should not promote genetic diversity? No, quite the contrary. We deduce that whenever posi- tive trait correlations are observed in multi-dimensional traits, other traits may trade off and so provide the right circumstances for diversity maintenance. Overall, this study reveals there are predictive limits when data only account for pairwise trait correlations and it argues that a wider range of circumstances than previously anticipated can promote genetic and species diversity.
Abstract.
Beardmore R, Gudelj I, Reding C, Wood E, Bergmiller T, Schulenberg H, Philip R, Catalan P (In Press). The Antibiotic Dosage of Fastest Resistance Evolution: gene amplifications underpinning the inverted-U. Molecular Biology and Evolution
2023
Wood E, Schulenburg H, Rosenstiel P, Bergmiller T, Ankrett D, Gudelj I, Beardmore R (2023). Ribosome-binding antibiotics increase bacterial longevity and growth efficiency.
Proc Natl Acad Sci U S A,
120(40).
Abstract:
Ribosome-binding antibiotics increase bacterial longevity and growth efficiency.
Antibiotics, by definition, reduce bacterial growth rates in optimal culture conditions; however, the real-world environments bacteria inhabit see rapid growth punctuated by periods of low nutrient availability. How antibiotics mediate population decline during these periods is poorly understood. Bacteria cannot optimize for all environmental conditions because a growth-longevity tradeoff predicts faster growth results in faster population decline, and since bacteriostatic antibiotics slow growth, they should also mediate longevity. We quantify how antibiotics, their targets, and resistance mechanisms influence longevity using populations of Escherichia coli and, as the tradeoff predicts, populations are maintained for longer if they encounter ribosome-binding antibiotics doxycycline and erythromycin, a finding that is not observed using antibiotics with alternative cellular targets. This tradeoff also predicts resistance mechanisms that increase growth rates during antibiotic treatment could be detrimental during nutrient stresses, and indeed, we find resistance by ribosomal protection removes benefits to longevity provided by doxycycline. We therefore liken ribosomal protection to a "Trojan horse" because it provides protection from an antibiotic but, during nutrient stresses, it promotes the demise of the bacteria. Seeking mechanisms to support these observations, we show doxycycline promotes efficient metabolism and reduces the concentration of reactive oxygen species. Seeking generality, we sought another mechanism that affects longevity and we found the number of doxycycline targets, namely, the ribosomal RNA operons, mediates growth and longevity even without antibiotics. We conclude that slow growth, as observed during antibiotic treatment, can help bacteria overcome later periods of nutrient stress.
Abstract.
Author URL.
2022
Catalán P, Wood E, Blair JMA, Gudelj I, Iredell JR, Beardmore RE (2022). Seeking patterns of antibiotic resistance in ATLAS, an open, raw MIC database with patient metadata.
Nat Commun,
13(1).
Abstract:
Seeking patterns of antibiotic resistance in ATLAS, an open, raw MIC database with patient metadata.
Antibiotic resistance represents a growing medical concern where raw, clinical datasets are under-exploited as a means to track the scale of the problem. We therefore sought patterns of antibiotic resistance in the Antimicrobial Testing Leadership and Surveillance (ATLAS) database. ATLAS holds 6.5M minimal inhibitory concentrations (MICs) for 3,919 pathogen-antibiotic pairs isolated from 633k patients in 70 countries between 2004 and 2017. We show most pairs form coherent, although not stationary, timeseries whose frequencies of resistance are higher than other databases, although we identified no systematic bias towards including more resistant strains in ATLAS. We sought data anomalies whereby MICs could shift for methodological and not clinical or microbiological reasons and found artefacts in over 100 pathogen-antibiotic pairs. Using an information-optimal clustering methodology to classify pathogens into low and high antibiotic susceptibilities, we used ATLAS to predict changes in resistance. Dynamics of the latter exhibit complex patterns with MIC increases, and some decreases, whereby subpopulations' MICs can diverge. We also identify pathogens at risk of developing clinical resistance in the near future.
Abstract.
Author URL.
2021
Reding C, Catalán P, Jansen G, Bergmiller T, Wood E, Rosenstiel P, Schulenberg H, Philip R, Gudelj I, Beardmore R, et al (2021). Antibiotic dosages of fastest resistance evolution: gene amplifications underpinning the inverted U.
Nev OA, Lindsay RJ, Jepson A, Butt L, Beardmore RE, Gudelj I (2021). Predicting microbial growth dynamics in response to nutrient availability.
PLoS Comput Biol,
17(3).
Abstract:
Predicting microbial growth dynamics in response to nutrient availability.
Developing mathematical models to accurately predict microbial growth dynamics remains a key challenge in ecology, evolution, biotechnology, and public health. To reproduce and grow, microbes need to take up essential nutrients from the environment, and mathematical models classically assume that the nutrient uptake rate is a saturating function of the nutrient concentration. In nature, microbes experience different levels of nutrient availability at all environmental scales, yet parameters shaping the nutrient uptake function are commonly estimated for a single initial nutrient concentration. This hampers the models from accurately capturing microbial dynamics when the environmental conditions change. To address this problem, we conduct growth experiments for a range of micro-organisms, including human fungal pathogens, baker's yeast, and common coliform bacteria, and uncover the following patterns. We observed that the maximal nutrient uptake rate and biomass yield were both decreasing functions of initial nutrient concentration. While a functional form for the relationship between biomass yield and initial nutrient concentration has been previously derived from first metabolic principles, here we also derive the form of the relationship between maximal nutrient uptake rate and initial nutrient concentration. Incorporating these two functions into a model of microbial growth allows for variable growth parameters and enables us to substantially improve predictions for microbial dynamics in a range of initial nutrient concentrations, compared to keeping growth parameters fixed.
Abstract.
Author URL.
2020
Catalan P, Reding C, Blair J, Gudelj I, Iredell J, Beardmore R (2020). Clinical Antibiotic Resistance Patterns Across 70 Countries.
Peña-Miller R, Arnoldini M, Ackermann M, Beardmore RE (2020). Dynamic phenotypic heterogeneity generated by delayed genetic oscillations.
Duxbury SJN, Bates S, Beardmore RE, Gudelj I (2020). Evolution of drug-resistant and virulent small colonies in phenotypically diverse populations of the human fungal pathogen. <i>Candida glabrata</i>.
Proceedings of the Royal Society B: Biological Sciences,
287(1931), 20200761-20200761.
Abstract:
Evolution of drug-resistant and virulent small colonies in phenotypically diverse populations of the human fungal pathogen. Candida glabrata
. Antimicrobial resistance frequently carries a fitness cost to a pathogen, measured as a reduction in growth rate compared to the sensitive wild-type, in the absence of antibiotics. Existing empirical evidence points to the following relationship between cost of resistance and virulence. If a resistant pathogen suffers a fitness cost in terms of reduced growth rate it commonly has lower virulence compared to the sensitive wild-type. If this cost is absent so is the reduction in virulence. Here we show, using experimental evolution of drug resistance in the fungal human pathogen
. Candida glabrata,
. that reduced growth rate of resistant strains need not result in reduced virulence. Phenotypically heterogeneous populations were evolved in parallel containing highly resistant sub-population small colony variants (SCVs) alongside sensitive sub-populations. Despite their low growth rate in the absence of an antifungal drug, the SCVs did not suffer a marked alteration in virulence compared with the wild-type ancestral strain, or their co-isolated sensitive strains. This contrasts with classical theory that assumes growth rate to positively correlate with virulence. Our work thus highlights the complexity of the relationship between resistance, basic life-history traits and virulence.
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Abstract.
Nev O, Jepson A, Beardmore R, Gudelj I (2020). Predicting community dynamics of antibiotic sensitive and resistant species in fluctuating environments (dataset). Journal of the Royal Society Interface
Nev OA, Jepson A, Beardmore RE, Gudelj I (2020). Predicting community dynamics of antibiotic-sensitive and -resistant species in fluctuating environments.
J R Soc Interface,
17(166).
Abstract:
Predicting community dynamics of antibiotic-sensitive and -resistant species in fluctuating environments.
Microbes occupy almost every niche within and on their human hosts. Whether colonizing the gut, mouth or bloodstream, microorganisms face temporal fluctuations in resources and stressors within their niche but we still know little of how environmental fluctuations mediate certain microbial phenotypes, notably antimicrobial-resistant ones. For instance, do rapid or slow fluctuations in nutrient and antimicrobial concentrations select for, or against, resistance? We tackle this question using an ecological approach by studying the dynamics of a synthetic and pathogenic microbial community containing two species, one sensitive and the other resistant to an antibiotic drug where the community is exposed to different rates of environmental fluctuation. We provide mathematical models, supported by experimental data, to demonstrate that simple community outcomes, such as competitive exclusion, can shift to coexistence and ecosystem bistability as fluctuation rates vary. Theory gives mechanistic insight into how these dynamical regimes are related. Importantly, our approach highlights a fundamental difference between resistance in single-species populations, the context in which it is usually assayed, and that in communities. While fast environmental changes are known to select against resistance in single-species populations, here we show that they can promote the resistant species in mixed-species communities. Our theoretical observations are verified empirically using a two-species Candida community.
Abstract.
Author URL.
Nev O, Jepson A, Butt L, Lindsay R, Beardmore R, Gudelj I (2020). Raw growth data for the manuscript Predicting microbial growth dynamics in changing environments.
2019
Beardmore R, Hewlett M, Peña-Miller R, Reding C, Gudelj I, Meyer JR (2019). Canonical host-pathogen tradeoffs subverted by mutations with dual benefits.
Reding C, Hewlett M, Bergmiller T, Gudelj I, Beardmore R (2019). Fluorescence photography of patterns and waves of bacterial adaptation at high antibiotic doses.
Abstract:
Fluorescence photography of patterns and waves of bacterial adaptation at high antibiotic doses
Fisher suggested advantageous genes would spread through populations as a wave so we sought genetic waves in evolving populations, as follows. By fusing a fluorescent marker to a drug efflux protein (AcrB) whose expression providesEscherichia coliwith resistance to some antibiotics, we quantified the evolution and spread of drug-resistantE. colithrough spacetime using image analysis and quantitative PCR. As is done in hospitals routinely, we exposed the bacterium to a gradient of antibiotic in a ‘disk diffusion’ drug susceptibility test that we videoed. The videos show complex spatio-genomic patterns redolent of, yet more complex than, Fisher’s predictions whereby a decelerating wave front of advantageous genes colonises towards the antibiotic source, forming bullseye patterns en route and leaving a wave back of bacterial sub-populations expressing AcrB at decreasing levels away from the drug source. qPCR data show thatE. colisited at rapidly-adapting spatial hotspots gain 2 additional copies ofacr, the operon that encodes AcrB, within 24h and imaging data show resistant sub-populations thrive most near the antibiotic source due to non-monotone relationships between inhibition due to antibiotic and distance from the source. In the spirit of Fisher, we provide an explicitly spatial nonlinear diffusion equation that exhibits these properties too. Finally, linear diffusion theory quantifies how the spatial extent of bacterial killing scales with increases in antibiotic dosage, predicting that microbes can survive chemotherapies that have been escalated to 250× the clinical dosage if the antibiotic is diffusion-limited.
Abstract.
Tonneau C (2019). Host-pathogen-drug interactions in the context of antibiotic resistance: How host xenobiotic metabolism can affect antibiotic efficacy in a Methicillin-Resistant Staphylococcus aureus infection.
Abstract:
Host-pathogen-drug interactions in the context of antibiotic resistance: How host xenobiotic metabolism can affect antibiotic efficacy in a Methicillin-Resistant Staphylococcus aureus infection
Our arsenal of weapons to fight against bacterial infections is weakening: bacteria are gaining resistance to the common antibiotics, while industries are struggling to develop new effective ones. To avoid triggering de-novo antibiotic resistance, we need the right antibiotic for the specific bacteria, at a dose adapted to the patient genetics. Genes driving the degradation of antibiotics have indeed known genetic variants that can dramatically affect the kinetics of antibiotic metabolism from one patient to another. This could lead to treatment failure, excessive side effects or emergence of resistance.
I first investigated the clinical relevance of the vancomycin-rifampicin combination to treat Methicillin-Resistant Staphylococcus aureus infections (Chapter 3). I showed in various experimental settings that these two antibiotics may promote an environment prone for antibiotic resistance. Their interaction might be unstable in vitro because of environmental factors, one could wonder how the host environment might generate such instability.
I then explored how interactions between antibiotics and host xenobiotic genetics could influence antibiotic concentrations, potentially triggering increased treatment failure, side-effects and antibiotic resistance in patients carrying particular variants. In silico, I estimated the effects of genetic variants of the Cytochrome P450 3A4 gene to its enzyme, and, as they are unequally distributed in the world, their global relevance (Chapter 4). In vivo, I focused on the Carboxylesterase 2 gene and I found two of its variants, rs11075646 and rs8192925, capable of significantly altering the degradation of various drugs, including rifampicin and mycophenolate mofetil. A clinical study was designed, to explore possible correlations between genotype for these variants and treatment response in patients (Chapter 5).
Altogether, this body of work highlights the prescribing importance of considering not only the strain in bacterial infections, but also the genetics of the human host. This raises a need to make sure the right antibiotics are used in practices, at doses adapted to the patients. As part of personalised medicine, checking their genotype for these biomarkers could tailor their therapy, improving recovery while avoiding antibiotic resistance.
Abstract.
Reding C, Catalán P, Jansen G, Bergmiller T, Rosenstiel P, Schulenburg H, Gudelj I, Beardmore R (2019). Hotspot Dosages of Most Rapid Antibiotic Resistance Evolution.
2018
Barbosa C, Beardmore R, Schulenburg H, Jansen G (2018). Antibiotic combination efficacy (ACE) networks for a Pseudomonas aeruginosa model.
PLoS Biol,
16(4).
Abstract:
Antibiotic combination efficacy (ACE) networks for a Pseudomonas aeruginosa model.
The spread of antibiotic resistance is always a consequence of evolutionary processes. The consideration of evolution is thus key to the development of sustainable therapy. Two main factors were recently proposed to enhance long-term effectiveness of drug combinations: evolved collateral sensitivities between the drugs in a pair and antagonistic drug interactions. We systematically assessed these factors by performing over 1,600 evolution experiments with the opportunistic nosocomial pathogen Pseudomonas aeruginosa in single- and multidrug environments. Based on the growth dynamics during these experiments, we reconstructed antibiotic combination efficacy (ACE) networks as a new tool for characterizing the ability of the tested drug combinations to constrain bacterial survival as well as drug resistance evolution across time. Subsequent statistical analysis of the influence of the factors on ACE network characteristics revealed that (i) synergistic drug interactions increased the likelihood of bacterial population extinction-irrespective of whether combinations were compared at the same level of inhibition or not-while (ii) the potential for evolved collateral sensitivities between 2 drugs accounted for a reduction in bacterial adaptation rates. In sum, our systematic experimental analysis allowed us to pinpoint 2 complementary determinants of combination efficacy and to identify specific drug pairs with high ACE scores. Our findings can guide attempts to further improve the sustainability of antibiotic therapy by simultaneously reducing pathogen load and resistance evolution.
Abstract.
Author URL.
Schuetz P, Beardmore RE (2018). Antibiotic strategies in critical care: back to square one?.
Lancet Infect Dis,
18(4), 360-361.
Author URL.
Beardmore RE, Cook E, Nilsson S, Smith AR, Tillmann A, Esquivel BD, Haynes K, Gow NAR, Brown AJP, White TC, et al (2018). Drug-mediated metabolic tipping between antibiotic resistant states in a mixed-species community.
Nat Ecol Evol,
2(8), 1312-1320.
Abstract:
Drug-mediated metabolic tipping between antibiotic resistant states in a mixed-species community.
Microbes rarely exist in isolation, rather, they form intricate multi-species communities that colonize our bodies and inserted medical devices. However, the efficacy of antimicrobials is measured in clinical laboratories exclusively using microbial monocultures. Here, to determine how multi-species interactions mediate selection for resistance during antibiotic treatment, particularly following drug withdrawal, we study a laboratory community consisting of two microbial pathogens. Single-species dose responses are a poor predictor of community dynamics during treatment so, to better understand those dynamics, we introduce the concept of a dose-response mosaic, a multi-dimensional map that indicates how species' abundance is affected by changes in abiotic conditions. We study the dose-response mosaic of a two-species community with a 'Gene × Gene × Environment × Environment' ecological interaction whereby Candida glabrata, which is resistant to the antifungal drug fluconazole, competes for survival with Candida albicans, which is susceptible to fluconazole. The mosaic comprises several zones that delineate abiotic conditions where each species dominates. Zones are separated by loci of bifurcations and tipping points that identify what environmental changes can trigger the loss of either species. Observations of the laboratory communities corroborated theory, showing that changes in both antibiotic concentration and nutrient availability can push populations beyond tipping points, thus creating irreversible shifts in community composition from drug-sensitive to drug-resistant species. This has an important consequence: resistant species can increase in frequency even if an antibiotic is withdrawn because, unwittingly, a tipping point was passed during treatment.
Abstract.
Author URL.
Beardmore RE, Cook E, Nilsson S, Smith AR, Tillmann A, Esquivel BD, Haynes K, Gow NAR, Brown AJP, White TC, et al (2018). Erratum to: Drug-mediated metabolic tipping between antibiotic resistant states in a mixed-species community (Nature Ecology & Evolution, (2018), 2, 8, (1312-1320), 10.1038/s41559-018-0582-7).
Nature Ecology and Evolution,
2(11).
Abstract:
Erratum to: Drug-mediated metabolic tipping between antibiotic resistant states in a mixed-species community (Nature Ecology & Evolution, (2018), 2, 8, (1312-1320), 10.1038/s41559-018-0582-7)
In the version of this Article originally published, the following sentence was missing from the Acknowledgements: “R.E.B. is an EPSRC Healthcare Technologies Impact Fellow EP/N033671/1; I.G. is funded by ERC Consolidator grant 647292 MathModExp; A.J.P.B. N.A.R.G. and A.T. were funded by BBSRC grant BB/F00513X/1; K.H. I.G. S.N. and E.C. were funded by BBSRC grant BB/F005210/2.” This text has now been added.
Abstract.
2017
Barbosa C, Trebosc V, Kemmer C, Rosenstiel P, Beardmore R, Schulenburg H, Jansen G (2017). Alternative Evolutionary Paths to Bacterial Antibiotic Resistance Cause Distinct Collateral Effects.
Mol Biol Evol,
34(9), 2229-2244.
Abstract:
Alternative Evolutionary Paths to Bacterial Antibiotic Resistance Cause Distinct Collateral Effects.
When bacteria evolve resistance against a particular antibiotic, they may simultaneously gain increased sensitivity against a second one. Such collateral sensitivity may be exploited to develop novel, sustainable antibiotic treatment strategies aimed at containing the current, dramatic spread of drug resistance. To date, the presence and molecular basis of collateral sensitivity has only been studied in few bacterial species and is unknown for opportunistic human pathogens such as Pseudomonas aeruginosa. In the present study, we assessed patterns of collateral effects by experimentally evolving 160 independent populations of P. aeruginosa to high levels of resistance against eight commonly used antibiotics. The bacteria evolved resistance rapidly and expressed both collateral sensitivity and cross-resistance. The pattern of such collateral effects differed to those previously reported for other bacterial species, suggesting interspecific differences in the underlying evolutionary trade-offs. Intriguingly, we also identified contrasting patterns of collateral sensitivity and cross-resistance among the replicate populations adapted to the same drug. Whole-genome sequencing of 81 independently evolved populations revealed distinct evolutionary paths of resistance to the selective drug, which determined whether bacteria became cross-resistant or collaterally sensitive towards others. Based on genomic and functional genetic analysis, we demonstrate that collateral sensitivity can result from resistance mutations in regulatory genes such as nalC or mexZ, which mediate aminoglycoside sensitivity in β-lactam-adapted populations, or the two-component regulatory system gene pmrB, which enhances penicillin sensitivity in gentamicin-resistant populations. Our findings highlight substantial variation in the evolved collateral effects among replicates, which in turn determine their potential in antibiotic therapy.
Abstract.
Author URL.
Beardmore RE, Peña-Miller R, Gori F, Iredell J (2017). Antibiotic Cycling and Antibiotic Mixing: Which One Best Mitigates Antibiotic Resistance?.
Mol Biol Evol,
34(4), 802-817.
Abstract:
Antibiotic Cycling and Antibiotic Mixing: Which One Best Mitigates Antibiotic Resistance?
UNLABELLED: can we exploit our burgeoning understanding of molecular evolution to slow the progress of drug resistance? One role of an infection clinician is exactly that: to foresee trajectories to resistance during antibiotic treatment and to hinder that evolutionary course. But can this be done at a hospital-wide scale? Clinicians and theoreticians tried to when they proposed two conflicting behavioral strategies that are expected to curb resistance evolution in the clinic, these are known as "antibiotic cycling" and "antibiotic mixing." However, the accumulated data from clinical trials, now approaching 4 million patient days of treatment, is too variable for cycling or mixing to be deemed successful. The former implements the restriction and prioritization of different antibiotics at different times in hospitals in a manner said to "cycle" between them. In antibiotic mixing, appropriate antibiotics are allocated to patients but randomly. Mixing results in no correlation, in time or across patients, in the drugs used for treatment which is why theorists saw this as an optimal behavioral strategy. So while cycling and mixing were proposed as ways of controlling evolution, we show there is good reason why clinical datasets cannot choose between them: by re-examining the theoretical literature we show prior support for the theoretical optimality of mixing was misplaced. Our analysis is consistent with a pattern emerging in data: neither cycling or mixing is a priori better than the other at mitigating selection for antibiotic resistance in the clinic. KEY WORDS: : antibiotic cycling, antibiotic mixing, optimal control, stochastic models.
Abstract.
Author URL.
Reding-Roman C, Hewlett M, Duxbury S, Gori F, Gudelj I, Beardmore R (2017). The unconstrained evolution of fast and efficient antibiotic-resistant bacterial genomes. Nature Ecology & Evolution, 1(3).
2016
Rashkov P, Barrett IP, Beardmore RE, Bendtsen C, Gudelj I (2016). Kinase Inhibition Leads to Hormesis in a Dual Phosphorylation-Dephosphorylation Cycle.
PLoS Comput Biol,
12(11).
Abstract:
Kinase Inhibition Leads to Hormesis in a Dual Phosphorylation-Dephosphorylation Cycle.
Many antimicrobial and anti-tumour drugs elicit hormetic responses characterised by low-dose stimulation and high-dose inhibition. While this can have profound consequences for human health, with low drug concentrations actually stimulating pathogen or tumour growth, the mechanistic understanding behind such responses is still lacking. We propose a novel, simple but general mechanism that could give rise to hormesis in systems where an inhibitor acts on an enzyme. At its core is one of the basic building blocks in intracellular signalling, the dual phosphorylation-dephosphorylation motif, found in diverse regulatory processes including control of cell proliferation and programmed cell death. Our analytically-derived conditions for observing hormesis provide clues as to why this mechanism has not been previously identified. Current mathematical models regularly make simplifying assumptions that lack empirical support but inadvertently preclude the observation of hormesis. In addition, due to the inherent population heterogeneities, the presence of hormesis is likely to be masked in empirical population-level studies. Therefore, examining hormetic responses at single-cell level coupled with improved mathematical models could substantially enhance detection and mechanistic understanding of hormesis.
Abstract.
Author URL.
2015
Meyer JR, Gudelj I, Beardmore R (2015). Biophysical mechanisms that maintain biodiversity through trade-offs.
Nat Commun,
6Abstract:
Biophysical mechanisms that maintain biodiversity through trade-offs.
Trade-offs are thought to arise from inevitable, biophysical limitations that prevent organisms from optimizing multiple traits simultaneously. A leading explanation for biodiversity maintenance is a theory that if the shape, or geometry, of a trade-off is right, then multiple species can coexist. Testing this theory, however, is difficult as trait data is usually too noisy to discern shape, or trade-offs necessary for the theory are not observed in vivo. To address this, we infer geometry directly from the biophysical mechanisms that cause trade-offs, deriving the geometry of two by studying nutrient uptake and metabolic properties common to all living cells. To test for their presence in vivo we isolated Escherichia coli mutants that vary in a nutrient transporter, LamB, and found evidence for both trade-offs. Consistent with data, population genetics models incorporating the trade-offs successfully predict the co-maintenance of three distinct genetic lineages, demonstrating that trade-off geometry can be deduced from fundamental principles of living cells and used to predict stable genetic polymorphisms.
Abstract.
Author URL.
Roemhild R, Barbosa C, Beardmore RE, Jansen G, Schulenburg H (2015). Temporal variation in antibiotic environments slows down resistance evolution in pathogenic Pseudomonas aeruginosa.
Evolutionary Applications,
8(10), 945-955.
Abstract:
Temporal variation in antibiotic environments slows down resistance evolution in pathogenic Pseudomonas aeruginosa
Antibiotic resistance is a growing concern to public health. New treatment strategies may alleviate the situation by slowing down the evolution of resistance. Here, we evaluated sequential treatment protocols using two fully independent laboratory-controlled evolution experiments with the human pathogen Pseudomonas aeruginosa PA14 and two pairs of clinically relevant antibiotics (doripenem/ciprofloxacin and cefsulodin/gentamicin). Our results consistently show that the sequential application of two antibiotics decelerates resistance evolution relative to monotherapy. Sequential treatment enhanced population extinction although we applied antibiotics at sublethal dosage. In both experiments, we identified an order effect of the antibiotics used in the sequential protocol, leading to significant variation in the long-term efficacy of the tested protocols. These variations appear to be caused by asymmetric evolutionary constraints, whereby adaptation to one drug slowed down adaptation to the other drug, but not vice versa. An understanding of such asymmetric constraints may help future development of evolutionary robust treatments against infectious disease.
Abstract.
Roemhild R, Barbosa C, Beardmore RE, Jansen G, Schulenburg H (2015). Temporal variation in antibiotic environments slows down resistance evolution in pathogenic Pseudomonas aeruginosa.
Evolutionary ApplicationsAbstract:
Temporal variation in antibiotic environments slows down resistance evolution in pathogenic Pseudomonas aeruginosa
Antibiotic resistance is a growing concern to public health. New treatment strategies may alleviate the situation by slowing down the evolution of resistance. Here, we evaluated sequential treatment protocols using two fully independent laboratory-controlled evolution experiments with the human pathogen Pseudomonas aeruginosa PA14 and two pairs of clinically relevant antibiotics (doripenem/ciprofloxacin and cefsulodin/gentamicin). Our results consistently show that the sequential application of two antibiotics decelerates resistance evolution relative to monotherapy. Sequential treatment enhanced population extinction although we applied antibiotics at sublethal dosage. In both experiments, we identified an order effect of the antibiotics used in the sequential protocol, leading to significant variation in the long-term efficacy of the tested protocols. These variations appear to be caused by asymmetric evolutionary constraints, whereby adaptation to one drug slowed down adaptation to the other drug, but not vice versa. An understanding of such asymmetric constraints may help future development of evolutionary robust treatments against infectious disease.
Abstract.
Fuentes-Hernandez A, Plucain J, Gori F, Pena-Miller R, Reding C, Jansen G, Schulenburg H, Gudelj I, Beardmore R (2015). Using a sequential regimen to eliminate bacteria at sublethal antibiotic dosages.
PLoS Biol,
13(4).
Abstract:
Using a sequential regimen to eliminate bacteria at sublethal antibiotic dosages.
We need to find ways of enhancing the potency of existing antibiotics, and, with this in mind, we begin with an unusual question: how low can antibiotic dosages be and yet bacterial clearance still be observed? Seeking to optimise the simultaneous use of two antibiotics, we use the minimal dose at which clearance is observed in an in vitro experimental model of antibiotic treatment as a criterion to distinguish the best and worst treatments of a bacterium, Escherichia coli. Our aim is to compare a combination treatment consisting of two synergistic antibiotics to so-called sequential treatments in which the choice of antibiotic to administer can change with each round of treatment. Using mathematical predictions validated by the E. coli treatment model, we show that clearance of the bacterium can be achieved using sequential treatments at antibiotic dosages so low that the equivalent two-drug combination treatments are ineffective. Seeking to treat the bacterium in testing circumstances, we purposefully study an E. coli strain that has a multidrug pump encoded in its chromosome that effluxes both antibiotics. Genomic amplifications that increase the number of pumps expressed per cell can cause the failure of high-dose combination treatments, yet, as we show, sequentially treated populations can still collapse. However, dual resistance due to the pump means that the antibiotics must be carefully deployed and not all sublethal sequential treatments succeed. A screen of 136 96-h-long sequential treatments determined five of these that could clear the bacterium at sublethal dosages in all replicate populations, even though none had done so by 24 h. These successes can be attributed to a collateral sensitivity whereby cross-resistance due to the duplicated pump proves insufficient to stop a reduction in E. coli growth rate following drug exchanges, a reduction that proves large enough for appropriately chosen drug switches to clear the bacterium.
Abstract.
Author URL.
2014
Arnoldini M, Vizcarra IA, Peña-Miller R, Stocker N, Diard M, Vogel V, Beardmore RE, Hardt W-D, Ackermann M (2014). Bistable expression of virulence genes in salmonella leads to the formation of an antibiotic-tolerant subpopulation.
PLoS Biol,
12(8).
Abstract:
Bistable expression of virulence genes in salmonella leads to the formation of an antibiotic-tolerant subpopulation.
Phenotypic heterogeneity can confer clonal groups of organisms with new functionality. A paradigmatic example is the bistable expression of virulence genes in Salmonella typhimurium, which leads to phenotypically virulent and phenotypically avirulent subpopulations. The two subpopulations have been shown to divide labor during S. typhimurium infections. Here, we show that heterogeneous virulence gene expression in this organism also promotes survival against exposure to antibiotics through a bet-hedging mechanism. Using microfluidic devices in combination with fluorescence time-lapse microscopy and quantitative image analysis, we analyzed the expression of virulence genes at the single cell level and related it to survival when exposed to antibiotics. We found that, across different types of antibiotics and under concentrations that are clinically relevant, the subpopulation of bacterial cells that express virulence genes shows increased survival after exposure to antibiotics. Intriguingly, there is an interplay between the two consequences of phenotypic heterogeneity. The bet-hedging effect that arises through heterogeneity in virulence gene expression can protect clonal populations against avirulent mutants that exploit and subvert the division of labor within these populations. We conclude that bet-hedging and the division of labor can arise through variation in a single trait and interact with each other. This reveals a new degree of functional complexity of phenotypic heterogeneity. In addition, our results suggest a general principle of how pathogens can evade antibiotics: Expression of virulence factors often entails metabolic costs and the resulting growth retardation could generally increase tolerance against antibiotics and thus compromise treatment.
Abstract.
Author URL.
Laehnemann D, Peña-Miller R, Rosenstiel P, Beardmore R, Jansen G, Schulenburg H (2014). Genomics of rapid adaptation to antibiotics: convergent evolution and scalable sequence amplification.
Genome Biol Evol,
6(6), 1287-1301.
Abstract:
Genomics of rapid adaptation to antibiotics: convergent evolution and scalable sequence amplification.
Evolutionary adaptation can be extremely fast, especially in response to high selection intensities. A prime example is the surge of antibiotic resistance in bacteria. The genomic underpinnings of such rapid changes may provide information on the genetic processes that enhance fast responses and the particular trait functions under selection. Here, we use experimentally evolved Escherichia coli for a detailed dissection of the genomics of rapid antibiotic resistance evolution. Our new analyses demonstrate that amplification of a sequence region containing several known antibiotic resistance genes represents a fast genomic response mechanism under high antibiotic stress, here exerted by drug combination. In particular, higher dosage of such antibiotic combinations coincided with higher copy number of the sequence region. The amplification appears to be evolutionarily costly, because amplification levels rapidly dropped after removal of the drugs. Our results suggest that amplification is a scalable process, as copy number rapidly changes in response to the selective pressure encountered. Moreover, repeated patterns of convergent evolution were found across the experimentally evolved bacterial populations, including those with lower antibiotic selection intensities. Intriguingly, convergent evolution was identified on different organizational levels, ranging from the above sequence amplification, high variant frequencies in specific genes, prevalence of individual nonsynonymous mutations to the unusual repeated occurrence of a particular synonymous mutation in Glycine codons. We conclude that constrained evolutionary trajectories underlie rapid adaptation to antibiotics. of the identified genomic changes, sequence amplification seems to represent the most potent, albeit costly genomic response mechanism to high antibiotic stress.
Abstract.
Author URL.
Peña-Miller R, Fuentes-Hernandez A, Reding C, Gudelj I, Beardmore R (2014). Testing the optimality properties of a dual antibiotic treatment in a two-locus, two-allele model.
J R Soc Interface,
11(96).
Abstract:
Testing the optimality properties of a dual antibiotic treatment in a two-locus, two-allele model.
Mathematically speaking, it is self-evident that the optimal control of complex, dynamical systems with many interacting components cannot be achieved with 'non-responsive' control strategies that are constant through time. Although there are notable exceptions, this is usually how we design treatments with antimicrobial drugs when we give the same dose and the same antibiotic combination each day. Here, we use a frequency- and density-dependent pharmacogenetics mathematical model based on a standard, two-locus, two-allele representation of how bacteria resist antibiotics to probe the question of whether optimal antibiotic treatments might, in fact, be constant through time. The model describes the ecological and evolutionary dynamics of different sub-populations of the bacterium Escherichia coli that compete for a single limiting resource in a two-drug environment. We use in vitro evolutionary experiments to calibrate and test the model and show that antibiotic environments can support dynamically changing and heterogeneous population structures. We then demonstrate, theoretically and empirically, that the best treatment strategies should adapt through time and constant strategies are not optimal.
Abstract.
Author URL.
2013
Maharjan R, Nilsson S, Sung J, Haynes K, Beardmore RE, Hurst LD, Ferenci T, Gudelj I (2013). The form of a trade-off determines the response to competition.
Ecology Letters,
16(10), 1267-1276.
Abstract:
The form of a trade-off determines the response to competition
Understanding how populations and communities respond to competition is a central concern of ecology. A seminal theoretical solution first formalised by Levins (and re-derived in multiple fields) showed that, in theory, the form of a trade-off should determine the outcome of competition. While this has become a central postulate in ecology it has evaded experimental verification, not least because of substantial technical obstacles. We here solve the experimental problems by employing synthetic ecology. We engineer strains of Escherichia coli with fixed resource allocations enabling accurate measurement of trade-off shapes between bacterial survival and multiplication in multiple environments. A mathematical chemostat model predicts different, and experimentally verified, trajectories of gene frequency changes as a function of condition-specific trade-offs. The results support Levins' postulate and demonstrates that otherwise paradoxical alternative outcomes witnessed in subtly different conditions are predictable. © 2013 John Wiley & Sons Ltd/CNRS.
Abstract.
Maharjan R, Nilsson S, Sung J, Haynes K, Beardmore RE, Hurst LD, Ferenci T, Gudelj I (2013). The form of a trade-off determines the response to competition.
Ecol Lett,
16(10), 1267-1276.
Abstract:
The form of a trade-off determines the response to competition.
Understanding how populations and communities respond to competition is a central concern of ecology. A seminal theoretical solution first formalised by Levins (and re-derived in multiple fields) showed that, in theory, the form of a trade-off should determine the outcome of competition. While this has become a central postulate in ecology it has evaded experimental verification, not least because of substantial technical obstacles. We here solve the experimental problems by employing synthetic ecology. We engineer strains of Escherichia coli with fixed resource allocations enabling accurate measurement of trade-off shapes between bacterial survival and multiplication in multiple environments. A mathematical chemostat model predicts different, and experimentally verified, trajectories of gene frequency changes as a function of condition-specific trade-offs. The results support Levins' postulate and demonstrates that otherwise paradoxical alternative outcomes witnessed in subtly different conditions are predictable.
Abstract.
Author URL.
Beardmore RE, Pena-Miller R, Laehnemann D, Jansen G, Fuentes-Hernandez A, Rosenstiel P, Schulenburg H (2013). When the most potent combination of antibiotics selects for the greatest bacterial load: the Smile-Frown transition. PLoS Biology, 11(4), 1-13.
2012
Peña-Miller R, Lähnemann D, Schulenburg H, Ackermann M, Beardmore R (2012). Selecting against antibiotic-resistant pathogens: optimal treatments in the presence of commensal bacteria.
Bull Math Biol,
74(4), 908-934.
Abstract:
Selecting against antibiotic-resistant pathogens: optimal treatments in the presence of commensal bacteria.
Using optimal control theory as the basic theoretical tool, we investigate the efficacy of different antibiotic treatment protocols in the most exacting of circumstances, described as follows. Viewing a continuous culture device as a proxy for a much more complex host organism, we first inoculate the device with a single bacterial species and deem this the 'commensal' bacterium of our host. We then force the commensal to compete for a single carbon source with a rapidly evolving and fitter 'pathogenic bacterium', the latter so-named because we wish to use a bacteriostatic antibiotic to drive the pathogen toward low population densities. Constructing a mathematical model to mimic the biology, we do so in such a way that the commensal would be eventually excluded by the pathogen if no antibiotic treatment were given to the host or if the antibiotic were over-deployed. Indeed, in our model, all fixed-dose antibiotic treatment regimens will lead to the eventual loss of the commensal from the host proxy. Despite the obvious gravity of the situation for the commensal bacterium, we show by example that it is possible to design drug deployment protocols that support the commensal and reduce the pathogen load. This may be achieved by appropriately fluctuating the concentration of drug in the environment; a result that is to be anticipated from the theory optimal control where bang-bang solutions may be interpreted as intermittent periods of either maximal and minimal drug deployment. While such 'antibiotic pulsing' is near-optimal for a wide range of treatment objectives, we also use this model to evaluate the efficacy of different antibiotic usage strategies to show that dynamically changing antimicrobial therapies may be effective in clearing a bacterial infection even when every 'static monotherapy' fails.
Abstract.
Author URL.
Peña-Miller R, Lähnemann D, Schulenburg H, Ackermann M, Beardmore R (2012). The optimal deployment of synergistic antibiotics: a control-theoretic approach.
J R Soc Interface,
9(75), 2488-2502.
Abstract:
The optimal deployment of synergistic antibiotics: a control-theoretic approach.
Medical and pharmacological communities have long searched for antimicrobial drugs that increase their effect when used in combination, an interaction known as synergism. These drug combinations, however, impose selective pressures in favour of multi-drug resistance and as a result, the benefit of synergy may be lost after only a few bacterial generations. Furthermore, there is experimental evidence that antibiotic treatment can disrupt colonization resistance by shifting the balance between enteropathogenic and commensal bacteria in favour of the pathogens, with the potential to increase the risk of infections. So, we ask, what is the best way of using synergistic drugs? We pose an evolutionary model of commensal and pathogenic bacteria competing in a continuous culture device for a single limiting carbon source under the effect of two bacteriostatic and synergistic antibiotics. This model allows us to evaluate the efficacy of different drug deployment strategies and, using ideas from optimal control theory, to understand whether there are circumstances in which other types of therapy might be favoured over those based on fixed-dose multi-drug combinations. Our main result can be stated thus: the optimal deployment of synergistic antibiotics to remove a pathogen in the presence of commensal bacteria in our model system occurs not in combination, but by deploying them sequentially.
Abstract.
Author URL.
2011
Beardmore RE, Gudelj I, Lipson DA, Hurst LD (2011). Metabolic trade-offs and the maintenance of the fittest and the flattest.
Nature,
472(7343), 342-346.
Abstract:
Metabolic trade-offs and the maintenance of the fittest and the flattest.
How is diversity maintained? Environmental heterogeneity is considered to be important, yet diversity in seemingly homogeneous environments is nonetheless observed. This, it is assumed, must either be owing to weak selection, mutational input or a fitness advantage to genotypes when rare. Here we demonstrate the possibility of a new general mechanism of stable diversity maintenance, one that stems from metabolic and physiological trade-offs. The model requires that such trade-offs translate into a fitness landscape in which the most fit has unfit near-mutational neighbours, and a lower fitness peak also exists that is more mutationally robust. The 'survival of the fittest' applies at low mutation rates, giving way to 'survival of the flattest' at high mutation rates. However, as a consequence of quasispecies-level negative frequency-dependent selection and differences in mutational robustness we observe a transition zone in which both fittest and flattest coexist. Although diversity maintenance is possible for simple organisms in simple environments, the more trade-offs there are, the wider the maintenance zone becomes. The principle may be applied to lineages within a species or species within a community, potentially explaining why competitive exclusion need not be observed in homogeneous environments. This principle predicts the enigmatic richness of metabolic strategies in clonal bacteria and questions the safety of lethal mutagenesis as an antimicrobial treatment.
Abstract.
Author URL.
2010
Beardmore RE, Pena-Miller R (2010). Antibiotic cycling versus mixing: the difficulty of using mathematical models to definitively quantify their relative merits.
Math Biosci Eng,
7(4), 923-933.
Abstract:
Antibiotic cycling versus mixing: the difficulty of using mathematical models to definitively quantify their relative merits.
We ask the question Which antibiotic deployment protocols select best against drug-resistant microbes: mixing or periodic cycling? and demonstrate that the statistical distribution of the performances of both sets of protocols, mixing and periodic cycling, must have overlapping supports. In other words, it is a general, mathematical result that there must be mixing policies that outperform cycling policies and vice versa. As a result, we agree with the tenet of Bonhoefer et al. [1] that one should not apply the results of [2] to conclude that an antibiotic cycling policy that implements cycles of drug restriction and prioritisation on an ad-hoc basis can select against drug-resistant microbial pathogens in a clinical setting any better than random drug use. However, nor should we conclude that a random, per-patient drug-assignment protocol is the de facto optimal method for allocating antibiotics to patients in any general sense.
Abstract.
Author URL.
Peters R, Beckett N, Beardmore R, Peña-Miller R, Rockwood K, Mitnitski A, Mt-Isa S, Bulpitt C (2010). Modelling cognitive decline in the Hypertension in the Very Elderly Trial [HYVET] and proposed risk tables for population use.
PLoS One,
5(7).
Abstract:
Modelling cognitive decline in the Hypertension in the Very Elderly Trial [HYVET] and proposed risk tables for population use.
INTRODUCTION: Although, on average, cognition declines with age, cognition in older adults is a dynamic process. Hypertension is associated with greater decline in cognition with age, but whether treatment of hypertension affects this is uncertain. Here, we modelled dynamics of cognition in relation to the treatment of hypertension, to see if treatment effects might better be discerned by a model that included baseline measures of cognition and consequent mortality METHODOLOGY/PRINCIPAL FINDINGS: This is a secondary analysis of the Hypertension in the Very Elderly Trial (HYVET), a double blind, placebo controlled trial of indapamide, with or without perindopril, in people aged 80+ years at enrollment. Cognitive states were defined in relation to errors on the Mini-Mental State Examination, with more errors signifying worse cognition. Change in cognitive state was evaluated using a dynamic model of cognitive transition. In the model, the probabilities of transitions between cognitive states is represented by a Poisson distribution, with the Poisson mean dependent on the baseline cognitive state. The dynamic model of cognitive transition was good (R(2) = 0.74) both for those on placebo and (0.86) for those on active treatment. The probability of maintaining cognitive function, based on baseline function, was slightly higher in the actively treated group (e.g. for those with the fewest baseline errors, the chance of staying in that state was 63% for those on treatment, compared with 60% for those on placebo). Outcomes at two and four years could be predicted based on the initial state and treatment. CONCLUSIONS/SIGNIFICANCE: a dynamic model of cognition that allows all outcomes (cognitive worsening, stability improvement or death) to be categorized simultaneously detected small but consistent differences between treatment and control groups (in favour of treatment) amongst very elderly people treated for hypertension. The model showed good fit, and suggests that most change in cognition in very elderly people is small, and depends on their baseline state and on treatment. Additional work is needed to understand whether this modelling approach is well suited to the valuation of small effects, especially in the face of mortality differences between treatment groups. TRIAL REGISTRATION: ClinicalTrials.gov NCT0012281.
Abstract.
Author URL.
Beardmore RE, Peña-Miller R (2010). Rotating antibiotics selects optimally against antibiotic resistance, in theory.
Math Biosci Eng,
7(3), 527-552.
Abstract:
Rotating antibiotics selects optimally against antibiotic resistance, in theory.
The purpose of this paper is to use mathematical models to investigate the claim made in the medical literature over a decade ago that the routine rotation of antibiotics in an intensive care unit (ICU) will select against the evolution and spread of antibiotic-resistant pathogens. In contrast, previous theoretical studies addressing this question have demonstrated that routinely changing the drug of choice for a given pathogenic infection may in fact lead to a greater incidence of drug resistance in comparison to the random deployment of different drugs. Using mathematical models that do not explicitly incorporate the spatial dynamics of pathogen transmission within the ICU or hospital and assuming the antibiotics are from distinct functional groups, we use a control theoretic-approach to prove that one can relax the medical notion of what constitutes an antibiotic rotation and so obtain protocols that are arbitrarily close to the optimum. Finally, we show that theoretical feedback control measures that rotate between different antibiotics motivated directly by the outcome of clinical studies can be deployed to good effect to reduce the prevalence of antibiotic resistance below what can be achieved with random antibiotic use.
Abstract.
Author URL.
2008
Beardmore R, Webster K (2008). A Hopf bifurcation theorem for singular differential-algebraic equations.
Abstract:
A Hopf bifurcation theorem for singular differential-algebraic equations
Abstract.
Beardmore R, Webster K (2008). Normal forms, quasi-invariant manifolds, and bifurcations of nonlinear difference-algebraic equations.
SIAM Journal on Mathematical Analysis,
40(1), 413-441.
Abstract:
Normal forms, quasi-invariant manifolds, and bifurcations of nonlinear difference-algebraic equations
We study the existence of quasi-invariant manifolds in a neighborhood of a fixed point of the difference-algebraic equation (ΔAE) F(zn, zn+1) = 0, where F : &Rdbl;2m → ℝm is a smooth map satisfying F(0, 0) = 0. We demonstrate the existence of quasi-invariant manifolds on which one can define forward and backward orbits of the ΔAE under mild assumptions on its linearization at the fixed point Z = 0. Indeed, by assuming this linearization to be a regular matrix pencil, one obtains a functional equation satisfied by invariant manifolds which can be solved using an extension of the contraction mapping to spaces that satisfy an interpolation property. If the ΔAE under study is permitted to depend smoothly on a parameter, we then obtain a Neimark-Sacker bifurcation theorem as a corollary that can be deduced from the existence of a normal form for nonlinear ΔAEs. © 2008 Society for Industrial and Applied Mathematics.
Abstract.
Forde SE, Beardmore RE, Gudelj I, Arkin SS, Thompson JN, Hurst LD (2008). Understanding the limits to generalizability of experimental evolutionary models.
Nature,
455(7210), 220-223.
Abstract:
Understanding the limits to generalizability of experimental evolutionary models.
Given the difficulty of testing evolutionary and ecological theory in situ, in vitro model systems are attractive alternatives; however, can we appraise whether an experimental result is particular to the in vitro model, and, if so, characterize the systems likely to behave differently and understand why? Here we examine these issues using the relationship between phenotypic diversity and resource input in the T7-Escherichia coli co-evolving system as a case history. We establish a mathematical model of this interaction, framed as one instance of a super-class of host-parasite co-evolutionary models, and show that it captures experimental results. By tuning this model, we then ask how diversity as a function of resource input could behave for alternative co-evolving partners (for example, E. coli with lambda bacteriophages). In contrast to populations lacking bacteriophages, variation in diversity with differences in resources is always found for co-evolving populations, supporting the geographic mosaic theory of co-evolution. The form of this variation is not, however, universal. Details of infectivity are pivotal: in T7-E. coli with a modified gene-for-gene interaction, diversity is low at high resource input, whereas, for matching-allele interactions, maximal diversity is found at high resource input. A combination of in vitro systems and appropriately configured mathematical models is an effective means to isolate results particular to the in vitro system, to characterize systems likely to behave differently and to understand the biology underpinning those alternatives.
Abstract.
Author URL.
2007
Beardmore RE, Peplow A (2007). A bifurcation analysis of the Ornstein-Zernike equation with hypernetted chain closure.
Journal of Mathematical Analysis and Applications,
333(2), 919-942.
Abstract:
A bifurcation analysis of the Ornstein-Zernike equation with hypernetted chain closure
Motivated by the large number of solutions obtained when applying bifurcation algorithms to the Ornstein-Zernike (OZ) equation with the hypernetted chain (HNC) closure from liquid state theory, we provide existence and bifurcation results for a computationally-motivated version of the problem. We first establish the natural result that if the potential satisfies a short-range condition then a low-density branch of smooth solutions exists. We then consider the so-called truncated OZ HNC equation that is obtained when truncating the region occupied by the fluid in the original OZ equation to a finite ball, as is often done in the physics literature before applying a numerical technique. On physical grounds one expects to find one or two solution branches corresponding to vapour and liquid phases of the fluid. However, we are able to demonstrate the existence of infinitely many solution branches and bifurcation points at very low temperatures for the truncated one-dimensional problem provided that the potential is purely repulsive and homogeneous. © 2006 Elsevier Inc. All rights reserved.
Abstract.
Beardmore RE, Peplow AT, Bresme F (2007). A numerical bifurcation analysis of the ornstein-zernike equation with hypernetted chain closure.
SIAM Journal on Scientific Computing,
29(6), 2442-2463.
Abstract:
A numerical bifurcation analysis of the ornstein-zernike equation with hypernetted chain closure
We study the codimension-one and -two bifurcations of the Ornstein-Zernike equation with hypernetted chain (HNC) closure with Lennard-Jones intermolecular interaction potential. The main purpose of the paper is to present the results of a numerical study undertaken using a suite of algorithms implemented in MATLAB and based on pseudo arc-length continuation for the codimension-one case and a Newton-GMRES method for the codimension-two case. Through careful consideration of the results of our computations, an argument is formulated which shows that spinodal isothermal solution branches arising in this model cannot be reproduced numerically. Furthermore, we show that the existence of an upper bound on the density that can be realized on a vapor isothermal solution branch, which must be present at a spinodal, causes the existence of at least one fold bifurcation along that vapor branch when density is used as the bifurcation parameter. This provides an explanation for previous inconclusive attempts to compute solutions using Newton-Picard methods that are popular in the physical chemistry literature. © 2007 Society for Industrial and Applied Mathematics.
Abstract.
Gudelj I, Beardmore RE, Arkin SS, MacLean RC (2007). Constraints on microbial metabolism drive evolutionary diversification in homogeneous environments.
J Evol Biol,
20(5), 1882-1889.
Abstract:
Constraints on microbial metabolism drive evolutionary diversification in homogeneous environments.
Understanding the evolution of microbial diversity is an important and current problem in evolutionary ecology. In this paper, we investigated the role of two established biochemical trade-offs in microbial diversification using a model that connects ecological and evolutionary processes with fundamental aspects of biochemistry. The trade-offs that we investigated are as follows:(1) a trade-off between the rate and affinity of substrate transport; and (2) a trade-off between the rate and yield of ATP production. Our model shows that these biochemical trade-offs can drive evolutionary diversification under the simplest possible ecological conditions: a homogeneous environment containing a single limiting resource. We argue that the results of a number of microbial selection experiments are consistent with the predictions of our model.
Abstract.
Author URL.
2006
Peplow AT, Beardmore RE, Bresme F (2006). Algorithms for the computation of solutions of the Ornstein-Zernike equation.
Phys Rev E Stat Nonlin Soft Matter Phys,
74(4 Pt 2).
Abstract:
Algorithms for the computation of solutions of the Ornstein-Zernike equation.
We introduce a robust and efficient methodology to solve the Ornstein-Zernike integral equation using the pseudoarc length (PAL) continuation method that reformulates the integral equation in an equivalent but nonstandard form. This enables the computation of solutions in regions where the compressibility experiences large changes or where the existence of multiple solutions and so-called branch points prevents Newton's method from converging. We illustrate the use of the algorithm with a difficult problem that arises in the numerical solution of integral equations, namely the evaluation of the so-called no-solution line of the Ornstein-Zernike hypernetted chain (HNC) integral equation for the Lennard-Jones potential. We are able to use the PAL algorithm to solve the integral equation along this line and to connect physical and nonphysical solution branches (both isotherms and isochores) where appropriate. We also show that PAL continuation can compute solutions within the no-solution region that cannot be computed when Newton and Picard methods are applied directly to the integral equation. While many solutions that we find are new, some correspond to states with negative compressibility and consequently are not physical.
Abstract.
Author URL.
Gudelj I, Coman CD, Beardmore RE (2006). Classifying the role of trade-offs in the evolutionary diversity of pathogens.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,
462(2065), 97-116.
Abstract:
Classifying the role of trade-offs in the evolutionary diversity of pathogens
In this paper we use a system of non-local reaction-diffusion equations to study the effect of host heterogeneity on the phenotypic evolution of a pathogen population. The evolving phenotype is taken to be the transmission rate of the pathogen on the different hosts, and in our system there are two host populations present. The central feature of our model is a trade-off relationship between the transmission rates on these hosts, which means that an increase in the pathogen transmission on one host will lead to a decrease in the pathogen transmission on the other. The purpose of the paper is to develop a classification of phenotypic diversity as a function of the shape of the trade-off relationship and this is achieved by determining the maximum number of phenotypes a pathogen population can support in the long term, for a given form of the trade-off. Our findings are then compared with results obtained by applying classical theory from evolutionary ecology and the more recent adaptive dynamics method to the same host-pathogen system. We find our work to be in good agreement with these two approaches. © 2005 the Royal Society.
Abstract.
2005
Laister R, Beardmore RE (2005). Bifurcations in degenerate elliptic equations.
Author URL.
Beardmore RE, Peletier MA, Budd CJ, Wadee MA (2005). Bifurcations of periodic solutions satisfying the zero-Hamiltonian constraint in reversible differential equations.
SIAM Journal on Mathematical Analysis,
36(5), 1461-1488.
Abstract:
Bifurcations of periodic solutions satisfying the zero-Hamiltonian constraint in reversible differential equations
This is a study of the existence of bifurcation branches for the problem of finding even, periodic solutions in fourth-order, reversible Hamiltonian systems such that the Hamiltonian evaluates to zero along each solution on the branch. The class considered here is a generalization of both the Swift-Hohenberg and extended Fisher-Kolmogorov equations that have been studied in several recent papers. We obtain the existence of local bifurcations from a trivial solution under mild restrictions on the nonlinearity and obtain existence and disjointness results regarding the global nature of the resulting bifurcating continua for the case where the Hamiltonian has a single-well potential. The local results rest on two abstract bifurcation theorems which also have applications to sixth-order problems and which show that the curves of zero-Hamiltonian solutions are contained within two-dimensional manifolds of solutions of both negative and positive Hamiltonian. © 2005 Society for Industrial and Applied Mathematics.
Abstract.
2004
Laister R, Peplow AT, Beardmore RE (2004). Finite time extinction in nonlinear diffusion equations.
Applied Mathematics Letters,
17(6), 561-567.
Abstract:
Finite time extinction in nonlinear diffusion equations
We consider a class of degenerate diffusion equations where the nonlinearity is assumed to be singular (non-Lipschitz) at zero. It is shown that solutions with compactly supported initial data become identically zero in finite time. Such extinction follows by comparison with newly constructed finite travelling waves connecting two stable equilibria. © 2004 Elsevier Ltd. All rights reserved.
Abstract.
Laister R, Peplow AT, Beardmore RE (2004). Finite time extinction in nonlinear diffusion equations.
Applied Mathematics Letters,
17(5), 561-567.
Abstract:
Finite time extinction in nonlinear diffusion equations
We consider a class of degenerate diffusion equations where the nonlinearity is assumed to be singular (non-Lipschitz) at zero. It is shown that solutions with compactly supported initial data become identically zero in finite time. Such extinction follows by comparison with newly constructed finite travelling waves connecting two stable equilibria. © 2004 Elsevier Ltd. All rights reserved.
Abstract.
Beardmore RE, Laister R (2004). Sequential and continuum bifurcations in degenerate elliptic equations.
Proceedings of the American Mathematical Society,
132(1), 165-174.
Abstract:
Sequential and continuum bifurcations in degenerate elliptic equations
We examine the bifurcations to positive and sign-changing solutions of degenerate elliptic equations. In the problems we study, which do not represent Fredholm operators, we show that there is a critical parameter value at which an infinity of bifurcations occur from the trivial solution. Moreover, a bifurcation occurs at each point in some unbounded interval in parameter space. We apply our results to non-monotone eigenvalue problems, degenerate semi-linear elliptic equations, boundary value differential-algebraic equations and fully non-linear elliptic equations.
Abstract.
Beardmore RE, Laister R, Peplow A (2004). Trajectories of a DAE near a pseudo-equilibrium.
Nonlinearity,
17(1), 253-279.
Abstract:
Trajectories of a DAE near a pseudo-equilibrium
We consider a class of differential-algebraic equations (DAEs) defined by analytic nonlinearities and study its singular solutions. The main assumption used is that the linearization of the DAE represents a Kronecker index-2 matrix pencil and that the constraint manifold has a quadratic fold along its singularity. From these assumptions we obtain a normal form for the DAE where the presence of the singularity and its effects on the dynamics of the problem are made explicit in the form of a quasi-linear differential equation. Subsequently, two distinct types of singular points are identified through which there pass exactly two analytic solutions: pseudo-nodes and pseudo-saddles. We also demonstrate that a singular point called a pseudo-node supports an uncountable infinity of solutions which are not analytic in general. Moreover, akin to known results in the literature for DAEs with singular equilibria, a degenerate singularity is found through which there passes one analytic solution such that the singular point in question is contained within a quasi-invariant manifold of solutions. We call this type of singularity a pseudo-centre and it provides not only a manifold of solutions which intersects the singularity, but also a local flow on that manifold which solves the DAE.
Abstract.
2003
Beardmore RE, Laister R (2003). The flow of a DAE near a singular equilibrium.
SIAM Journal on Matrix Analysis and Applications,
24(1), 106-120.
Abstract:
The flow of a DAE near a singular equilibrium
We extend the differential-algebraic equation (DAE) taxonomy by assuming that the linearization of a DAE about a singular equilibrium has a particular index-2 Kronecker normal form. A Lyapunov-Schmidt procedure is used to reduce the DAE to a quasilinear normal form which is shown to posses quasi-invariant manifolds which intersect the singularity. In turn, this provides solutions of the DAE which pass through the singularity.
Abstract.
Beardmore I, Beardmore R (2003). The global structure of a spatial model of infectious disease.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,
459(2034), 1427-1448.
Abstract:
The global structure of a spatial model of infectious disease
In this paper we study an SI model of infectious diseases that takes into account spatial inhomogeneities, resulting in a system of reaction-convection- diffusion equations on a bounded domain. The convection process is included to account for social interaction, as modelled by the location of a focal point or den where the population will tend to aggregate. We show that a vertical bifurcation of steady-state solutions occurs in this model when birth rate is taken as the bifurcation parameter, from which emanates a global secondary branch, which then bifurcates at infinity. Subsequently, we use singular perturbation techniques to give a description of the limiting spatial structure along this branch in large and small parameter limits. Finally, the results are illustrated numerically on some biologically relevant cases.
Abstract.
Laister R, Beardmore RE (2003). Transversality and separation of zeros in second order differential equations.
Proceedings of the American Mathematical Society,
131(1), 209-218.
Abstract:
Transversality and separation of zeros in second order differential equations
Sufficient conditions on the non-linearity f are given which ensure that non-trivial solutions of second order differential equations of the form Lu = f(t, u) have a finite number of transverse zeros in a given finite time interval. We also obtain a priori lower bounds on the separation of zeros of solutions. In particular our results apply to non-Lipschitz non-linearities. Applications to non-linear porous medium equations are considered, yielding information on the existence and strict positivity of equilibrium solutions in some important classes of equations.
Abstract.
2002
Beardmore RE (2002). The singularity-induced bifurcation and its Kronecker normal form.
SIAM Journal on Matrix Analysis and Applications,
23(1), 126-137.
Abstract:
The singularity-induced bifurcation and its Kronecker normal form
It is shown that the singularity-induced bifurcation theorem due to Venkatasubramanian, Schattler, and Zaborszky [Proceedings of the IEEE. 83 (1995), pp. 1530-1558] can be expressed as the perturbation of an infinite eigenvalue of a particular class of parameterized index-1 matrix pencil, denoted (M, L(λ)). It is shown that the matrix pencil at the singularity-induced bifurcation point, (M, L(λ0)), has Kronecker index-2. It is also shown that a two-parameter unfolding of a singularity-induced bifurcation point results in a locus of index-0 pencils, denoted (M(ε), L(λ(ε))), which has two purely imaginary eigenvalues near infinity.
Abstract.
2001
Beardmore RE (2001). A singularity-induced bifurcation theorem for infinite-dimensional implicit dynamical systems.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,
457(2010), 1295-1305.
Abstract:
A singularity-induced bifurcation theorem for infinite-dimensional implicit dynamical systems
In this note we study a class of partial differential equations whose formal structure resembles the semi-explicit, index-1 form from the theory of differential-algebraic equations. It is known from the theory of such problems that one can encounter simple poles in the eigenvalue loci of parameter-dependent linearizations, leading to the loss of linear stability without the presence of steady-state bifurcations. We demonstrate how this carries over to implicitly defined problems on a Hilbert space.
Abstract.
2000
Beardmore RE (2000). Double singularity-induced bifurcation points and singular Hopf bifurcations.
Dynamical Systems,
15(4), 319-342.
Abstract:
Double singularity-induced bifurcation points and singular Hopf bifurcations
The singularity-induced bifurcation and singular Hopf bifurcation theorems and the degeneracies that arise when Newton's laws are coupled to Kirchhoff's laws are explored. Such models are used in the electrical engineering literature to describe electrical power systems and they can take the form of either an index-1 differential-algebraic equation (DAE) or a singularly perturbed ordinary differential equation (ODE). As a consequence of the debate in the engineering literature as to which class of system is the 'true' representation of power systems, a discussion is included of the consequences of the power engineer's 'load-flow singularity' for both ODE and DAE.
Abstract.
1998
Beardmore RE, Song YH (1998). Differential-algebraic equations: a tutorial review.
International Journal of Bifurcation and Chaos in Applied Sciences and Engineering,
8(7), 1399-1411.
Abstract:
Differential-algebraic equations: a tutorial review
This article1 explores some introductory principles of differential-algebraic equations (DAEs) and makes a connection with the theory of dynamical systems. Some results which are new in the field of DAEs are also surveyed. Most treatments on DAE emphasize the differences that exist when compared with the ODE case. Here we seek to underline the similarities so that readers with a very basic knowledge of nonlinear dynamics can understand some of their consequences in this more general context.
Abstract.
Beardmore RE (1998). Stability and bifurcation properties of index-1 DAEs.
Numerical Algorithms,
19(1-4), 43-53.
Abstract:
Stability and bifurcation properties of index-1 DAEs
It is well known that an equilibrium of a semi-explicit, index-1 differential-algebraic equation under a parameter variation may encounter the singularity manifold. It is a generic property of this encounter that one eigenvalue of the linear stability mapping associated with the equilibrium will pass from one half of the complex plane to the other without passing through the imaginary axis. This is known as singularity-induced bifurcation and an equivalent result is proven in this paper. While this property is generic, it is shown how more than one eigenvalue can diverge in an analogous manner, with applications in electrical power systems.
Abstract.
