PeerJ:Mathematical Biologyhttps://peerj.com/articles/index.atom?journal=peerj&subject=1900Mathematical Biology articles published in PeerJCoccolith arrangement follows Eulerian mathematics in the coccolithophore Emiliania huxleyihttps://peerj.com/articles/46082018-04-092018-04-09Kai XuDavid HutchinsKunshan Gao
Background
The globally abundant coccolithophore, Emiliania huxleyi, plays an important ecological role in oceanic carbon biogeochemistry by forming a cellular covering of plate-like CaCO3 crystals (coccoliths) and fixing CO2. It is unknown how the cells arrange different-sized coccoliths to maintain full coverage, as the cell surface area of the cell changes during daily cycle.
Methods
We used Euler’s polyhedron formula and CaGe simulation software, validated with the geometries of coccoliths, to analyze and simulate the coccolith topology of the coccosphere and to explore the arrangement mechanisms.
Results
There were only small variations in the geometries of coccoliths, even when the cells were cultured under variable light conditions. Because of geometric limits, small coccoliths tended to interlock with fewer and larger coccoliths, and vice versa. Consequently, to sustain a full coverage on the surface of cell, each coccolith was arranged to interlock with four to six others, which in turn led to each coccosphere contains at least six coccoliths.
Conclusion
The number of coccoliths per coccosphere must keep pace with changes on the cell surface area as a result of photosynthesis, respiration and cell division. This study is an example of natural selection following Euler’s polyhedral formula, in response to the challenge of maintaining a CaCO3 covering on coccolithophore cells as cell size changes.
Background
The globally abundant coccolithophore, Emiliania huxleyi, plays an important ecological role in oceanic carbon biogeochemistry by forming a cellular covering of plate-like CaCO3 crystals (coccoliths) and fixing CO2. It is unknown how the cells arrange different-sized coccoliths to maintain full coverage, as the cell surface area of the cell changes during daily cycle.
Methods
We used Euler’s polyhedron formula and CaGe simulation software, validated with the geometries of coccoliths, to analyze and simulate the coccolith topology of the coccosphere and to explore the arrangement mechanisms.
Results
There were only small variations in the geometries of coccoliths, even when the cells were cultured under variable light conditions. Because of geometric limits, small coccoliths tended to interlock with fewer and larger coccoliths, and vice versa. Consequently, to sustain a full coverage on the surface of cell, each coccolith was arranged to interlock with four to six others, which in turn led to each coccosphere contains at least six coccoliths.
Conclusion
The number of coccoliths per coccosphere must keep pace with changes on the cell surface area as a result of photosynthesis, respiration and cell division. This study is an example of natural selection following Euler’s polyhedral formula, in response to the challenge of maintaining a CaCO3 covering on coccolithophore cells as cell size changes.Uncertainty and sensitivity analysis of the basic reproduction number of diphtheria: a case study of a Rohingya refugee camp in Bangladesh, November–December 2017https://peerj.com/articles/45832018-04-022018-04-02Ryota MatsuyamaAndrei R. AkhmetzhanovAkira EndoHyojung LeeTakayuki YamaguchiShinya TsuzukiHiroshi Nishiura
Background
A Rohingya refugee camp in Cox’s Bazar, Bangladesh experienced a large-scale diphtheria epidemic in 2017. The background information of previously immune fraction among refugees cannot be explicitly estimated, and thus we conducted an uncertainty analysis of the basic reproduction number, R0.
Methods
A renewal process model was devised to estimate the R0 and ascertainment rate of cases, and loss of susceptible individuals was modeled as one minus the sum of initially immune fraction and the fraction naturally infected during the epidemic. To account for the uncertainty of initially immune fraction, we employed a Latin Hypercube sampling (LHS) method.
Results
R0 ranged from 4.7 to 14.8 with the median estimate at 7.2. R0 was positively correlated with ascertainment rates. Sensitivity analysis indicated that R0 would become smaller with greater variance of the generation time.
Discussion
Estimated R0 was broadly consistent with published estimate from endemic data, indicating that the vaccination coverage of 86% has to be satisfied to prevent the epidemic by means of mass vaccination. LHS was particularly useful in the setting of a refugee camp in which the background health status is poorly quantified.
Background
A Rohingya refugee camp in Cox’s Bazar, Bangladesh experienced a large-scale diphtheria epidemic in 2017. The background information of previously immune fraction among refugees cannot be explicitly estimated, and thus we conducted an uncertainty analysis of the basic reproduction number, R0.
Methods
A renewal process model was devised to estimate the R0 and ascertainment rate of cases, and loss of susceptible individuals was modeled as one minus the sum of initially immune fraction and the fraction naturally infected during the epidemic. To account for the uncertainty of initially immune fraction, we employed a Latin Hypercube sampling (LHS) method.
Results
R0 ranged from 4.7 to 14.8 with the median estimate at 7.2. R0 was positively correlated with ascertainment rates. Sensitivity analysis indicated that R0 would become smaller with greater variance of the generation time.
Discussion
Estimated R0 was broadly consistent with published estimate from endemic data, indicating that the vaccination coverage of 86% has to be satisfied to prevent the epidemic by means of mass vaccination. LHS was particularly useful in the setting of a refugee camp in which the background health status is poorly quantified.Genome rearrangements and phylogeny reconstruction in Yersinia pestishttps://peerj.com/articles/45452018-03-272018-03-27Olga O. BochkarevaNatalia O. DranenkoElena S. OcheredkoGerman M. KanevskyYaroslav N. LozinskyVera A. KhalaychevaIrena I. ArtamonovaMikhail S. Gelfand
Genome rearrangements have played an important role in the evolution of Yersinia pestis from its progenitor Yersinia pseudotuberculosis. Traditional phylogenetic trees for Y. pestis based on sequence comparison have short internal branches and low bootstrap supports as only a small number of nucleotide substitutions have occurred. On the other hand, even a small number of genome rearrangements may resolve topological ambiguities in a phylogenetic tree. We reconstructed phylogenetic trees based on genome rearrangements using several popular approaches such as Maximum likelihood for Gene Order and the Bayesian model of genome rearrangements by inversions. We also reconciled phylogenetic trees for each of the three CRISPR loci to obtain an integrated scenario of the CRISPR cassette evolution. Analysis of contradictions between the obtained evolutionary trees yielded numerous parallel inversions and gain/loss events. Our data indicate that an integrated analysis of sequence-based and inversion-based trees enhances the resolution of phylogenetic reconstruction. In contrast, reconstructions of strain relationships based on solely CRISPR loci may not be reliable, as the history is obscured by large deletions, obliterating the order of spacer gains. Similarly, numerous parallel gene losses preclude reconstruction of phylogeny based on gene content.
Genome rearrangements have played an important role in the evolution of Yersinia pestis from its progenitor Yersinia pseudotuberculosis. Traditional phylogenetic trees for Y. pestis based on sequence comparison have short internal branches and low bootstrap supports as only a small number of nucleotide substitutions have occurred. On the other hand, even a small number of genome rearrangements may resolve topological ambiguities in a phylogenetic tree. We reconstructed phylogenetic trees based on genome rearrangements using several popular approaches such as Maximum likelihood for Gene Order and the Bayesian model of genome rearrangements by inversions. We also reconciled phylogenetic trees for each of the three CRISPR loci to obtain an integrated scenario of the CRISPR cassette evolution. Analysis of contradictions between the obtained evolutionary trees yielded numerous parallel inversions and gain/loss events. Our data indicate that an integrated analysis of sequence-based and inversion-based trees enhances the resolution of phylogenetic reconstruction. In contrast, reconstructions of strain relationships based on solely CRISPR loci may not be reliable, as the history is obscured by large deletions, obliterating the order of spacer gains. Similarly, numerous parallel gene losses preclude reconstruction of phylogeny based on gene content.Dynamics of Zika virus outbreaks: an overview of mathematical modeling approacheshttps://peerj.com/articles/45262018-03-222018-03-22Anuwat WiratsudakulParinya SuparitCharin Modchang
Background
The Zika virus was first discovered in 1947. It was neglected until a major outbreak occurred on Yap Island, Micronesia, in 2007. Teratogenic effects resulting in microcephaly in newborn infants is the greatest public health threat. In 2016, the Zika virus epidemic was declared as a Public Health Emergency of International Concern (PHEIC). Consequently, mathematical models were constructed to explicitly elucidate related transmission dynamics.
Survey Methodology
In this review article, two steps of journal article searching were performed. First, we attempted to identify mathematical models previously applied to the study of vector-borne diseases using the search terms “dynamics,” “mathematical model,” “modeling,” and “vector-borne” together with the names of vector-borne diseases including chikungunya, dengue, malaria, West Nile, and Zika. Then the identified types of model were further investigated. Second, we narrowed down our survey to focus on only Zika virus research. The terms we searched for were “compartmental,” “spatial,” “metapopulation,” “network,” “individual-based,” “agent-based” AND “Zika.” All relevant studies were included regardless of the year of publication. We have collected research articles that were published before August 2017 based on our search criteria. In this publication survey, we explored the Google Scholar and PubMed databases.
Results
We found five basic model architectures previously applied to vector-borne virus studies, particularly in Zika virus simulations. These include compartmental, spatial, metapopulation, network, and individual-based models. We found that Zika models carried out for early epidemics were mostly fit into compartmental structures and were less complicated compared to the more recent ones. Simple models are still commonly used for the timely assessment of epidemics. Nevertheless, due to the availability of large-scale real-world data and computational power, recently there has been growing interest in more complex modeling frameworks.
Discussion
Mathematical models are employed to explore and predict how an infectious disease spreads in the real world, evaluate the disease importation risk, and assess the effectiveness of intervention strategies. As the trends in modeling of infectious diseases have been shifting towards data-driven approaches, simple and complex models should be exploited differently. Simple models can be produced in a timely fashion to provide an estimation of the possible impacts. In contrast, complex models integrating real-world data require more time to develop but are far more realistic. The preparation of complicated modeling frameworks prior to the outbreaks is recommended, including the case of future Zika epidemic preparation.
Background
The Zika virus was first discovered in 1947. It was neglected until a major outbreak occurred on Yap Island, Micronesia, in 2007. Teratogenic effects resulting in microcephaly in newborn infants is the greatest public health threat. In 2016, the Zika virus epidemic was declared as a Public Health Emergency of International Concern (PHEIC). Consequently, mathematical models were constructed to explicitly elucidate related transmission dynamics.
Survey Methodology
In this review article, two steps of journal article searching were performed. First, we attempted to identify mathematical models previously applied to the study of vector-borne diseases using the search terms “dynamics,” “mathematical model,” “modeling,” and “vector-borne” together with the names of vector-borne diseases including chikungunya, dengue, malaria, West Nile, and Zika. Then the identified types of model were further investigated. Second, we narrowed down our survey to focus on only Zika virus research. The terms we searched for were “compartmental,” “spatial,” “metapopulation,” “network,” “individual-based,” “agent-based” AND “Zika.” All relevant studies were included regardless of the year of publication. We have collected research articles that were published before August 2017 based on our search criteria. In this publication survey, we explored the Google Scholar and PubMed databases.
Results
We found five basic model architectures previously applied to vector-borne virus studies, particularly in Zika virus simulations. These include compartmental, spatial, metapopulation, network, and individual-based models. We found that Zika models carried out for early epidemics were mostly fit into compartmental structures and were less complicated compared to the more recent ones. Simple models are still commonly used for the timely assessment of epidemics. Nevertheless, due to the availability of large-scale real-world data and computational power, recently there has been growing interest in more complex modeling frameworks.
Discussion
Mathematical models are employed to explore and predict how an infectious disease spreads in the real world, evaluate the disease importation risk, and assess the effectiveness of intervention strategies. As the trends in modeling of infectious diseases have been shifting towards data-driven approaches, simple and complex models should be exploited differently. Simple models can be produced in a timely fashion to provide an estimation of the possible impacts. In contrast, complex models integrating real-world data require more time to develop but are far more realistic. The preparation of complicated modeling frameworks prior to the outbreaks is recommended, including the case of future Zika epidemic preparation.Use and misuse of temperature normalisation in meta-analyses of thermal responses of biological traitshttps://peerj.com/articles/43632018-02-092018-02-09Dimitrios - Georgios KontopoulosBernardo García-CarrerasSofía SalThomas P. SmithSamraat Pawar
There is currently unprecedented interest in quantifying variation in thermal physiology among organisms, especially in order to understand and predict the biological impacts of climate change. A key parameter in this quantification of thermal physiology is the performance or value of a rate, across individuals or species, at a common temperature (temperature normalisation). An increasingly popular model for fitting thermal performance curves to data—the Sharpe-Schoolfield equation—can yield strongly inflated estimates of temperature-normalised rate values. These deviations occur whenever a key thermodynamic assumption of the model is violated, i.e., when the enzyme governing the performance of the rate is not fully functional at the chosen reference temperature. Using data on 1,758 thermal performance curves across a wide range of species, we identify the conditions that exacerbate this inflation. We then demonstrate that these biases can compromise tests to detect metabolic cold adaptation, which requires comparison of fitness or rate performance of different species or genotypes at some fixed low temperature. Finally, we suggest alternative methods for obtaining unbiased estimates of temperature-normalised rate values for meta-analyses of thermal performance across species in climate change impact studies.
There is currently unprecedented interest in quantifying variation in thermal physiology among organisms, especially in order to understand and predict the biological impacts of climate change. A key parameter in this quantification of thermal physiology is the performance or value of a rate, across individuals or species, at a common temperature (temperature normalisation). An increasingly popular model for fitting thermal performance curves to data—the Sharpe-Schoolfield equation—can yield strongly inflated estimates of temperature-normalised rate values. These deviations occur whenever a key thermodynamic assumption of the model is violated, i.e., when the enzyme governing the performance of the rate is not fully functional at the chosen reference temperature. Using data on 1,758 thermal performance curves across a wide range of species, we identify the conditions that exacerbate this inflation. We then demonstrate that these biases can compromise tests to detect metabolic cold adaptation, which requires comparison of fitness or rate performance of different species or genotypes at some fixed low temperature. Finally, we suggest alternative methods for obtaining unbiased estimates of temperature-normalised rate values for meta-analyses of thermal performance across species in climate change impact studies.Complex versus simple models: ion-channel cardiac toxicity predictionhttps://peerj.com/articles/43522018-02-052018-02-05Hitesh B. Mistry
There is growing interest in applying detailed mathematical models of the heart for ion-channel related cardiac toxicity prediction. However, a debate as to whether such complex models are required exists. Here an assessment in the predictive performance between two established large-scale biophysical cardiac models and a simple linear model Bnet was conducted. Three ion-channel data-sets were extracted from literature. Each compound was designated a cardiac risk category using two different classification schemes based on information within CredibleMeds. The predictive performance of each model within each data-set for each classification scheme was assessed via a leave-one-out cross validation. Overall the Bnet model performed equally as well as the leading cardiac models in two of the data-sets and outperformed both cardiac models on the latest. These results highlight the importance of benchmarking complex versus simple models but also encourage the development of simple models.
There is growing interest in applying detailed mathematical models of the heart for ion-channel related cardiac toxicity prediction. However, a debate as to whether such complex models are required exists. Here an assessment in the predictive performance between two established large-scale biophysical cardiac models and a simple linear model Bnet was conducted. Three ion-channel data-sets were extracted from literature. Each compound was designated a cardiac risk category using two different classification schemes based on information within CredibleMeds. The predictive performance of each model within each data-set for each classification scheme was assessed via a leave-one-out cross validation. Overall the Bnet model performed equally as well as the leading cardiac models in two of the data-sets and outperformed both cardiac models on the latest. These results highlight the importance of benchmarking complex versus simple models but also encourage the development of simple models.Sicegar: R package for sigmoidal and double-sigmoidal curve fittinghttps://peerj.com/articles/42512018-01-162018-01-16M. Umut CaglarAshley I. TeufelClaus O. Wilke
Sigmoidal and double-sigmoidal dynamics are commonly observed in many areas of biology. Here we present sicegar, an R package for the automated fitting and classification of sigmoidal and double-sigmoidal data. The package categorizes data into one of three categories, “no signal,” “sigmoidal,” or “double-sigmoidal,” by rigorously fitting a series of mathematical models to the data. The data is labeled as “ambiguous” if neither the sigmoidal nor double-sigmoidal model fit the data well. In addition to performing the classification, the package also reports a wealth of metrics as well as biologically meaningful parameters describing the sigmoidal or double-sigmoidal curves. In extensive simulations, we find that the package performs well, can recover the original dynamics even under fairly high noise levels, and will typically classify curves as “ambiguous” rather than misclassifying them. The package is available on CRAN and comes with extensive documentation and usage examples.
Sigmoidal and double-sigmoidal dynamics are commonly observed in many areas of biology. Here we present sicegar, an R package for the automated fitting and classification of sigmoidal and double-sigmoidal data. The package categorizes data into one of three categories, “no signal,” “sigmoidal,” or “double-sigmoidal,” by rigorously fitting a series of mathematical models to the data. The data is labeled as “ambiguous” if neither the sigmoidal nor double-sigmoidal model fit the data well. In addition to performing the classification, the package also reports a wealth of metrics as well as biologically meaningful parameters describing the sigmoidal or double-sigmoidal curves. In extensive simulations, we find that the package performs well, can recover the original dynamics even under fairly high noise levels, and will typically classify curves as “ambiguous” rather than misclassifying them. The package is available on CRAN and comes with extensive documentation and usage examples.On the exponent in the Von Bertalanffy growth modelhttps://peerj.com/articles/42052018-01-042018-01-04Katharina Renner-MartinNorbert BrunnerManfred KühleitnerWerner Georg NowakKlaus Scheicher
Von Bertalanffy proposed the differential equation m′(t) = p × m(t)a − q × m(t) for the description of the mass growth of animals as a function m(t) of time t. He suggested that the solution using the metabolic scaling exponent a = 2/3 (Von Bertalanffy growth function VBGF) would be universal for vertebrates. Several authors questioned universality, as for certain species other models would provide a better fit. This paper reconsiders this question. Based on 60 data sets from literature (37 about fish and 23 about non-fish species) it optimizes the model parameters, in particular the exponent 0 ≤ a < 1, so that the model curve achieves the best fit to the data. The main observation of the paper is the large variability in the exponent, which can vary over a very large range without affecting the fit to the data significantly, when the other parameters are also optimized. The paper explains this by differences in the data quality: variability is low for data from highly controlled experiments and high for natural data. Other deficiencies were biologically meaningless optimal parameter values or optimal parameter values attained on the boundary of the parameter region (indicating the possible need for a different model). Only 11 of the 60 data sets were free of such deficiencies and for them no universal exponent could be discerned.
Von Bertalanffy proposed the differential equation m′(t) = p × m(t)a − q × m(t) for the description of the mass growth of animals as a function m(t) of time t. He suggested that the solution using the metabolic scaling exponent a = 2/3 (Von Bertalanffy growth function VBGF) would be universal for vertebrates. Several authors questioned universality, as for certain species other models would provide a better fit. This paper reconsiders this question. Based on 60 data sets from literature (37 about fish and 23 about non-fish species) it optimizes the model parameters, in particular the exponent 0 ≤ a < 1, so that the model curve achieves the best fit to the data. The main observation of the paper is the large variability in the exponent, which can vary over a very large range without affecting the fit to the data significantly, when the other parameters are also optimized. The paper explains this by differences in the data quality: variability is low for data from highly controlled experiments and high for natural data. Other deficiencies were biologically meaningless optimal parameter values or optimal parameter values attained on the boundary of the parameter region (indicating the possible need for a different model). Only 11 of the 60 data sets were free of such deficiencies and for them no universal exponent could be discerned.Neither slim nor fat: estimating the mass of the dodo (Raphus cucullatus, Aves, Columbiformes) based on the largest sample of dodo bones to datehttps://peerj.com/articles/41102017-12-052017-12-05Anneke H. van HeterenRoland C.H. van DierendonckMaria A.N.E. van EgmondSjang L. ten HagenJippe Kreuning
The dodo (Raphus cucullatus) might be the most enigmatic bird of all times. It is, therefore, highly remarkable that no consensus has yet been reached on its body mass; previous scientific estimates of its mass vary by more than 100%. Until now, the vast amount of bones stored at the Natural History Museum in Mauritius has not yet been studied morphometrically nor in relation to body mass. Here, a new estimate of the dodo’s mass is presented based on the largest sample of dodo femora ever measured (n = 174). In order to do this, we have used the regression method and chosen our variables based on biological, mathematical and physical arguments. The results indicate that the mean mass of the dodo was circa 12 kg, which is approximately five times as heavy as the largest living Columbidae (pigeons and doves), the clade to which the dodo belongs.
The dodo (Raphus cucullatus) might be the most enigmatic bird of all times. It is, therefore, highly remarkable that no consensus has yet been reached on its body mass; previous scientific estimates of its mass vary by more than 100%. Until now, the vast amount of bones stored at the Natural History Museum in Mauritius has not yet been studied morphometrically nor in relation to body mass. Here, a new estimate of the dodo’s mass is presented based on the largest sample of dodo femora ever measured (n = 174). In order to do this, we have used the regression method and chosen our variables based on biological, mathematical and physical arguments. The results indicate that the mean mass of the dodo was circa 12 kg, which is approximately five times as heavy as the largest living Columbidae (pigeons and doves), the clade to which the dodo belongs.On the relationship between tumour growth rate and survival in non-small cell lung cancerhttps://peerj.com/articles/41112017-11-292017-11-29Hitesh B. Mistry
A recurrent question within oncology drug development is predicting phase III outcome for a new treatment using early clinical data. One approach to tackle this problem has been to derive metrics from mathematical models that describe tumour size dynamics termed re-growth rate and time to tumour re-growth. They have shown to be strong predictors of overall survival in numerous studies but there is debate about how these metrics are derived and if they are more predictive than empirical end-points. This work explores the issues raised in using model-derived metric as predictors for survival analyses. Re-growth rate and time to tumour re-growth were calculated for three large clinical studies by forward and reverse alignment. The latter involves re-aligning patients to their time of progression. Hence, it accounts for the time taken to estimate re-growth rate and time to tumour re-growth but also assesses if these predictors correlate to survival from the time of progression. I found that neither re-growth rate nor time to tumour re-growth correlated to survival using reverse alignment. This suggests that the dynamics of tumours up until disease progression has no relationship to survival post progression. For prediction of a phase III trial I found the metrics performed no better than empirical end-points. These results highlight that care must be taken when relating dynamics of tumour imaging to survival and that bench-marking new approaches to existing ones is essential.
A recurrent question within oncology drug development is predicting phase III outcome for a new treatment using early clinical data. One approach to tackle this problem has been to derive metrics from mathematical models that describe tumour size dynamics termed re-growth rate and time to tumour re-growth. They have shown to be strong predictors of overall survival in numerous studies but there is debate about how these metrics are derived and if they are more predictive than empirical end-points. This work explores the issues raised in using model-derived metric as predictors for survival analyses. Re-growth rate and time to tumour re-growth were calculated for three large clinical studies by forward and reverse alignment. The latter involves re-aligning patients to their time of progression. Hence, it accounts for the time taken to estimate re-growth rate and time to tumour re-growth but also assesses if these predictors correlate to survival from the time of progression. I found that neither re-growth rate nor time to tumour re-growth correlated to survival using reverse alignment. This suggests that the dynamics of tumours up until disease progression has no relationship to survival post progression. For prediction of a phase III trial I found the metrics performed no better than empirical end-points. These results highlight that care must be taken when relating dynamics of tumour imaging to survival and that bench-marking new approaches to existing ones is essential.