On the exclusion of exponential autocatalysts by sub-exponential autocatalysts

Selection among autocatalytic species fundamentally depends on their growth law: exponential species, whose number of copies grows exponentially, are mutually exclusive, while sub-exponential ones, whose number of copies grows polynomially, can coexist. Here we consider competitions between autocatalytic species with different growth laws and make the simple yet counterintuitive observation that sub-exponential species can exclude exponential ones while the reverse is, in principle, impossible. This observation has implications for scenarios pertaining to the emergence of natural selection.

Simulating tumor volume dynamics in response to radiotherapy: Implications of model selection

From the beginning of the usage of radiotherapy (RT) for cancer treatment, mathematical modeling has been integral to understanding radiobiology and for designing treatment approaches and schedules. There has been extensive modeling of response to RT with the inclusion of various degrees of biological complexity. In this study, we compare three models of tumor volume dynamics: (1) exponential growth with RT directly reducing tumor volume, (2) logistic growth with direct tumor volume reduction, and (3) logistic growth with RT reducing the tumor carrying capacity with the objective of understanding the implications of model selection and informing the process of model calibration and parameterization. For all three models, we: examined the rates of change in tumor volume during and RT treatment course; performed parameter sensitivity and identifiability analyses; and investigated the impact of the parameter sensitivity on the tumor volume trajectories. In examining the tumor volume dynamics trends, we coined a new metric - the point of maximum reduction of tumor volume (MRV) - to quantify the magnitude and timing of the expected largest impact of RT during a treatment course. We found distinct timing differences in MRV, dependent on model selection. The parameter identifiability and sensitivity analyses revealed the interdependence of the different model parameters and that it is only possible to independently identify tumor growth and radiation response parameters if the underlying tumor growth rate is sufficiently large. Ultimately, the results of these analyses help us to better understand the implications of model selection while simultaneously generating falsifiable hypotheses about MRV timing that can be tested on longitudinal measurements of tumor volume from pre-clinical or clinical data with high acquisition frequency. Although, our study only compares three particular models, the results demonstrate that caution is necessary in selecting models of response to RT, given the artifacts imposed by each model.

Doubling time of infectious diseases

The concept of doubling time has been increasingly used since the onset of the coronavirus disease 2019 (COVID-19) pandemic, but its characteristics are not well understood, especially as applied to infectious disease epidemiology. The present study aims to be a practical guide to monitoring the doubling time of infectious diseases. Via simulation exercise, we clarify the epidemiological characteristics of doubling time, allowing possible interpretations. We show that the commonly believed relationship between the doubling time and intrinsic growth rate in population ecology does not strictly apply to infectious diseases, and derive the correct relationship between the two. We examined the impact of varying (i) the growth rate, (ii) the starting point of counting cumulative number of cases, and (iii) the length of observation on statistical estimation of doubling time. It was difficult to recover values of growth rate from doubling time, especially when the growth rate was small. Starting time period is critical when the statistical estimation of doubling time occurs during the course of an epidemic. The length of observation was critical in determining the overall magnitude of doubling time, and when only the latest 1-2 weeks' data were used, the resulting doubling time was very short, regardless of the intrinsic growth rate r. We suggest that doubling time estimates of infectious disease epidemics should at a minimum be accompanied by descriptions of (i) the starting time at which the cumulative count is initiated and (ii) the length of observation.

Growth parameters and responses of green algae across a gradient of phototrophic, mixotrophic and heterotrophic conditions

Many studies have shown that algal growth is enhanced by organic carbon and algal mixotrophy is relevant for physiology and commercial cultivation. Most studies have tested only a single organic carbon concentration and report different growth parameters which hampers comparisons and improvements to algal cultivation methodology. This study compared growth of green algae Chlorella vulgaris and Chlamydomonas reinhardtii across a gradient of photoautotrophic-mixotrophic-heterotrophic culture conditions, with five acetate concentrations. Culture growth rates and biomass achieved were compared using different methods of biomass estimation. Both species grew faster and produced the most biomass when supplied with moderate acetate concentrations (1-4 g L^-1), but light was required to optimize growth rates, biomass yield, cell size and cell chlorophyll content. Higher acetate concentration (10 g L^-1) inhibited algal production. The choice of growth parameter and method to estimate biomass (optical density (OD), chlorophyll a fluorescence, flow cytometry, cell counts) affected apparent responses to organic carbon, but use of OD at 600, 680 or 750 nm was consistent. There were apparent trade-offs among exponential growth rate, maximum biomass, and culture time spent in exponential phase. Different cell responses over 1-10 g L^-1 acetate highlight profound physiological acclimation across a gradient of mixotrophy. In both species, cell size vs cell chlorophyll relationships were more constrained in photoautotrophic and heterotrophic cultures, but under mixotrophy, and outside exponential growth phase, these relationships were more variable. This study provides insights into algal physiological responses to mixotrophy but also has practical implications for choosing parameters for monitoring commercial algal cultivation.

The impact of policy timing on the spread of COVID-19

We model COVID-19 data for 89 nations and US states with a recently developed formalism that describes mathematically any pattern of growth with the minimum number of parameters. The results show that the disease has a typical duration of 18 days, with a significant increase in fatality when it lasts longer than about 4 months. Searching for correlations between "flattening of the curve" and preventive public policies, we find strong statistical evidence for the impact of the first implemented policy on decreasing the pandemic growth rate; a delay of one week in implementation nearly triples the size of the infected population, on average. Without any government action, the initial outburst still slows down after 36 days, possibly thanks to changes in public behavior in response to the pandemic toll. Stay-at-home (lockdown) was not the first policy of any sample member, and we could not find statistically meaningful evidence for its added impact, similar to a recent study that employed an entirely different approach. However, lockdown was mostly imposed only shortly before the exponential rise was arrested by other measures, too late for a meaningful impact. A third of the sample members that did implement lockdown imposed it only after the outburst had already started to slow down. The possibility remains that lockdown might have significantly shortened the initial exponential rise had it been employed as first resort rather than last.

Kinetic analysis of microcalorimetric data derived from microbial growth: Basic theoretical, practical and industrial considerations

We report here a mathematical framework for the quantitative interpretation of exponential bacterial growth measured with isothermal microcalorimetry. The method allows determination of many parameters that define the exponential growth phase. To automate the analysis, we also wrote a coding program, so that the approach could be embedded in a commercial setting. As an exemplar, we apply the method to a commercial probiotic product. The outcome was that we could identify characteristic parameters of growth (including rate constant and doubling time), and hence authenticate product quality, within 15 h. This compares favourably with the current 7-10 days required for conventional microbiological assessment (to allow release of product for bottling and marketing) via plating methods. The method would lend itself to growth analysis of single and mixed bacterial cultures.

Exploring Israeli high school graduates' explanations for the spread of the coronavirus

The aim of this study is to explore Israeli high school graduates' mathematical explanations for the spread of the coronavirus, given that the mathematics required to do so was part of their school curriculum. An online questionnaire consisting of two sections provided a variety of potential framings for explaining the phenomenon. The first section invited the participants to explain the spread of the coronavirus in terms of their school majors in general, with no specific reference to mathematics. The second section asked explicitly to explain the mathematical context underlying the phenomenon. In this section, the participants were asked to discuss the Prime Minister's speech given in the media a few weeks earlier, in which he described the spread of the coronavirus as a geometric series. Data analysis of 87 participants' responses to the questionnaire revealed 11 different mathematical ideas used to explain the spread of the coronavirus. These ideas included are as follows: doubling, sequence, exponential growth, using powers, tree diagram, recursion, fast-growing rate with covariation, probability, parabola and quadratic function, acceleration, and factorial. It was also found that the second section of the questionnaire elicited a wider range of mathematical ideas than the first one. We suggest possible explanations for the emergence of the mathematical ideas, which seem to reflect the graduates' intuitive knowledge, influenced not only by their mathematics track level but also by their chosen high school majors. Possible implications are discussed.

COVID-19 prevalence estimation: Four most affected African countries

The world at large has been confronted with several disease outbreak which has posed and still posing a serious menace to public health globally. Recently, COVID-19 a new kind of coronavirus emerge from Wuhan city in China and was declared a pandemic by the World Health Organization. There has been a reported case of about 8622985 with global death of 457,355 as of 15.05 GMT, June 19, 2020. South-Africa, Egypt, Nigeria and Ghana are the most affected African countries with this outbreak. Thus, there is a need to monitor and predict COVID-19 prevalence in this region for effective control and management. Different statistical tools and time series model such as the linear regression model and autoregressive integrated moving average (ARIMA) models have been applied for disease prevalence/incidence prediction in different diseases outbreak. However, in this study, we adopted the ARIMA model to forecast the trend of COVID-19 prevalence in the aforementioned African countries. The datasets examined in this analysis spanned from February 21, 2020, to June 16, 2020, and was extracted from the World Health Organization website. ARIMA models with minimum Akaike information criterion correction (AICc) and statistically significant parameters were selected as the best models. Accordingly, the ARIMA (0,2,3), ARIMA (0,1,1), ARIMA (3,1,0) and ARIMA (0,1,2) models were chosen as the best models for SA, Nigeria, and Ghana and Egypt, respectively. Forecasting was made based on the best models. It is noteworthy to claim that the ARIMA models are appropriate for predicting the prevalence of COVID-19. We noticed a form of exponential growth in the trend of this virus in Africa in the days to come. Thus, the government and health authorities should pay attention to the pattern of COVID-19 in Africa. Necessary plans and precautions should be put in place to curb this pandemic in Africa.

Differences in metal tolerance among strains, populations, and species of marine diatoms - Importance of exponential growth for quantification

Strains of microalgae vary in traits between species and populations due to adaptation or stochastic processes. Traits of individual strains may also vary depending on the acclimatization state and external forces, such as abiotic stress. In this study we tested how metal tolerance differs among marine diatoms at three organizational levels: species, populations, and strains. At the species level we compared two pelagic Baltic Sea diatoms (Skeletonema marinoi and Thalassiosira baltica). We found that the between-species differences in tolerance (EC50) to the biologically active metals (Cu, Co, Ni, and Zn) was similar to that within-species. In contrast, the two species differed significantly in tolerance towards the non-essential metals, Ag (three-fold higher in T. baltica), Pb and Cd (two and three-fold higher in S. marinoi). At the population level, we found evidence that increased tolerance against Cu and Co (17 and 41 % higher EC50 on average, respectively) had evolved in a S. marinoi population subjected to historical mining activity. On a strain level we demonstrate how the growth phase of cultures (i.e., cellular densities above exponential growth) modulated dose-response relationships to Ag, Cd, Co, Cu, and Zn. Specifically, the EC50's were reduced by 10-60 % in non-exponentially growing S. marinoi (strain RO5AC), depending on metal. For the essential metals these differences were often larger than the average differences between the two species and populations. Consequently, without careful experimental design, interactions between nutrient limitation and metal stress may interfere with detection of small, but evolutionary and ecologically important, differences in tolerance between microalgae. To avoid such artifacts, we outline a semi-continuous cultivation approach that maintains, and empirically tests, that exponential growth is achieved. We argue that such an approach is essential to enable comparison of population or strain differences in tolerance using dose-response tests on cultures of microalgae.

Calling for an exponential escalation scheme in vaccine development for COVID-19

Purpose

COVID-19 as a pandemic calls for rapid development of vaccines.

Methods

Here a proposal of a seamless, adaptive, phase 1–3 trial for accelerated vaccine development is described.

Results

Starting at 10, the number of vaccinated volunteers would exponentially increase by tenfold at an interval of 2 weeks; close surveillance of antibody responses, safety and efficacy is necessary. After only 16 weeks, general vaccination would be feasible if supply meets the demand.

Conclusion

A COVID-19 vaccine would be rapidly available at a slightly increased risk for undetected late side effects or insufficient efficacy if compared with standard vaccine development schemes.

Epidemiological control measures and predicted number of infections for SARS-CoV-2 pandemic: case study Serbia march-april 2020

BACKGROUND

In this paper, we are studying the response of the Serbian government and health authorities to the SARS-CoV-2 pandemic in the early stage of the local outbreak between Mar. 15th and Apr. 15th, 2020 by predictive numerical models. Such a study should be helpful to access the effectiveness of measures conducted to suppress the pandemic at a local scale.

METHODS

We have performed extrapolation of the number of SARS-CoV-2 infections with the first stable set of data exploiting exponential growth (linear in logarithmic scale). Based on obtained coefficients it is performed prediction of a number of cases until the end of March. After initial exponential growth, we have changed predictive model to the generalized gamma function. Obtained results are compared with the number of infections and the prediction for the remainder of the outbreak is given.

FINDINGS

We have found that the daily growth rate was above 21.5% at the beginning of the period, increased slightly after the introduction of the State of Emergency and the first set of strict epidemical control measures. It took about 13 days after the first set of strict measures to smooth daily growth. It seems that early government measures had an only moderate impact to reduce growth due to the social behavior of citizens and influx of diaspora returning to Serbia from highly affected areas, i.e., the exponential growth of infected persons is kept but with a reduced slope of about 14-15%. Anyway, it is demonstrated that period required that any measure has effect is up to 15 days after introduction, firstly to exponential growth with a smaller rate and after to smooth function representing the number of infected persons below exponential growth rate.

CONCLUSIONS

Obtained results are consistent with findings from other countries, i.e., initial exponential growth slows down within the presumed incubation period of 2 weeks after adopting lockdown and other non-pharmaceutical epidemiological measures. However, it is also shown that the exponential growth can continue after this period with a smaller slope. Therefore, quarantine and other social distancing measures should be adopted as soon as possible in a case of any similar outbreak since alternatives mean prolonged epidemical situation and growing costs in human life, pressure on the health system, economy, etc. For modeling the remainder of the outbreak generalized gamma function is used showing accurate results but requiring more samples and pre-processing (data filtering) concerning exponential part of the outbreak. We have estimated the number of infected persons for the remaining part of the outbreak until the end of June.

Crossover in spreading behavior due to memory in population dynamics

The reaction-diffusion equation is one of the possible ways for modeling animal movement, where the reactive part stands for the population growth and the diffusive part for random dispersal of the population. However, a reaction-diffusion model may not represent all aspects of the spatial dynamics, because of the existence of distinct mechanisms that can affect the movement, such as spatial memory, which results in a bias for one direction of dispersal. This bias is modeled through an advective term on an advection-reaction-diffusion equation. Thus, considering the effects of memory on the population spread, we propose a model composed of a coupled partial differential equation system with two equations: one for the population dynamics and the other for the memory density distribution. For the population growth, we use either the exponential or logistic growth function. The analytic approach shows that for the exponential and logistic growth, the minimum traveling wave speeds are the same with or without memory dynamics in which the variation of memory is infinitesimal. From the numerical analysis, we explore how our parameters, memory, growth rate, and carrying capacity, affect the population redistribution. The combinations of these parameters result in a redistribution pattern of the population associated with either diffusive or superdiffusive and imply the dispersal is faster than the diffusion. Further, in the parameter-space defined by memory and growth rate, we have shown that memory is a factor that switches the dynamics between two spreading behaviors, one faster than the other.

(p)ppGpp: the magic governor of bacterial growth economy

A fundamental question in microbiology is how bacterial cells manage to coordinate gene expression with cell growth during adapting to various environmental conditions. Although the cellular responses to changing environments have been extensively studied using transcriptomic and proteomic approaches, it remains poorly understood regarding the molecular strategy enabling bacteria to manipulate the global gene expression patterns. The alarmone (p)ppGpp is a key secondary messenger involved in regulating various biochemical and physiological processes of bacterial cells. However, despite of the extensive studies of (p)ppGpp signaling in stringent response during the past 50 years, the connection between (p)ppGpp and exponential growth remains poorly understood. Our recent work demonstrates that (p)ppGpp is strongly involved in regulating cell growth of Escherichia coli through balancing the cellular investment on metabolic proteins and ribosomes, highlighting itself as a magic governor of bacterial global resource allocation. In this mini-review, we briefly summarize some historical perspectives and current progress of the relation between (p)ppGpp and bacterial exponential growth. Two important future directions are also highlighted: the first direction is to elucidate the cellular signal that triggers (p)ppGpp accumulation during poor growth conditions; the second direction is to investigate the relation between (p)ppGpp and exponential growth for bacterial species other than E. coli.

Autocatalytic confusion clarified

There is frequent confusion about the terms autocatalytic reaction, autocatalytic cycle, and autocatalytic set. As the use of the same adjective implies, these three systems do indeed share common properties, in particular their potential for exponential growth. However, the ways in which they achieve this potential are different, giving rise to different internal network structures and dynamics. Therefore, care should be taken which term is used in which context. Here, we explain and discuss the similarities and differences between the three systems in detail, in an effort to avoid any further confusion. We then also discuss the relevance of these autocatalytic systems for possible origin of life scenarios, with an emphasis on how autocatalytic sets may have played an important role in this.

Perspectives on model forecasts of the 2014-2015 Ebola epidemic in West Africa: lessons and the way forward

The unprecedented impact and modeling efforts associated with the 2014-2015 Ebola epidemic in West Africa provides a unique opportunity to document the performances and caveats of forecasting approaches used in near-real time for generating evidence and to guide policy. A number of international academic groups have developed and parameterized mathematical models of disease spread to forecast the trajectory of the outbreak. These modeling efforts often relied on limited epidemiological data to derive key transmission and severity parameters, which are needed to calibrate mechanistic models. Here, we provide a perspective on some of the challenges and lessons drawn from these efforts, focusing on (1) data availability and accuracy of early forecasts; (2) the ability of different models to capture the profile of early growth dynamics in local outbreaks and the importance of reactive behavior changes and case clustering; (3) challenges in forecasting the long-term epidemic impact very early in the outbreak; and (4) ways to move forward. We conclude that rapid availability of aggregated population-level data and detailed information on a subset of transmission chains is crucial to characterize transmission patterns, while ensemble-forecasting approaches could limit the uncertainty of any individual model. We believe that coordinated forecasting efforts, combined with rapid dissemination of disease predictions and underlying epidemiological data in shared online platforms, will be critical in optimizing the response to current and future infectious disease emergencies.

Useful Tips, Widely Used Techniques, and Quantifying Cell Metabolic Behavior

The insect cell culture/baculovirus system has three primary applications: (1) recombinant protein synthesis, (2) biopesticide synthesis, and (3) as a model system (e.g., for studying apoptosis). The fundamental techniques involved in these applications are described throughout this book. In this chapter the most widely used techniques are summarized and the reader is directed to detailed information found elsewhere in this book. Furthermore, many useful tips and my personal preferences that are rarely published are discussed in this chapter along with quantitative methods to characterize cell growth, baculovirus infection, and metabolism.

On growth measurements of abdominal aortic aneurysms using maximally inscribed spheres

The maximum diameter, total volume of the abdominal aorta, and its growth rate are usually regarded as key factors for making a decision on the therapeutic operation time for an abdominal aortic aneurysm (AAA) patient. There is, however, a debate on what is the best standard method to measure the diameter. Currently, two dominant methods for measuring the maximum diameter are used. One is measured on the planes perpendicular to the aneurism's central line (orthogonal diameter) and the other one is measured on the axial planes (axial diameter). In this paper, another method called 'inscribed-spherical diameter' is proposed to measure the diameter. The main idea is to find the diameter of the largest sphere that fits within the aorta. An algorithm is employed to establish a centerline for the AAA geometries obtained from a set of longitudinal scans obtained from South Korea. This centerline, besides being the base of the inscribed spherical method, is used for the determination of orthogonal and axial diameter. The growth rate parameters are calculated in different diameters and the total volume and the correlations between them are studied. Furthermore, an exponential growth pattern is sought for the maximum diameters over time to examine a nonlinear growth pattern of AAA expansion both globally and locally. The results present the similarities and discrepancies of these three methods. We report the shortcomings and the advantages of each method and its performance in the quantification of expansion rates. While the orthogonal diameter measurement has an ability of capturing a realistic diameter, it fluctuated. On the other hand, the inscribed sphere diameter method tends to underestimate the diameter measurement but the growth rate can be bounded in a narrow region for aiding prediction capability. Moreover, expansion rate parameters derived from this measurement exhibit good correlation with each other and with growth rate of volume. In conclusion, although the orthogonal method remains the main method of measuring the diameter of an abdominal aorta, employing the idea of maximally inscribed spheres provides both a tool for generation of the centerline, and an additional parameter for quantification of aneurysmal growth rates.

Development and growth of fruit bodies and crops of the button mushroom, Agaricus bisporus

We studied the appearance of fruit body primordia, the growth of individual fruit bodies and the development of the consecutive flushes of the crop. Relative growth, measured as cap expansion, was not constant. It started extremely rapidly, and slowed down to an exponential rate with diameter doubling of 1.7 d until fruit bodies showed maturation by veil breaking. Initially many outgrowing primordia were arrested, indicating nutritional competition. After reaching 10 mm diameter, no growth arrest occurred; all growing individuals, whether relatively large or small, showed an exponential increase of both cap diameter and biomass, until veil breaking. Biomass doubled in 0.8 d. Exponential growth indicates the absence of competition. Apparently there exist differential nutritional requirements for early growth and for later, continuing growth. Flushing was studied applying different picking sizes. An ordinary flushing pattern occurred at an immature picking size of 8 mm diameter (picking mushrooms once a day with a diameter above 8 mm). The smallest picking size yielded the highest number of mushrooms picked, confirming the competition and arrested growth of outgrowing primordia: competition seems less if outgrowing primordia are removed early. The flush duration (i.e. between the first and last picking moments) was not affected by picking size. At small picking size, the subsequent flushes were not fully separated in time but overlapped. Within 2 d after picking the first individuals of the first flush, primordia for the second flush started outgrowth. Our work supports the view that the acquisition of nutrients by the mycelium is demand rather than supply driven. For formation and early outgrowth of primordia, indications were found for an alternation of local and global control, at least in the casing layer. All these data combined, we postulate that flushing is the consequence of the depletion of some unknown specific nutrition required by outgrowing primordia.