Growth and depletion in linear stochastic reaction networks.

This paper is about a class of stochastic reaction networks. Of interest are the dynamics of interconversion among a finite number of substances through reactions that consume some of the substances and produce others. The models we consider are continuous-time Markov jump processes, intended as idealizations of a broad class of biological networks. Reaction rates depend linearly on "enzymes," which are among the substances produced, and a reaction can occur only in the presence of sufficient upstream material. We present rigorous results for this class of stochastic dynamical systems, the mean-field behaviors of which are described by ordinary differential equations (ODEs). Under the assumption of exponential network growth, we identify certain ODE solutions as being potentially traceable and give conditions on network trajectories which, when rescaled, can with high probability be approximated by these ODE solutions. This leads to a complete characterization of the ω-limit sets of such network solutions (as points or random tori). Dimension reduction is noted depending on the number of enzymes. The second half of this paper is focused on depletion dynamics, i.e., dynamics subsequent to the "phase transition" that occurs when one of the substances becomes unavailable. The picture can be complex, for the depleted substance can be produced intermittently through other network reactions. Treating the model as a slow-fast system, we offer a mean-field description, a first step to understanding what we believe is one of the most natural bifurcations for reaction networks.

On getting it right by being wrong: A case study of how flawed research may become self-fulfilling at last.

Scientists prominently argue that the COVID-19 pandemic stems not least from people's inability to understand exponential growth. They increasingly cite evidence from a classic psychological experiment published some 45 years prior to the first case of COVID-19. Despite­or precisely because of­becoming such a canonical study (more often cited than read), its critical design flaws went completely unnoticed. They are discussed here as a cautionary tale against uncritically enshrining unsound research in the "lore" of a field of research. In hindsight, this is a unique case study of researchers falling prey to just the cognitive bias they set out to study­undermining an experiment's methodology while, ironically, still supporting its conclusion.

Rapid growth rate of Enterobacter sp. SM3 determined using several methods.

BACKGROUND: Bacterial growth rate, commonly reported in terms of doubling time, is frequently determined by one of two techniques: either by measuring optical absorption of a growing culture or by taking samples at different times during their growth phase, diluting them, spreading them on agar plates, incubating them, and counting the colonies that form. Both techniques require measurements of multiple repeats, as well careful assessment of reproducibility and consistency. Existing literature using either technique gives a wide range of growth rate values for even the most extensively studied species of bacteria, such as Escherichia coli, Pseudomonas aeruginosa, and  Staphylococcus aureus. This work aims to apply several methods to reliably determine the growth rate of a recently identified species of Enterobacteriaceae, called Enterobacter sp. SM3, and to compare that rate with that of a well-known wildtype E. coli strain KP437. RESULTS: We extend conventional optical density (OD) measurements to determine the growth rate of Enterobacter sp. SM3. To assess the reliability of this technique, we compare growth rates obtained by fitting the OD data to exponential growth, applying a relative density method, and measuring shifts in OD curves following set factors of dilution. The main source of error in applying the OD technique is due to the reliance on an exponential growth phase with a short span. With proper choice of parameter range, however, we show that these three methods yield consistent results. We also measured the SM3 division rate by counting colony-forming units (CFU) versus time, yielding results consistent with the OD measurements. In lysogeny broth at 37oC, SM3 divides every 21 ± 3 min, notably faster than the RP437 strain of E. coli, which divides every 29 ± 2 min. CONCLUSION: The main conclusion of this report is that conventional optical density (OD) measurements and the colony-forming units (CFU) method can yield consistent values of bacterial growth rate. However, to ensure the reproducibility and reliability of the measured growth rate of each bacterial strain, different methods ought to be applied in close comparison. The effort of checking for consistency among multiple techniques, as we have done in this study, is necessary to avoid reporting variable values of doubling time for particular species or strains of bacteria, as seen in the literature.

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.

Mutations in artificial self-replicating tiles: A step toward Darwinian evolution.

Artificial self-replication and exponential growth holds the promise of gaining a better understanding of fundamental processes in nature but also of evolving new materials and devices with useful properties. A system of DNA origami dimers has been shown to exhibit exponential growth and selection. Here we introduce mutation and growth advantages to study the possibility of Darwinian-like evolution. We seed and grow one dimer species, AB, from A and B monomers that doubles in each cycle. A similar species from C and D monomers can replicate at a controlled growth rate of two or four per cycle but is unseeded. Introducing a small mutation rate so that AB parents infrequently template CD offspring we show experimentally that the CD species can take over the system in approximately six generations in an advantageous environment. This demonstration opens the door to the use of evolution in materials design.

Breast Cancer Growth on Serial MRI: Volume Doubling Time Based on 3-Dimensional Tumor Volume Assessment.

BACKGROUND: The volume doubling time (VDT) of breast cancer was most frequently calculated using the two-dimensional (2D) diameter, which is not reliable for irregular tumors. It was rarely investigated using three-dimensional (3D) imaging with tumor volume on serial magnetic resonance imaging (MRI). PURPOSE: To investigate the VDT of breast cancer using 3D tumor volume assessment on serial breast MRIs. STUDY TYPE: Retrospective. SUBJECTS: Sixty women (age at diagnosis: 57 ± 10 years) with breast cancer, assessed by two or more breast MRI examinations. The median interval time was 791 days (range: 70-3654 days). FIELD STRENGTH/SEQUENCE: 3-T, fast spin-echo T2-weighted imaging (T2WI), single-shot echo-planar diffusion-weighted imaging (DWI), and gradient echo dynamic contrast-enhanced imaging. ASSESSMENT: Three radiologists independently reviewed the morphological, DWI, and T2WI features of lesions. The whole tumor was segmented to measure the volume on contrast-enhanced images. The exponential growth model was fitted in the 11 patients with at least three MRI examinations. The VDT of breast cancer was calculated using the modified Schwartz equation. STATISTICAL TESTS: Mann-Whitney U test, Kruskal-Wallis test, Chi-squared test, intraclass correlation coefficients, and Fleiss kappa coefficients. A P-value <0.05 was considered statistically significant. The exponential growth model was evaluated using the adjusted R2 and root mean square error (RMSE). RESULTS: The median tumor diameter was 9.7 mm and 15.2 mm on the initial and final MRI, respectively. The median adjusted R2 and RMSE of the 11 exponential models were 0.97 and 15.8, respectively. The median VDT was 540 days (range: 68-2424 days). For invasive ductal carcinoma (N = 33), the median VDT of the non-luminal type was shorter than that of the luminal type (178 days vs. 478 days). On initial MRI, breast cancer manifesting as a focus or mass lesion showed a shorter VDT than that of a non-mass enhancement (NME) lesion (median VDT: 426 days vs. 665 days). DATA CONCLUSION: A shorter VDT was observed in breast cancer manifesting as focus or mass as compared to an NME lesion. LEVEL OF EVIDENCE: 3 TECHNICAL EFFICACY: Stage 2.

Origin of exponential growth in nonlinear reaction networks.

Exponentially growing systems are prevalent in nature, spanning all scales from biochemical reaction networks in single cells to food webs of ecosystems. How exponential growth emerges in nonlinear systems is mathematically unclear. Here, we describe a general theoretical framework that reveals underlying principles of long-term growth: scalability of flux functions and ergodicity of the rescaled systems. Our theory shows that nonlinear fluxes can generate not only balanced growth but also oscillatory or chaotic growth modalities, explaining nonequilibrium dynamics observed in cell cycles and ecosystems. Our mathematical framework is broadly useful in predicting long-term growth rates from natural and synthetic networks, analyzing the effects of system noise and perturbations, validating empirical and phenomenological laws on growth rate, and studying autocatalysis and network evolution.

Correcting misperceptions of exponential coronavirus growth increases support for social distancing.

The most effective way to stem the spread of a pandemic such as coronavirus disease 2019 (COVID-19) is social distancing, but the introduction of such measures is hampered by the fact that a sizeable part of the population fails to see their need. Three studies conducted during the mass spreading of the virus in the United States toward the end of March 2020 show that this results partially from people's misperception of the virus's exponential growth in linear terms and that overcoming this bias increases support for social distancing. Study 1 shows that American participants mistakenly perceive the virus's exponential growth in linear terms (conservatives more so than liberals). Studies 2 and 3 show that instructing people to avoid the exponential growth bias significantly increases perceptions of the virus's growth and thereby increases support for social distancing. Together, these results show the importance of statistical literacy to recruit support for fighting pandemics such as the coronavirus.

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.

Litters of self-replicating origami cross-tiles.

Self-replication and exponential growth are ubiquitous in nature but until recently there were few examples of artificial self-replication. Often replication is a templated process where a parent produces a single offspring, doubling the population in each generation. Many species however produce more than one offspring at a time, enabling faster population growth and higher probability of species perpetuation. We have made a system of cross-shaped origami tiles that yields a number of offspring, four to eight or more, depending on the concentration of monomer units to be assembled. The parent dimer template serves as a seed to crystallize a one-dimensional crystal, a ladder. The ladder rungs are then UV-cross-linked and the offspring are then released by heating, to yield a litter of autonomous daughters. In the complement study, we also optimize the growth conditions to speed up the process and yield a 103 increase in the growth rate for the single-offspring replication system. Self-replication and exponential growth of autonomous motifs is useful for fundamental studies of selection and evolution as well as for materials design, fabrication, and directed evolution. Methods that increase the growth rate, the primary evolutionary drive, not only speed up experiments but provide additional mechanisms for evolving materials toward desired functionalities.

Nst1, Densely Associated to P-Body in the Post-Exponential Phases of Saccharomyces cerevisiae, Shows an Intrinsic Potential of Producing Liquid-Like Condensates of P-Body Components in Cells.

Membrane-less biomolecular compartmentalization is a core phenomenon involved in many physiological activities that occur ubiquitously in cells. Condensates, such as promyelocytic leukemia (PML) bodies, stress granules, and P-bodies (PBs), have been investigated to understand the process of membrane-less cellular compartmentalization. In budding yeast, PBs dispersed in the cytoplasm of exponentially growing cells rapidly accumulate in response to various stresses such as osmotic stress, glucose deficiency, and heat stress. In addition, cells start to accumulate PBs chronically in post-exponential phases. Specific protein-protein interactions are involved in accelerating PB accumulation in each circumstance, and discovering the regulatory mechanism for each is the key to understanding cellular condensation. Here, we demonstrate that Nst1 of budding yeast Saccharomyces cerevisiae is far more densely associated with PBs in post-exponentially growing phases from the diauxic shift to the stationary phase than during glucose deprivation of exponentially growing cells, while the PB marker Dcp2 exhibits a similar degree of condensation under these conditions. Similar to Edc3, ectopic Nst1 overexpression induces self-condensation and the condensation of other PB components, such as Dcp2 and Dhh1, which exhibit liquid-like properties. Altogether, these results suggest that Nst1 has the intrinsic potential for self-condensation and the condensation of other PB components, specifically in post-exponential phases.

Speed and strength of an epidemic intervention.

An epidemic can be characterized by its strength (i.e., the reproductive number [Formula: see text]) and speed (i.e., the exponential growth rate r). Disease modellers have historically placed much more emphasis on strength, in part because the effectiveness of an intervention strategy is typically evaluated on this scale. Here, we develop a mathematical framework for the classic, strength-based paradigm and show that there is a dual speed-based paradigm which can provide complementary insights. In particular, we note that r = 0 is a threshold for disease spread, just like [Formula: see text] [ 1], and show that we can measure the strength and speed of an intervention on the same scale as the strength and speed of an epidemic, respectively. We argue that, while the strength-based paradigm provides the clearest insight into certain questions, the speed-based paradigm provides the clearest view in other cases. As an example, we show that evaluating the prospects of 'test-and-treat' interventions against the human immunodeficiency virus (HIV) can be done more clearly on the speed than strength scale, given uncertainty in the proportion of HIV spread that happens early in the course of infection. We also discuss evaluating the effects of the importance of pre-symptomatic transmission of the SARS-CoV-2 virus. We suggest that disease modellers should avoid over-emphasizing the reproductive number at the expense of the exponential growth rate, but instead look at these as complementary measures.

SARS-CoV-2 mutational cascades and the risk of hyper-exponential growth.

The emergence of novel SARS-CoV-2 variants of concern (VOC), in late 2020, with selective transmission advantage and partial immunity escape potential, has been driving further evolution in the pandemic. The timing of mutational evolution and its limits are thus of paramount importance in preparedness planning. Here, we present a model predicting the pattern of epidemic growth including the emergence of variants through mutation. It is based on the SEIR (Susceptible, Exposed, Infected, Removed) model, but its equations are modified according to the transmission parameters of novel variants. Since more transmissible strains will drive a further increase in the number of cases, they will also lead to further novel mutations. As one cannot predict whether there is a viral mutational evolutionary limit, we model a cascade that could lead to hyper-exponential growth (HEG) involving the emergence of even more transmissible mutants that could overwhelm any systematic response. Our results are consistent with the timing, since the beginning of the pandemic, of the concurrent and independent emergence of the VOCs. The current dominance of the Delta variant and the need for additional public health measures indicates some of the risks of a possible HEG. We examine conditions that favor the expected appearance of similar variants, thus enabling better preparedness and more targeted research.

Reconstruction of transcriptional regulatory networks of Fis and H-NS in Escherichia coli from genome-wide data analysis.

Fis (Factor for inversion stimulation) and H-NS (Histone-like nucleoid-structuring protein) are two well-known nucleoid-associated proteins (NAPs) in proteobacteria, which play crucial roles in genome organization and transcriptional regulation. We performed RNA-sequencing to identify genes regulated by these NAPs. Study reveals that Fis and H-NS affect expression of 462 and 88 genes respectively in Escherichia coli at mid-exponential growth phase. By integrating available ChIP-seq data, we identified direct and indirect regulons of Fis and H-NS proteins. Functional analysis reveals that Fis controls expression of genes involved in translation, oxidative phosphorylation, sugar metabolism and transport, amino acid metabolism, bacteriocin transport, cell division, two-component system, biofilm formation, pilus organization and lipopolysaccharide biosynthesis pathways. However, H-NS represses expression of genes in cell adhesion, recombination, biofilm formation and lipopolysaccharide biosynthesis pathways under mid-exponential growth condition. The current regulatory networks thus provide a global glimpse of coordinated regulatory roles for these two important NAPs.

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.

Inference of the generalized-growth model via maximum likelihood estimation: A reflection on the impact of overdispersion.

Recently, the generalized-growth model was introduced as a flexible approach to characterize growth dynamics of disease outbreaks during the early ascending phase. In this work, by using classical maximum likelihood estimation to obtain parameter estimates, we evaluate the impact of varying levels of overdispersion on the inference of the growth scaling parameter through comparing Poisson and Negative binomial models. In particular, under exponential and sub-exponential growth scenarios, we evaluate, via simulations, the error rate of making an incorrect characterization of early outbreak growth patterns. Simulation results show that the ability to correctly identify early outbreak growth patterns can be affected by overdispersion even when accounted for using the Negative binomial model. We exemplify our findings using data on five different outbreaks. Overall, our results show that estimates should be interpreted with caution when data are overdispersed.

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.

[Alteration in the Expression of Genes Encoding Primary Metabolism Enzymes and Plastid Transporters during the Culture Growth of Chlamydomonas reinhardtii].

In a mixotrophic Chlamydomonas reinhardtii culture, the expression levels of genes encoding primary metabolic enzymes and chloroplast plastid transporters were analyzed. For the majority of the genes studied, their expression levels decreased as they approached the final stages of culture growth. During the period of exponential growth, the expression profiles changed more intensively than during the stationary phase. In the middle of exponential growth, significant changes of mRNA profiles reflected reorganization of metabolism, with an emphasis on the induction of lipid synthesis, accompanied by alterations in carbon fluxes through biochemical pathways and a shift in the energy balance between the plastid and cytosol.

(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.