Comparison of mathematical frameworks for modeling erythropoiesis in the context of malaria infection

Systems biology of malaria explored with nonhuman primates

"The Primate Malarias" book has been a uniquely important resource for multiple generations of scientists, since its debut in 1971, and remains pertinent to the present day. Indeed, nonhuman primates (NHPs) have been instrumental for major breakthroughs in basic and pre-clinical research on malaria for over 50 years. Research involving NHPs have provided critical insights and data that have been essential for malaria research on many parasite species, drugs, vaccines, pathogenesis, and transmission, leading to improved clinical care and advancing research goals for malaria control, elimination, and eradication. Whilst most malaria scientists over the decades have been studying Plasmodium falciparum, with NHP infections, in clinical studies with humans, or using in vitro culture or rodent model systems, others have been dedicated to advancing research on Plasmodium vivax, as well as on phylogenetically related simian species, including Plasmodium cynomolgi, Plasmodium coatneyi, and Plasmodium knowlesi. In-depth study of these four phylogenetically related species over the years has spawned the design of NHP longitudinal infection strategies for gathering information about ongoing infections, which can be related to human infections. These Plasmodium-NHP infection model systems are reviewed here, with emphasis on modern systems biological approaches to studying longitudinal infections, pathogenesis, immunity, and vaccines. Recent discoveries capitalizing on NHP longitudinal infections include an advanced understanding of chronic infections, relapses, anaemia, and immune memory. With quickly emerging new technological advances, more in-depth research and mechanistic discoveries can be anticipated on these and additional critical topics, including hypnozoite biology, antigenic variation, gametocyte transmission, bone marrow dysfunction, and loss of uninfected RBCs. New strategies and insights published by the Malaria Host-Pathogen Interaction Center (MaHPIC) are recapped here along with a vision that stresses the importance of educating future experts well trained in utilizing NHP infection model systems for the pursuit of innovative, effective interventions against malaria.

Discrete Biochemical Systems Theory

Almost every biomedical systems analysis requires early decisions regarding the choice of the most suitable representations to be used. De facto the most prevalent choice is a system of ordinary differential equations (ODEs). This framework is very popular because it is flexible and fairly easy to use. It is also supported by an enormous array of stand-alone programs for analysis, including many distinct numerical solvers that are implemented in the main programming languages. Having selected ODEs, the modeler must then choose a mathematical format for the equations. This selection is not trivial as nearly unlimited options exist and there is seldom objective guidance. The typical choices include ad hoc representations, default models like mass-action or Lotka-Volterra equations, and generic approximations. Within the realm of approximations, linear models are typically successful for analyses of engineered systems, but they are not as appropriate for biomedical phenomena, which often display nonlinear features such as saturation, threshold effects or limit cycle oscillations, and possibly even chaos. Power-law approximations are simple but overcome these limitations. They are the key ingredient of Biochemical Systems Theory (BST), which uses ODEs exclusively containing power-law representations for all processes within a model. BST models cover a vast repertoire of nonlinear responses and, at the same time, have structural properties that are advantageous for a wide range of analyses. Nonetheless, as all ODE models, the BST approach has limitations. In particular, it is not always straightforward to account for genuine discreteness, time delays, and stochastic processes. As a new option, we therefore propose here an alternative to BST in the form of discrete Biochemical Systems Theory (dBST). dBST models have the same generality and practicality as their BST-ODE counterparts, but they are readily implemented even in situations where ODEs struggle. As a case study, we illustrate dBST applied to the dynamics of the aryl hydrocarbon receptor (AhR), a signal transduction system that simultaneously involves time delays and stochasticity.

Understanding dynamics of Plasmodium falciparum gametocytes production: Insights from an age-structured model

Many models of within-host malaria infection dynamics have been formulated since the pioneering work of Anderson et al. in 1989. Biologically, the goal of these models is to understand what governs the severity of infections, the patterns of infectiousness, and the variation thereof across individual hosts. Mathematically, these models are based on dynamical systems, with standard approaches ranging from K-compartments ordinary differential equations (ODEs) to delay differential equations (DDEs), to capture the relatively constant duration of replication and bursting once a parasite infects a host red blood cell. Using malariatherapy data, which offers fine-scale resolution on the dynamics of infection across a number of individual hosts, we compare the fit and robustness of one of these standard approaches (K-compartments ODE) with a partial differential equations (PDEs) model, which explicitly tracks the "age" of an infected cell. While both models perform quite similarly in terms of goodness-of-fit for suitably chosen K, the K-compartments ODE model particularly overestimates parasite densities early on in infections when the number of repeated compartments is not large enough. Finally, the K-compartments ODE model (for suitably chosen K) and the PDE model highlight a strong qualitative connection between the density of transmissible parasite stages (i.e., gametocytes) and the density of host-damaging (and asexually-replicating) parasite stages. This finding provides a simple tool for predicting which hosts are most infectious to mosquitoes -vectors of Plasmodium parasites- which is a crucial component of global efforts to control and eliminate malaria.

Integration of biology, mathematics and computing in the classroom through the creation and repeated use of transdisciplinary modules

The integration of biology with mathematics and computer science mandates the training of students capable of comfortably navigating among these fields. We address this formidable pedagogical challenge with the creation of transdisciplinary modules that guide students toward solving realistic problems with methods from different disciplines. Knowledge is gradually integrated as the same topic is revisited in biology, mathematics, and computer science courses. We illustrate this process with a module on the homeostasis and dynamic regulation of red blood cell production, which was first implemented in an introductory biology course and will be revisited in the mathematics and computer science curricula.

Clinical recovery of Macaca fascicularis infected with Plasmodium knowlesi

BACKGROUND

Kra monkeys (Macaca fascicularis), a natural host of Plasmodium knowlesi, control parasitaemia caused by this parasite species and escape death without treatment. Knowledge of the disease progression and resilience in kra monkeys will aid the effective use of this species to study mechanisms of resilience to malaria. This longitudinal study aimed to define clinical, physiological and pathological changes in kra monkeys infected with P. knowlesi, which could explain their resilient phenotype.

METHODS

Kra monkeys (n = 15, male, young adults) were infected intravenously with cryopreserved P. knowlesi sporozoites and the resulting parasitaemias were monitored daily. Complete blood counts, reticulocyte counts, blood chemistry and physiological telemetry data (n = 7) were acquired as described prior to infection to establish baseline values and then daily after inoculation for up to 50 days. Bone marrow aspirates, plasma samples, and 22 tissue samples were collected at specific time points to evaluate longitudinal clinical, physiological and pathological effects of P. knowlesi infections during acute and chronic infections.

RESULTS

As expected, the kra monkeys controlled acute infections and remained with low-level, persistent parasitaemias without anti-malarial intervention. Unexpectedly, early in the infection, fevers developed, which ultimately returned to baseline, as well as mild to moderate thrombocytopenia, and moderate to severe anaemia. Mathematical modelling and the reticulocyte production index indicated that the anaemia was largely due to the removal of uninfected erythrocytes and not impaired production of erythrocytes. Mild tissue damage was observed, and tissue parasite load was associated with tissue damage even though parasite accumulation in the tissues was generally low.

CONCLUSIONS

Kra monkeys experimentally infected with P. knowlesi sporozoites presented with multiple clinical signs of malaria that varied in severity among individuals. Overall, the animals shared common mechanisms of resilience characterized by controlling parasitaemia 3-5 days after patency, and controlling fever, coupled with physiological and bone marrow responses to compensate for anaemia. Together, these responses likely minimized tissue damage while supporting the establishment of chronic infections, which may be important for transmission in natural endemic settings. These results provide new foundational insights into malaria pathogenesis and resilience in kra monkeys, which may improve understanding of human infections.

A Comparative Study between Discrete Stochastic Arithmetic and Floating-Point Arithmetic to Validate the Results of Fractional Order Model of Malaria Infection

The researchers aimed to study the nonlinear fractional order model of malaria infection based on the Caputo-Fabrizio fractional derivative. The homotopy analysis transform method (HATM) is applied based on the floating-point arithmetic (FPA) and the discrete stochastic arithmetic (DSA). In the FPA, to show the accuracy of the method we use the absolute error which depends on the exact solution and a positive value ε. Because in real life problems we do not have the exact solution and the optimal value of ε, we need to introduce a new condition and arithmetic to show the efficiency of the method. Thus the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library are applied. The CESTAC method is based on the DSA. Also, a new termination criterion is used which is based on two successive approximations. Using the CESTAC method we can find the optimal approximation, the optimal error and the optimal iteration of the method. The main theorem of the CESTAC method is proved to show that the number of common significant digits (NCSDs) between two successive approximations are almost equal to the NCSDs of the exact and approximate solutions. Plotting several graphs, the regions of convergence are demonstrated for different number of iterations k = 5, 10. The numerical results based on the simulated data show the advantages of the DSA in comparison with the FPA.

Replicative cellular age distributions in compartmentalized tissues

The cellular age distribution of hierarchically organized tissues can reveal important insights into the dynamics of cell differentiation and self-renewal and associated cancer risks. Here, we examine the effect of progenitor compartments with varying differentiation and self-renewal capacities on the resulting observable distributions of replicative cellular ages. We find that strongly amplifying progenitor compartments, i.e. compartments with high self-renewal capacities, substantially broaden the age distributions which become skewed towards younger cells with a long tail of few old cells. For several of these strongly amplifying compartments, the age distribution becomes virtually independent of the influx from the stem cell compartment. By contrast, if tissues are organized into many downstream compartments with low self-renewal capacity, the shape of the replicative cell distribution in more differentiated compartments is dominated by stem cell dynamics with little added variation. In the limiting case of a strict binary differentiation tree without self-renewal, the shape of the output distribution becomes indistinguishable from that of the input distribution. Our results suggest that a comparison of cellular age distributions between healthy and cancerous tissues may inform about dynamical changes within the hierarchical tissue structure, i.e. an acquired increased self-renewal capacity in certain tumours. Furthermore, we compare our theoretical results to telomere length distributions in granulocyte populations of 10 healthy individuals across different ages, highlighting that our theoretical expectations agree with experimental observations.

Analysis of erythrocyte dynamics in Rhesus macaque monkeys during infection with Plasmodium cynomolgi

BACKGROUND

Malaria is a major mosquito transmitted, blood-borne parasitic disease that afflicts humans. The disease causes anaemia and other clinical complications, which can lead to death. Plasmodium vivax is known for its reticulocyte host cell specificity, but many gaps in disease details remain. Much less is known about the closely related species, Plasmodium cynomolgi, although it is naturally acquired and causes zoonotic malaria. Here, a computational model is developed based on longitudinal analyses of P. cynomolgi infections in nonhuman primates to investigate the erythrocyte dynamics that is pertinent to understanding both P. cynomolgi and P. vivax malaria in humans.

METHODS

A cohort of five P. cynomolgi infected Rhesus macaques (Macaca mulatta) is studied, with individuals exhibiting a plethora of clinical outcomes, including varying levels of anaemia. A discrete recursive model with age structure is developed to replicate the dynamics of P. cynomolgi blood-stage infections. The model allows for parasitic reticulocyte preference and assumes an age preference among the mature RBCs. RBC senescence is modelled using a hazard function, according to which RBCs have a mean lifespan of 98 ± 21 days.

RESULTS

Based on in vivo data from three cohorts of macaques, the computational model is used to characterize the reticulocyte lifespan in circulation as 24 ± 5 h (n = 15) and the rate of RBC production as 2727 ± 209 cells/h/µL (n = 15). Analysis of the host responses reveals a pre-patency increase in the number of reticulocytes. It also allows the quantification of RBC removal through the bystander effect.

CONCLUSIONS

The evident pre-patency increase in reticulocytes is due to a shift towards the release of younger reticulocytes, which could result from a parasite-induced factor meant to increase reticulocyte availability and satisfy the parasite's tropism, which has an average value of 32:1 in this cohort. The number of RBCs lost due to the bystander effect relative to infection-induced RBC losses is 62% for P. cynomolgi infections, which is substantially lower than the value of 95% previously determined for another simian species, Plasmodium coatneyi.

Mathematical Modeling of RBC Count Dynamics after Blood Loss

The regeneration of red blood cells (RBCs) after blood loss is an individual complex process. We present a novel simple compartment model which is able to capture the most important features and can be personalized using parameter estimation. We compare predictions of the proposed and personalized model to a more sophisticated state-of-the-art model for erythropoiesis, and to clinical data from healthy subjects. We discuss the choice of model parameters with respect to identifiability. We give an outlook on how extensions of this novel mathematical model could have an important impact for personalized clinical decision support in the case of polycythemia vera (PV). PV is a slow-growing type of blood cancer, where especially the production of RBCs is increased. The principal treatment targeting the symptoms of PV is bloodletting (phlebotomy), at regular intervals that are based on personal experiences of the physicians. Model-based decision support might help to identify optimal and individualized phlebotomy schedules.

Examining the Reticulocyte Preference of Two Plasmodium berghei Strains during Blood-Stage Malaria Infection

The blood-stage of the Plasmodium parasite is one of the key phases within its life cycle that influences disease progression during a malaria infection. The efficiency of the parasite in infecting red blood cells (RBC) determines parasite load and parasite-induced hemolysis that is responsible for the development of anemia and potentially drives severe disease progression. However, the molecular factors defining the infectivity of Plasmodium parasites have not been completely identified so far. Using the Plasmodium berghei mouse model for malaria, we characterized and compared the blood-stage infection dynamics of PbANKA WT and a mutant parasite strain lacking a novel Plasmodium antigen, PbmaLS_05, that is well conserved in both human and animal Plasmodium parasite strains. Infection of mice with parasites lacking PbmaLS_05 leads to lower parasitemia levels and less severe disease progression in contrast to mice infected with the wildtype PbANKA strain. To specifically determine the effect of deleting PbmaLS_05 on parasite infectivity we developed a mathematical model describing erythropoiesis and malarial infection of RBC. By applying our model to experimental data studying infection dynamics under normal and drug-induced altered erythropoietic conditions, we found that both PbANKA and PbmaLS_05 (-) parasite strains differed in their infectivity potential during the early intra-erythrocytic stage of infection. Parasites lacking PbmaLS_05 showed a decreased ability to infect RBC, and immature reticulocytes in particular that are usually a preferential target of the parasite. These altered infectivity characteristics limit parasite burden and affect disease progression. Our integrative analysis combining mathematical models and experimental data suggests that deletion of PbmaLS_05 affects productive infection of reticulocytes, which makes this antigen a useful target to analyze the actual processes relating RBC preferences to the development of severe disease outcomes in malaria.

Metabolic modeling helps interpret transcriptomic changes during malaria

Disease represents a specific case of malfunctioning within a complex system. Whereas it is often feasible to observe and possibly treat the symptoms of a disease, it is much more challenging to identify and characterize its molecular root causes. Even in infectious diseases that are caused by a known parasite, it is often impossible to pinpoint exactly which molecular profiles of components or processes are directly or indirectly altered. However, a deep understanding of such profiles is a prerequisite for rational, efficacious treatments. Modern omics methodologies are permitting large-scale scans of some molecular profiles, but these scans often yield results that are not intuitive and difficult to interpret. For instance, the comparison of healthy and diseased transcriptome profiles may point to certain sets of involved genes, but a host of post-transcriptional processes and regulatory mechanisms renders predictions regarding metabolic or physiological consequences of the observed changes in gene expression unreliable. Here we present proof of concept that dynamic models of metabolic pathway systems may offer a tool for interpreting transcriptomic profiles measured during disease. We illustrate this strategy with the interpretation of expression data of genes coding for enzymes associated with purine metabolism. These data were obtained during infections of rhesus macaques (Macaca mulatta) with the malaria parasite Plasmodium cynomolgi or P. coatneyi. The model-based interpretation reveals clear patterns of flux redistribution within the purine pathway that are consistent between the two malaria pathogens and are even reflected in data from humans infected with P. falciparum. This article is part of a Special Issue entitled: Accelerating Precision Medicine through Genetic and Genomic Big Data Analysis edited by Yudong Cai & Tao Huang.

A model of Plasmodium vivax concealment based on Plasmodium cynomolgi infections in Macaca mulatta

BACKGROUND

Plasmodium vivax can cause severe malaria. The total parasite biomass during infections is correlated with the severity of disease but not necessarily quantified accurately by microscopy. This finding has raised the question whether there could be sub-populations of parasites that are not observed in peripheral blood smears but continue to contribute to the increase in parasite numbers that drive pathogenesis. Non-human primate infection models utilizing the closely related simian malaria parasite Plasmodium cynomolgi hold the potential for quantifying the magnitude of possibly unobserved infected red blood cell (iRBC) populations and determining how the presence of this hidden reservoir correlates with disease severity.

METHODS

Time series data tracking the longitudinal development of parasitaemia in five Macaca mulatta infected with P. cynomolgi were used to design a computational model quantifying iRBCs that circulate in the blood versus those that are not detectable and are termed here as 'concealed'. This terminology is proposed to distinguish such observations from the deep vascular and widespread 'sequestration' of Plasmodium falciparum iRBCs, which is governed by distinctly different molecular mechanisms.

RESULTS

The computational model presented here clearly demonstrates that the observed growth data of iRBC populations are not consistent with the known biology and blood-stage cycle of P. cynomolgi. However, the discrepancies can be resolved when a sub-population of concealed iRBCs is taken into account. The model suggests that the early growth of a hidden parasite sub-population has the potential to drive disease. As an alternative, the data could be explained by the sequential release of merozoites from the liver over a number of days, but this scenario seems less likely.

CONCLUSIONS

Concealment of a non-circulating iRBC sub-population during P. cynomolgi infection of M. mulatta is an important aspect of this successful host-pathogen relationship. The data also support the likelihood that a sub-population of iRBCs of P. vivax has a comparable means to become withdrawn from the peripheral circulation. This inference has implications for understanding vivax biology and pathogenesis and stresses the importance of considering a concealed parasite reservoir with regard to vivax epidemiology and the quantification and treatment of P. vivax infections.

Mathematical modeling of erythrocyte chimerism informs genetic intervention strategies for sickle cell disease

Recent advances in gene therapy and genome-engineering technologies offer the opportunity to correct sickle cell disease (SCD), a heritable disorder caused by a point mutation in the β-globin gene. The developmental switch from fetal γ-globin to adult β-globin is governed in part by the transcription factor (TF) BCL11A. This TF has been proposed as a therapeutic target for reactivation of γ-globin and concomitant reduction of β-sickle globin. In this and other approaches, genetic alteration of a portion of the hematopoietic stem cell (HSC) compartment leads to a mixture of sickling and corrected red blood cells (RBCs) in periphery. To reverse the sickling phenotype, a certain proportion of corrected RBCs is necessary; the degree of HSC alteration required to achieve a desired fraction of corrected RBCs remains unknown. To address this issue, we developed a mathematical model describing aging and survival of sickle-susceptible and normal RBCs; the former can have a selective survival advantage leading to their overrepresentation. We identified the level of bone marrow chimerism required for successful stem cell-based gene therapies in SCD. Our findings were further informed using an experimental mouse model, where we transplanted mixtures of Berkeley SCD and normal murine bone marrow cells to establish chimeric grafts in murine hosts. Our integrative theoretical and experimental approach identifies the target frequency of HSC alterations required for effective treatment of sickling syndromes in humans. Our work replaces episodic observations of such target frequencies with a mathematical modeling framework that covers a large and continuous spectrum of chimerism conditions. Am. J. Hematol. 91:931-937, 2016. © 2016 Wiley Periodicals, Inc.

Quantifying the removal of red blood cells in Macaca mulatta during a Plasmodium coatneyi infection

Background

Malaria is the most deadly parasitic disease in humans globally, and the long-time coexistence with malaria has left indelible marks in the human genome that are the causes of a variety of genetic disorders. Although anaemia is a common clinical complication of malaria, the root causes and mechanisms involved in the pathogenesis of malarial anaemia are unclear and difficult to study in humans. Non-human primate (NHP) model systems enable the mechanistic study and quantification of underlying causative factors of malarial anaemia, and particularly the onset of severe anaemia.

Methods

Data were obtained in the course of Plasmodium coatneyi infections of malaria-naïve and semi-immune rhesus macaques (Macaca mulatta), whose red blood cells (RBCs) were labelled in situ with biotin at the time the infections were initiated. The data were used for a survival analysis that permitted, for the first time, an accurate estimation of the lifespan of erythrocytes in macaques. The data furthermore formed the basis for the development and parameterization of a recursive dynamic model of erythrocyte turnover, which was used for the quantification of RBC production and removal in each macaque.

Results

The computational analysis demonstrated that the lifespan of erythrocytes in macaques is 98 ± 21 days. The model also unambiguously showed that death due to senescence and parasitaemia is not sufficient to account for the extent of infection-induced anaemia. Specifically, the model permits, for the first time, the quantification of the different causes of RBC death, namely, normal senescence, age-independent random loss, parasitization, and bystander effects in uninfected cells. Such a dissection of the overall RBC removal process is hardly possible with experimental means alone. In the infected malaria-naïve macaques, death of erythrocytes by normal physiological senescence processes accounts for 20 % and parasitization for only 4 %, whereas bystander effects are associated with an astonishing 76 % of total RBC losses. Model-based comparisons of alternative mechanisms involved in the bystander effect revealed that most of the losses are likely due to a process of removing uninfected RBCs of all age classes and only minimally due to an increased rate of senescence of the uninfected RBCs.

Conclusions

A new malaria blood-stage model was developed for the analysis of data characterizing P. coatneyi infections of M. mulatta. The model used a discrete and recursive framework with age-structure that allowed the quantification of the most significant pathophysiological processes of RBC removal. The computational results revealed that the malarial anaemia caused by this parasite is mostly due to a loss of uninfected RBCs by an age-independent process. The biological identity and complete mechanism of this process is not fully understood and requires further investigation.

From within host dynamics to the epidemiology of infectious disease: Scientific overview and challenges

Since their earliest days, humans have been struggling with infectious diseases. Caused by viruses, bacteria, protozoa, or even higher organisms like worms, these diseases depend critically on numerous intricate interactions between parasites and hosts, and while we have learned much about these interactions, many details are still obscure. It is evident that the combined host-parasite dynamics constitutes a complex system that involves components and processes at multiple scales of time, space, and biological organization. At one end of this hierarchy we know of individual molecules that play crucial roles for the survival of a parasite or for the response and survival of its host. At the other end, one realizes that the spread of infectious diseases by far exceeds specific locales and, due to today's easy travel of hosts carrying a multitude of organisms, can quickly reach global proportions. The community of mathematical modelers has been addressing specific aspects of infectious diseases for a long time. Most of these efforts have focused on one or two select scales of a multi-level disease and used quite different computational approaches. This restriction to a molecular, physiological, or epidemiological level was prudent, as it has produced solid pillars of a foundation from which it might eventually be possible to launch comprehensive, multi-scale modeling efforts that make full use of the recent advances in biology and, in particular, the various high-throughput methodologies accompanying the emerging -omics revolution. This special issue contains contributions from biologists and modelers, most of whom presented and discussed their work at the workshop From within Host Dynamics to the Epidemiology of Infectious Disease, which was held at the Mathematical Biosciences Institute at Ohio State University in April 2014. These contributions highlight some of the forays into a deeper understanding of the dynamics between parasites and their hosts, and the consequences of this dynamics for the spread and treatment of infectious diseases.

WITHDRAWN: From within host dynamics to the epidemiology of infectious disease: Scientific overview and challenges

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