Linking functional and molecular mechanisms of host resilience to malaria infection
Abstract
It remains challenging to understand why some hosts suffer severe illnesses, while others are unscathed by the same infection. We fitted a mathematical model to longitudinal measurements of parasite and red blood cell density in murine hosts from diverse genetic backgrounds to identify aspects of withinhost interactions that explain variation in host resilience and survival during acute malaria infection. Among eight mouse strains that collectively span 90% of the common genetic diversity of laboratory mice, we found that high host mortality was associated with either weak parasite clearance, or a strong, yet imprecise response that inadvertently removes uninfected cells in excess. Subsequent crosssectional cytokine assays revealed that the two distinct functional mechanisms of poor survival were underpinned by low expression of either pro or antiinflammatory cytokines, respectively. By combining mathematical modelling and molecular immunology assays, our study uncovered proximate mechanisms of diverse infection outcomes across multiple host strains and biological scales.
Introduction
In human malaria, infection outcomes range widely from subclinical to fatal. While it is difficult to disentangle the factors contributing to this variation in resilience to malaria, host genetics is a major determinant (HernandezValladares et al., 2005; López et al., 2010; Hedrick, 2011). Even in experimental rodent malaria infections, where environment, diet, and as many other factors as possible are highly controlled, different mouse strains infected with the same strain of Plasmodium chabaudi demonstrate remarkable variation in infection dynamics and malaria mortality. Among eight strains of inbred laboratory mice (129S1/SvImJ, A/J, C57BL/6, CAST/EiJ, NOD/ShiLtJ, NZO/HILtJ, PWK/PhJ, WSB/EiJ) that collectively span 90% of the mouse genetic diversity commonly used in laboratory experiments (Roberts et al., 2007), survival from malaria infection ranges from less than 5–100% (Figure 1). Underlying this survival variation are likely differences in functional properties of withinhost ecology (i.e. parasite growth, parasite clearance, and replenishment of red blood cells [RBCs]), which are difficult to measure directly. However, longitudinal measurements of host health and parasite burden can inform processes of withinhost ecology (i.e. RBCs and infected red blood cells (iRBCs), respectively, in malaria infections). For example, a ‘disease curve’ — longitudinal data of health and parasites plotted against each other in a phase plane — helps visualise the process of parasite growth, host sickness and recovery at the individual host level (Figure 2; Schneider, 2011). Furthermore, a mathematical model fitted to these data can predict particular functional mechanisms (e.g. parasite proliferation [Mideo et al., 2011], specific versus nonspecific immunity [Wale et al., 2019] and dosedependent host responses [Haydon et al., 2003; Metcalf et al., 2011; Kamiya et al., 2020]) that lead to divergent infection dynamics (Mideo et al., 2008a; Metcalf et al., 2012). These model predictions can then be independently tested by followup experiments: for example, Mideo et al., 2011 chemically manipulated RBC replenishment to interrogate the role of RBC age structure in parasite growth predicted by a previous model (Mideo et al., 2008b). Crucially, however, model predictions of host immune responses against malaria are rarely examined together with immunological data. Thus, it remains unclear whether and how modelled responses at the level of withinhost ecology (e.g. rate of parasite clearance) are linked to observable quantities at the cellular and molecular levels (e.g. cytokines).
The immune system is a critical proximate mechanism of host genetic resilience to infection (López et al., 2010). Failure to mount a robust immune response can lead to unchecked parasite proliferation, while dysregulated responses may cause collateral damage, that is, immunopathology. While the benefit of immune protection often outweighs any costs associated with these responses (Sorci et al., 2017), severe outcomes of many infectious diseases are a consequence of immunopathology rather than direct damage caused by parasites (Graham et al., 2005). Thus, a ‘healthy’ immune response requires striking a delicate balance.
During the acute phase of bloodstage malaria infection, innate responses target and remove iRBCs as well as shortlived extracellular parasites known as merozoites (Stevenson and Riley, 2004). In addition, RBCs — regardless of infection status — are susceptible to clearance by immune effectors such as macrophages (Jakeman et al., 1999; Chua et al., 2013). While the targeted response removes more iRBCs, datadriven modelling studies highlight the importance of indiscriminate RBC clearance for lowering parasite burden (Wale et al., 2019; Kamiya et al., 2020; Metcalf et al., 2012; Miller et al., 2010). Functionally, this hostdriven destruction of RBCs can have both favourable and unfavourable consequences for host health. On the one hand, it has been proposed as a host adaptation in the presence of malaria parasites to clear the parasites directly (i.e. topdown effect) as well as to limit resources for the parasite (i.e. bottomup effect) (Wale et al., 2019; Haydon et al., 2003; Metcalf et al., 2012; Cromer et al., 2009). On the other hand, an excessive loss of RBCs brings forth adverse health implications. In immune naive infants and children, severe malarial anaemia is the most common severe manifestation of disease, and its associated mortality rate can reach 30% (Perkins et al., 2011). A variety of processes cause malariarelated anaemia, including loss due to parasite exploitation, RBC clearance (e.g. phagocytosis of both infected and uninfected cells), suppression of RBC production, and defective RBC development (Chua et al., 2013). Among them, clearance is the most important process, accounting for between 75% and 90% of the total RBC deficit during malaria infections (Jakeman et al., 1999). In comparison, direct exploitation by malaria parasites has been estimated to account for less than 10% of the RBC deficit (Jakeman et al., 1999; Price et al., 2001; Fonseca et al., 2016).
At the molecular level, vertebrate host responses are regulated by immune signalling molecules, known as cytokines (Lamb et al., 2006). Acute malaria infection induces proinflammatory cytokines required for mounting a timely and robust response while antiinflammatory cytokines inhibit excess immune reactions to safeguard against collateral damage (Lamb et al., 2006). For instance, tumour necrosis factor alpha ($\text{TNF}\alpha $) and interferongamma ($\text{IFN}\gamma $), are proinflammatory cytokines responsible for a myriad of inflammatory responses, including the production of nitric oxide and reactive oxygen species (BouharounTayoun et al., 1995; Bogdan et al., 2000), which are associated with rapid clearance of P. falciparum, the deadliest human malaria parasite (Rockett et al., 1992; Kremsner et al., 1995; Mordmüller et al., 1997; HernandezValladares et al., 2006; Franklin et al., 2007). However, the same inflammatory responses can also be damaging to the organisms that produce them (Clark et al., 1991; King and Lamb, 2015). For example, TNFα overproduction — which can result from a deficit of antiinflammatory cytokines like interleukin 10 (IL10) and transforming growth factorbeta (TGFβ) — could lead to adverse effects including worsened anaemia, weight loss and survival in the mouse model (Omer and Riley, 1998; Li et al., 2003; Long et al., 2006; Long et al., 2008). Balanced expression of these cytokines is likely a mechanism that promotes resilience (survival) to malaria infection. However, mechanistic studies usually focus on just one or two inbred mouse strains with similar cytokine responses, limiting our ability to link molecular signatures with functional variation in host traits (e.g. indiscriminate versus targeted RBC clearance) that impacts the infection dynamics.
To uncover the functional mechanisms underlying malaria survival and variation thereof, we formulated a mathematical model of withinhost malaria ecology that describes the asexual replicative cycle and qualitatively distinct components of host immunity (Figure 3). Rather than aiming for a mechanistically precise description of host immunity, we employed a simple mathematical model to track the net effects of host responses, that is, clearance rate of iRBCs and RBCs (Kamiya et al., 2020; Figure 3a). Using a hierarchical Bayesian approach, we fitted the model to longitudinal data of RBCs and iRBCs from eight mouse strains with varied resilience to P. chabaudi. We then examined crosssectional cytokine data from the same eight strains to uncover the molecular underpinnings of our model predictions.
Results and discussion
Functional mechanisms underlying resilience to malaria
Our mathematical model of withinhost malaria infection accurately described the timecourse of RBCs and iRBCs during the acute phase of malaria infection in all mouse strains (Appendices 1 and 2). Several estimated model parameters varied with mouse strain (Figure 4). To characterise these multivariate, withinhost ecological differences, we carried out principal component analysis (PCA) on the estimated parameter set, θ. We found several clusters that distinguished mouse strains revealing functional diversity of host resilience to malaria infection (Figure 5).
First, we identified C57BL/6 (80% survival; Figure 1) as the most ‘functionally average’ of the eight strains, indicated by the most central position in the PCA biplot (Figure 5) and nearzero estimates for strainspecificity, $s$ (Figure 4a). Two fully resilient (100% survival) strains, 129S1/SvImJ and NZO/HILtJ, were functionally similar to C57BL/6, but 129S1/SvImJ showed a slightly smaller parasite burst size, β, and higher background RBC clearance during infection, ${\mu}_{R}^{\prime}$, while NZO/HILtJ showed slightly higher activation of both the indiscriminate response (${\psi}_{{N}_{1}}$) and lower propensity to replenish RBCs, ρ (Figure 4b). These subtle functional differences contribute to suppressing parasite density and likely promote better resistance and survival outcomes, at least against this particular parasite genotype (Figure 1).
In addition to having the highest propensity to activate targeted clearance of iRBCs (highest ${\psi}_{{N}_{2}}$), WSB/EiJ, another fully resilient strain, demonstrated the smallest burst size, β (Figure 4). While we found a general negative association between β and host survival (Figure 4) — with a notable exception of PWK/PhJ mice — little is documented on the host’s contribution to variation in iRBC burst size. One host factor that could affect parasite burst size is intrinsic differences in RBC properties among mouse strains. For example, the flow cytometry marker, TER119, a standard marker for mouse erythroid cells, works poorly with WSB/EiJ, hinting at a possible difference in RBC surface proteins (Davis, personal observation). WSB/EiJ also showed the highest capacity to increase background RBC mortality during infection, ${\mu}_{R}^{\prime}$, which contributes to limiting parasite growth through lowered resource availability. Overall, WSB/EiJ excelled in every facet of resilience against P. chabaudi AJ and maintained comparatively much lower iRBC densities (Figure 2). Infection resilience in this mouse strain may generalise to other malaria parasites, including P. berghei (Bopp et al., 2010), yet interestingly, these mice are highly vulnerable to Salmonella infections (Zhang et al., 2019).
Three less resilient strains (CAST/EiJ, NOD/ShiLtJ, and PWK/PhJ) clustered together in the PCA biplot (Figure 5), indicating their functional similarity. We found a combination of topdown (immunemediated) and bottomup (resourcemediated) factors that distinguish these strains from the rest. Specifically, they showed low propensity to trigger a targeted immune response (low ${\psi}_{{N}_{2}}$, which hinders effective parasite clearance) and high erythropoiesis during infection (high ρ and ${\mu}_{R}^{\prime \prime}$, which may inadvertently fuel parasite growth; Figures 4 and 5). Distinguishing moderately poor surviving strains (CAST/EiJ and NOD/ShiLtJ; 72% and 40% survival, respectively) from an extremely fragile strain (PWK/PhJ; $<5\%$ survival) is likely the markedly lower immune activation, ${\psi}_{{N}_{2}}$, in the latter, since other traits were similar among the three strains.
Finally, the A/J mouse strain showed the strongest activation of indiscriminate RBC clearance, ${\psi}_{{N}_{1}}$ and the secondhighest activation of the targeted response, ${\psi}_{{N}_{2}}$ (Figure 4b). Given the poor resilience of these mice (20% survival; Figure 1), general RBC clearance is likely harmful, at least in this strain of mice. While the host potentially stands to benefit from destroying RBCs by removing some iRBCs and taking the resource away from malaria parasites, this ‘scorchedearth tactic’ (Wale et al., 2019) could remove healthy RBCs in excess and trigger severe anaemia that causes host mortality. The significant role of indiscriminate RBC clearance on severe anaemia is empirically supported by a study that demonstrated a high turnover of transfused RBCs in BALB/c mice infected with P. berghei (Evans et al., 2006). The clearance was likely immunemediated as severe anaemia was alleviated by depletion of immune cells (Evans et al., 2006). Another study observed lower young RBC (i.e., reticulocyte) counts in A/J mice and postulated that these mice are defective in the production of new RBCs during malaria infection (Chang et al., 2004). This is consistent with our model prediction that, among poorly resilient strains, A/J mice showed the lowest propensity to upregulate erythropoiesis (lower ρ values, relative to CAST/EiJ, NOD/ShiLtJ, and PWK/PhJ; Figure 4). Together with our finding that A/J mice mount a stronger response than other poorly resilient strains (Figure 4), the lower ρ values in A/J mice may reflect the inhibitory effect of inflammation on the steadystate erythropoiesis that takes place in the bone marrow (Morceau et al., 2009). As the interactions between inflammation and erythropoiesis are complex (e.g., inflammatory responses also induce stress erythropoiesis in the spleen [Paulson et al., 2020] and erythropoietin [EPO, cytokine primarily responsible for RBC production] inhibits inflammation [Nairz et al., 2012]), further investigation is needed to better understand the net impact of malariainduced immune responses on RBC production.
The potentially negative impact of RBC clearance highlights vital implications for clinical interventions against malaria. First, blood transfusion or EPO injection to replenish depleted RBCs in severe anaemia may be most effective against patients that are particularly prone to indiscriminate RBC clearance. In fact, timely EPO injection alleviates anaemia and improves survival in A/J mice (Chang et al., 2004). However, the same treatment may be less effective against other poorly resilient patients (similar to CAST/EiJ, NOD/ShiLtJ, and PWK/PhJ) whose mortality is attributed to insufficient immune responses (Figures 4 and 5). Second, the potentially pathological consequence of indiscriminate RBC clearance should be considered during the development of a bloodstage malaria vaccine. Alarmingly, the possibility of immunopathology has so far been largely overlooked in the vaccine development process (Stanisic and Good, 2016). It is pertinent to ensure that vaccinetriggered immunity that helps clear malaria parasites also avoids immunopathology, including severe anaemia. In summary, our model predicted that poor survival was associated with hosts with the weakest activation of the targeted response (in CAST/EiJ, NOD/ShiLtJ and PWK/PhJ), or the strongest activation of the indiscriminate response (in A/J). Thus, the host’s ability to mount a precise response to clear parasites is likely a major determinant of host survival.
Cytokine assays uncover molecular variation in immune responses
As our mathematical model identified the strength and precision of host immunity as the key functional motif of malaria resilience, we carried out a separate crosssectional assay (with destructive sampling) and characterised the expression patterns of pro and antiinflammatory cytokines (i.e. immune signalling molecules), which play a pivotal role in regulating immune responses (Figure 6). Notably, proinflammatory cytokines such as TNFα and IFNγ impact malaria parasite clearance, while antiinflammatory cytokines like IL10 and TGFβ are crucial for limiting inflammation and immunopathology (ArtavanisTsakonas et al., 2003).
In general, resilient strains (WSB/EiJ, 129S1/SvImJ, NZO/HILtJ, C57BL/6) showed a higher level of cytokine activity (both pro and antiinflammatory; Figure 6a,b and c) while poorly resilient strains showed either a relatively stunted activity (CAST/EiJ and PWK/PhJ), or tendency towards proinflammatory biased expression (NOD/ShiLtJ and A/J). These findings provided explanations at the molecular level that dovetail with our model inference about the variation in the strength and precision of the net effect of immune responses in these strains. Specifically, the model predicted that the highly resilient mouse strains 129S1/SvImJ, NZO/HILtJ and to a large extent C57BL/6, activate the targeted response more strongly than the less resilient CAST/EiJ, NOD/ShiLtJ, and PWK/PhJ. This prediction was consistent with the higher expression of proinflammatory cytokines in these strains (Figure 6a,b and c), in comparison to two of the lesser resilient strains (CAST/EiJ and PWK/PhJ) for which we predicted weak immune responses (Figures 4 and 5). Importantly, in the three strains that show higher survival (129S1/SvImJ, NZO/HILtJ, and C57BL/6), the robust expression of TNFα and IFNγ was matched by equally robust expression of IL10 and TGFβ (Figure 6d), which inhibit overproduction of immune effectors (ArtavanisTsakonas et al., 2003). Less resilient CAST/EiJ and PWK/PhJ showed a comparatively low expression of both pro and antiinflammatory cytokines (Figure 6a,b and c). The lower proinflammatory cytokine expression draws a parallel with our modelling results that CAST/EiJ and PWK/PhJ show comparatively weaker immune activation (Figure 4).
We found signs of proinflammatory bias in the cytokine expressions of NOD/ShiLtJ and A/J (Figure 6), both of which are poorly resilient at 40% and 20% survival, respectively (Figure 1). In these strains, we observed elevated expression of proinflammatory cytokines (particularly IFNγ) without a matched increase in antiinflammatory responses (Figure 6b,c and d), which inhibit overproduction of immune effectors (ArtavanisTsakonas et al., 2003). For the A/J mice, our model predicted strong immune responses, both indiscriminate and targeted (high ${\psi}_{{N}_{1}}$ and ${\psi}_{{N}_{2}}$; Figure 4), consistent with the expectation that a proinflammatory bias leads to strong host responses (King and Lamb, 2015). However, immunoregulatory imbalance is also associated with immunopathology during malaria infections (King and Lamb, 2015). In particular, overproduction of immune effectors could cause collateral host tissue damage and excessive indiscriminate RBC clearance may lead to severe anaemia. Although generally less pronounced than A/J, resilient strains NZO/HILtJ and C57BL/6 also showed a relatively high ratio of pro to antoinflammatory cytokines (Figure 6d). Our model predicted that AJ, NZO/HILtJ, and C57BL/6 also exhibit strong activation of indiscriminate RBC clearance (high ${\psi}_{{N}_{1}}$; Figure 4). Thus, our modelling results and cytokine assays together suggest a causal link between the strength of hostdriven destruction of RBCs — which may be both beneficial and detrimental to the host (Wale et al., 2019; Perkins et al., 2011) — and the underlying proinflammatory bias (Figure 6d). At first glance, our model prediction that NOD/ShiLtJ mice trigger weaker than average immune activation (low ${\psi}_{{N}_{1}}$ and ${\psi}_{{N}_{2}}$; Figure 4) appears incongruent with the strong relative expression of proinflammatory cytokines (Figure 6d). However, NOD mice are documented for immunodeficiencies downstream of inflammatory cytokines: for example, severely reduced natural killer cell activity (Kataoka et al., 1983), hyporesponsiveness of macrophages to growth factors and IFNγ (Serreze et al., 1993), and defective development of antigenpresenting dendritic cells (Pearson et al., 2003). Thus, our model prediction and cytokine assay together indicate that inflammatory responses at the molecular level failed to translate functionally to effective clearance of malaria parasites.
We observed low raw intensity of both pro and antiinflammatory cytokines in WSB/EiJ (Figure 6b), for which our model predicted exceptional parasite control (through small parasite burst size and high parasite clearance, low β and high ${\psi}_{{N}_{2}}$) and resource suppression (heightened RBC mortality, high ${\mu}_{R}^{\prime}$; Figures 4 and 5). At first, low cytokine intensity appeared to be at odds with high parasite clearance. However, it is worth highlighting that the WSB/EiJ mice experience exceptionally low iRBC density (Figure 2) and hence the cue for triggering cytokine production remains low. When scaled by the iRBC density, it becomes apparent that the propensity of WSB/EiJ to express both pro and antiinflammatory cytokines generally exceeds that of other resilient strains (Figure 6d). Further studies are needed to empirically determine what explains the exceptionally low iRBC density in this strain. However, our modelling results suggest that mechanisms other than inflammation may also facilitate WSB/EiJ’s resilience: for example, small parasite burst size, low β, may be linked to natural RBC resistance to parasite invasion (Taylor and Fairhurst, 2014).
Overall, cytokine assays complemented mathematical modelling by uncovering variation in molecular mechanisms that underlies functional differences among host strains that show diverse infection outcomes. Host resilience to Plasmodium infection was linked to a balanced expression of pro and antiinflammatory cytokines. Poorly resilient strains either showed stunted activation of proinflammatory cytokines associated with insufficient parasite control, or proinflammatory bias that has been implicated in immunopathology (ArtavanisTsakonas et al., 2003).
Conclusion
It is well documented that immune responses are a key host factor influencing protection from malaria infections (López et al., 2010). However, it remains difficult to relate health outcomes to the underlying variation in host immunity because parasite load, immune regulation and host health are intertwined and are variable over time. Over the past two decades, several dynamical models have inferred mechanisms of complex withinhost ecological interactions from longitudinal data of parasite load and RBCs (i.e. parasite resource and indicator of host health) in the rodent malaria system (e.g. Haydon et al., 2003; Mideo et al., 2008b; Kochin et al., 2010; Miller et al., 2010; Mideo et al., 2011; Metcalf et al., 2011; Metcalf et al., 2012; Santhanam et al., 2014; Wale et al., 2019; Kamiya et al., 2020). However, a knowledge gap exists between what is empirically measurable (at the molecular and cellular level, for example immune molecules and cells) to what is functionally important to infection dynamics (at the withinhost ecological level, for example net effect of parasite clearance). At the withinhost ecological level, our dynamical modelling of infection revealed that better host survival during malaria infection was associated with precisely targeted, robust clearance of bloodstage parasites. Using crosssectional cytokine assays, we uncovered wellregulated inflammatory cytokine expressions are key molecular signatures of inbred mouse strains that survive malaria infections. By augmenting mathematical modelling of withinhost ecology with crosssectional cytokine assays, our study narrows the gap between functional and molecular mechanisms of host resilience to malaria infection.
Materials and methods
Data
Mouse strains
Request a detailed protocolMice were purchased from Jackson Laboratories (WSB/EiJ stock #001145), 129S1/SvImJ stock #002448, NZO/HILtJ stock #002105, CAST/EiJ stock #000928, A/J stock #000646, NOD/ ShiLtJ stock #001976, and PWK/PhJ stock #003715 and Charles River (C57BL/6). A subset of mice (WSB/EiJ, NZO/HILtJ, and PWK/PhJ) were also bred inhouse at Stanford University. Animals were housed in the Stanford Research Animal Facility according to Stanford University guidelines. All mouse experiments were approved by the Stanford Administrative Panel on Laboratory Care (APLAC).
Infection with P. chabaudi
Request a detailed protocolWe administered the AJ strain of Plasmodium chabaudi to the experimental animals at a dose of 10^{5} iRBCs and monitored infections longitudinally, for 15 days, as previously described (Torres et al., 2016). RBCs were quantified using a BD Accuri C6 Plus cytometer. We quantified parasitemia (i.e. proportion of RBCs infected) via thin blood smears and manual microscope counting. We report parasite density as the number of iRBCs per microliter of blood, which was calculated by multiplying parasitemia by the number of total RBCs per microliter of blood. Survival was monitored daily until day 15 postinfection.
Crosssectional cytokine assay
Request a detailed protocolWe infected mice with P. chabaudi as described above. For crosssectional sampling, between three and five infected mice of each strain were euthanised each day between 10 am and 2 pm from days 3 to 12 postinfection. For each mouse strain, two uninfected control animals were euthanised at baseline and generally on oddnumbered days. Following euthanasia, 75 microliters of plasma was used for immunoassay using the mouse 38plex kit (eBiosciences/Affymetrix). Further details are available from Davis et al., 2021.
Model
Dynamical model of malaria asexual cycle
Request a detailed protocolIn the experiment, mice were inoculated with iRBCs. Synchronously at midnight, the initial cohort of iRBCs rupture, releasing a new generation of merozoites into the bloodstream (Paul et al., 2003). These merozoites then rapidly invade further RBCs where the parasites develop to release the next generation of merozoites with a 24 hr interval. We assumed that regulation of immune responses takes place continuously throughout the day ($0<t\le 1$ where $t$ is the fraction of a day) and demographic processes of the host and parasite (i.e., turnover of RBCs, iRBC bursting and RBC invasion by merozoites) happen at the end of each day at midnight ($t=1$).
Regulation of host responses
Request a detailed protocolMalaria infection triggers a variety of host responses (Stevenson and Riley, 2004; Price et al., 2001; CastroGomes et al., 2014), of which we considered two previously identified as the most quantitatively important: indiscriminate clearance of RBCs and targeted clearance of iRBCs (Miller et al., 2010). Similar to previous studies (Kochin et al., 2010; Kamiya et al., 2020), consider a set of ordinary differential equations tracking the change in the rate of activity of each response ${N}_{i}$, where $i$ indicates the type of response (i.e. general RBC clearance, $i=1$; targeted iRBC clearance, $i=2$):
where ${\psi}_{{N}_{i}}$ and ${\varphi}_{{N}_{i}}$ are the activation and decay strength of ${N}_{i}$, respectively. Assuming that the abundance of iRBCs reflects that of pathogenassociated molecular patterns (PAMPs), we defined the scaled density of iRBCs, $\frac{{I}_{(t)}}{{I}_{max}}$, as the withinhost cue driving these responses, where ${I}_{max}$ is the maximum observed iRBC density in the experiment. We assumed that there is no activity in the absence of infection, consequently there is no constitutive immunity in this model. Because demographic events are formulated in discrete time with a unit of one day, the iRBC density on day $d$, is assumed a constant during the time scale of immune regulation (i.e. $0<t\le 1$). Previous estimates indicate that the response activity of indiscriminate clearance decays in approximately one day while the targeted response decays with a halflife an order of magnitude longer than the duration of the acute phase of infection (Kamiya et al., 2020): mean halflife of 0.96 and 332.6 days, respectively. Based on these estimates, we made the following simplifying assumptions, eliminating parameters ${\varphi}_{{N}_{1}}$ and ${\varphi}_{{N}_{2}}$: the indiscriminate activity decays at one day following the Diracdelta distribution and the targeted activity does not decay during the acute phase. Consequently, we can reformulate Equation 1 in discretetime, assuming that the former response resets daily while the latter accumulates over multiple days without any decay:
Turnover of RBCs
Request a detailed protocolThe first event at the end of the day ($t=1$), is clearance and replenishment of RBCs. In the absence of infection, we assumed that RBCs are subjected to background RBC mortality, ${\mu}_{R}$. The host replaces RBCs lost to baseline cell mortality by producing ${R}_{c}(1{e}^{{\mu}_{R}})$ new RBCs, where ${R}_{c}$ is the RBC density at homeostatic equilibrium, assumed equal to the RBC density measured per mouse before parasite inoculation. While we have prior information on baseline mortality and replenishment in the absence of infection (i.e. ${\mu}_{R}$; Van Putten and Croon, 1958; Foster et al., 2014), we fitted two new parameters (${\mu}_{R}^{\prime}$ and ${\mu}_{R}^{\prime \prime}$, respectively) that allow these processes to be qualitatively different during infection. Our motivation for this was the following. First, during malaria infection experiments, RBCs are lost to daily blood sampling and perhaps handling related stress. Second, fitting ${\mu}_{R}^{\prime}$ allows for the possibility that some indiscriminate clearance of RBCs is independent of iRBC dynamics (unlike in Equation 2). Third, there is evidence that erythropoiesis is downregulated during malaria infection (Wale et al., 2019), and fitting ${\mu}_{R}^{\prime \prime}$ allowed us to capture this possibility.
Indiscriminate RBC clearance, due to the action of the immune response, occurs at a daily rate ${N}_{1(d,t=1)}$. In addition to baseline replenishment of RBCs (governed by ${\mu}_{R}^{\prime \prime}$ as described above), RBCs are produced in a densitydependent manner during infection to restore the RBC population (Chang et al., 2004) with a timelag of 2–3 days before the newly produced RBCs are released in the bloodstream (Savill et al., 2009). Here assuming a 2day lag (indicated by $d2$), the host produces a fraction ρ of the deviation from RBC density at ${R}_{c}$. Infected cells incur an additional rate of mortality, ${N}_{2(d,t=1)}$ through targeted killing. Together, the turnover of RBCs is expressed as:
where ${R}_{(d,t=1)}$ and ${I}_{(d,t=1)}$ are the postturnover densities.
iRBC bursting
Request a detailed protocolGiven synchronous iRBC bursting and the short lifespan of merozoites relative to the length of a day, we modelled iRBC bursting and merozoite invasion as instantaneous events. As iRBCs rupture and release merozoites into the bloodstream at midnight ($t=1$), the density of merozoites, ${M}_{(d,t=1)}$ equals $\beta {I}_{(d,t=1)}$ where β is the parasite burst size per iRBC.
RBC invasion by merozoites
Request a detailed protocolUpon release, a merozoite either invades an uninfected red blood cell (uRBC), ${R}_{(d,t=1)}$, at a per capita invasion rate $p$, or it gets cleared before invasion, with a short halflife of 1/μ_{M} (∼30 min [McAlister, 1977]). For simplicity, we ignore infections of RBCs by multiple merozoites. Thus, the probability that a given merozoite successfully invades an uRBC is:
Multiplying the probability by the density of merozoites, and dividing by ${R}_{(d,t=1)}$, the average number of invading merozoites per uRBC, λ is:
We assumed that the probability of RBC invasion by merozoites is Poissondistributed with parameter λ, that is, $\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}(\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\mathrm{b}\mathrm{y}\phantom{\rule{thinmathspace}{0ex}}k\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\mathrm{e}\mathrm{z}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{s})=\frac{{\lambda}^{k}\phantom{\rule{thinmathspace}{0ex}}{e}^{\lambda}}{k!}$ (Miller et al., 2010; Mideo et al., 2011). Thus, the probability that a given uRBC gets invaded by a merozoite (i.e. $k=1$) is $\lambda {e}^{\lambda}$ and the probability that an uRBC escapes merozoite invasion altogether (i.e. $k=0$) is ${e}^{\lambda}$. Ignoring infections of RBCs by multiple merozoites, it follows that the numbers of uRBCs and iRBCs after merozoite invasion (i.e. ${R}_{(d,t=1)}^{*}$ and ${I}_{(d,t=1)}^{*}$ with the asterisk denoting the postinvasion densities) are:
Hierarchical Bayesian inference
Request a detailed protocolWe fitted the above withinhost infection model to the corresponding longitudinal data from 80 mice using a Bayesian statistical approach, which allows for parameter estimation in high dimensional spaces, for example, in hierarchical models where observations are organised in multiple levels of sampling units (Mugglin et al., 2002; Cressie et al., 2009). In this study, there are two levels of sampling units: mouse strains and subjects (i.e. individual mice).
Strainspecific and individual variation
Request a detailed protocolWe estimated host strain and individualspecific effects in a set of nine fitted parameters describing withinhost ecological processes: that is, samples $s$ and $u$ from $\mathcal{N}(0,{\sigma}_{s})$ and $\mathcal{N}(0,{\sigma}_{u})$, respectively. Below, we collectively refer to the parameter set as θ ($\theta \ni {\mu}_{R}^{\prime},{\mu}_{R}^{\prime \prime},\rho ,{\psi}_{{N}_{1}},{\psi}_{{N}_{2}},\beta $). The prior distributions for these parameters are provided in Table 1.
Likelihood
Request a detailed protocolA Bayesian approach requires a likelihood function to assess the probability of observing the data given model parameters and associated predictions. Our loglikelihood function assumed that the measurement error for the total density of RBCs (i.e. sum of uRBCs and iRBCs), and iRBCs is distributed normally and log_{10}normally, respectively (Mideo et al., 2008b; Mideo et al., 2011):
where ${D}_{i,d}^{\text{RBC}}$ and ${D}_{i,d}^{\text{iRBC}}$ are the observed count of total RBCs and iRBCs, ${M}_{i,d}^{\text{RBC}}$ and ${M}_{i,d}^{\text{iRBC}}$ are the model predictions of total RBCs and iRBCs for individual $i$ at day $d$. We estimated standard deviations, ${\sigma}_{\text{RBC}}$ and ${\sigma}_{\text{iRBC}}$ for the total RBC and iRBC count, respectively, with specific informative priors (Mideo et al., 2008b; Miller et al., 2010; Table 1). Our modelling focused on the first wave of infection, thus we fitted data up to two weeks postinfection (${n}_{\text{days}}=14$ at maximum). In mice that succumbed to infection, we fitted the model to data until the last sampling prior to death.
MCMC sampling
Request a detailed protocolEstimating the posterior probability density of parameters of a complex model requires a Markov Chain Monte Carlo (MCMC) sampling algorithm. Our model was written in Stan 2.21.2 and fitted through the RStan interface (Carpenter et al., 2017; Stan Development Team, 2019), which provides an efficient, generalpurpose MCMC sampler (NoUTurn Hamiltonian Monte Carlo) and a Bayesian inference environment. The model was fitted in parallel in four independent chains, each with 3000 sampled iterations and 1000 warmup iterations. For diagnostics, we confirmed over 400 effective samples and ensured convergence of independent chains using the $\widehat{R}$ metric (values below 1.1 are considered an indication of multichain convergence) for all parameters (Gelman et al., 2013; Stan Development Team, 2018). We assessed the goodness of fit to data using standardised residuals (Appendix 1). We also quantified the posterior zscore and posterior contraction to examine the accuracy and precision of posterior distributions, and the relative strength of data to prior information (Schad et al., 2021) (Appendix 2).
Appendix 1
Assessment of model fit
Our fitted model accurately describes the daily time course of RBCs and iRBCs during the acute phase of malaria infection in mice (Appendix 1—figure 1).
To provide a rigorous assessment of the model fit, we examined the standardised residuals for RBC and iRBC densities following Miller et al., 2010. By integrating over the probability density of each parameter, Φ, the marginal standardised residual of each data point $i$ was defined as:
where ${\sigma}_{x}$ is standard deviation of $x$, which is either RBC or iRBC density. The fit of the dynamical model to RBC and iRBC density was accurate without a significant sign of bias (Appendix 1—figure 2).
Appendix 2
Assessment of posterior and prior distributions
Posterior correlations
Visualising correlations between model parameters aids assessing practical identifiability of a parameter set (Gabry and Mahr, 2021), i.e., whether the likelihood function can distinguish between the contributions from multiple parameters in the set. The absolute correlation coefficients of 0.7 or larger are generally considered to indicate severe collinearity that adversely affects parameter estimation (Dormann et al., 2013). With no such strong correlation observed, the pairwise correlation plot indicates that the MCMC algorithm can sample the parameter set θ without heavy reliance on prior information (Appendix 2—figure 1).
Posterior accuracy, precision and prior contraction
To provide a rigorous assessment of our Bayesian inference, we leveraged the properties of posterior distributions to interrogate our modelling assumptions. To examine the accuracy and precision of posterior distributions, we first generated simulated observations based on the estimated posterior mean parameters. We then refitted our model to the simulated observations (i.e., secondary fitting) to compute the posterior zscore for each parameter, which measures how closely the posterior recovers the parameters of the true data generating process (Schad et al., 2021):
where ${\mathbb{E}}_{\text{post}}$ denotes the posterior mean of the fit to the actual data that we consider the ‘true’ parameter. ${\mathbb{E}}_{\text{sim}}$ and ${\sigma}_{\text{sim}}$ denote the mean and standard deviation of the posterior distribution of the secondary fitting. The smaller the absolute zscore, the closer the bulk of the posterior is to the true parameter: zscores beyond the absolute value of three to four may indicate substantial bias (Schad et al., 2021).
To examine the influence of the likelihood function in relation to prior information, we computed the posterior contraction:
where ${\sigma}_{\text{post}}^{2}$ and ${\sigma}_{\text{prior}}^{2}$ correspond to the variance of posterior and prior distributions, respectively. The posterior contraction values close to zero indicate that data contain little information (i.e., poor identifiability, rendering priors strongly informative). Conversely, values close to one indicate that data are much more informative than the prior (Schad et al., 2021).
We found that most of our model parameters — i.e., $\theta \ni ({\mu}_{R}^{\prime},{\mu}_{R}^{\prime \prime},\rho ,{\psi}_{{N}_{1}},{\psi}_{{N}_{2}},\beta )$ and hyperpriors ${\sigma}_{s}$ and ${\sigma}_{u}$ — were estimated with accuracy, precision and identifiability, with the absolute posterior zscores well below three and posterior contraction values beyond 75% for most parameters (Appendix 2—figure 2). The strainlevel variation, ${\sigma}_{s}$ for ρ (proportion of anaemia restored per day) tended towards overfitting with the posterior zscore of $2.99$ (Appendix 2—figure 2). Thus, caution might be warranted when interpreting strainspecific differences in this parameter. One parameter, ${\mu}_{R}^{\prime \prime}$ (densityindependent RBC replenishment rate during infection) showed a comparatively lower posterior contraction value, yet its posterior distribution contracted by 64.0%, meaning that data still provided substantial information over the prior distribution (Appendix 2—figure 2).
Data availability
We include the entire workflow (data and modelling, analysis and visualisation programs) as Supplementary file 1. Further details of the data can be found in Davis et al., 2021 (mBio; https://doi.org/10.1128/mBio.0242421).
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Decision letter

Jennifer FleggReviewing Editor; The University of Melbourne, Australia

Miles P DavenportSenior Editor; University of New South Wales, Australia

Jennifer FleggReviewer; The University of Melbourne, Australia
In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.
Acceptance summary:
This paper will be of interest to parasitologists, immunologists and modellers working in the field of malaria. It provides a modelbased approach to quantify different host responses to malaria infection and identify the response(s) best explaining the infection outcomes (i.e., survival rate and host resilience to malaria infection). The methods for model construction and fitting and data analysis are appropriate and rigorous, however, extra work is required for validation of model predictions, which are critical for the interpretation of the results.
Decision letter after peer review:
Thank you for submitting your article "Withinhost dynamics reveal functional diversity of host resilience to malaria infection" for consideration by eLife. Your article has been reviewed by 3 peer reviewers, including Jennifer Flegg as the Reviewing Editor and Reviewer #1, and the evaluation has been overseen by Miles Davenport as the Senior Editor.
The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.
Essential revisions:
1) Parameter correlations. The results of the manuscript are primarily based on the parameter estimates, which are calculated by the samples of the posterior distributions, so the correlations between model parameters (based on the posterior samples) should be examined. One example for the correlated parameter would be the activation rate and decay rate of immune response in equation 1. Since the immune response N_{i} was not measured and used in model fitting, the two parameters should show a strong positive correlation, which would prevent accurate estimates. A clear view of the parameter correlations is critical for the reliability of the estimates and in turn any conclusion based on the estimates.
2) Prior distributions. These were not shown on top of the posteriors such that it is difficult to judge how much information was from the priors and how much was from the data. Since the priors are relatively informative, e.g. 2.5 and sqrt(2.5) (see Table 1), it is also necessary to examine the impact of the choice of the priors on the posteriors by a sensitivity analysis. The choice of priors is not welljustified. How were the hyperparameters for the host response parameters chosen? Is the model sensitive to these?
3) The authors showed that the too strong and too weak immune activation predicted by the model was associated with high host mortality and concluded that a balanced immune response may be a major determinant of host survival. However, a stronger immune activation in the model refers to a higher activation rate (i.e., a higher value of parameter psi in equation 1). This is not necessarily related to a balance immune response, which is a balanced level of expression of proinflammatory and anti inflammatory cytokines and is mathematically interpreted by that a strong/weak activation should be associated with a strong/weak decay such that the net effect (N_{i} in equation 1) is maintained at a healthy level. Further clarification on this point is necessary.
4) Cytokine dynamics. In the results, the authors selected and discussed some model predictions that are consistent with what the cytokine data suggest. However, it lacks a comprehensive comparison between the data and the model prediction. I am wondering if it is possible to create a figure similar to Figure 3b (only psi's and phi's) using the cytokine data and compare with the model prediction. It will tell clearly how well the model predictions are consistent with the data. Could you connect the cytokine dynamics with the mechanistic model more directly? This seems like a missed opportunity – you have beautiful timecourses here – could you reframe the model to directly include processes reflected by the cytokines?
5) Figures. It would be useful to have a detailed graphical representation of the model in the main text. The temporal information in Figure 5 is hard to process.
Reviewer #1:
This paper presents a dynamic mathematical model of malaria parasites and red blood cells in a mouse model. A strength of the paper is the collection of experimental data to fit to the mathematical model, however the model is overly simplified and is therefore unlikely to have high impact in the field.
Reviewer #2:
Kamiya, Davis et al., studied the associations between the diversity of host responses to malaria infection and infection outcomes (i.e., survival rate and host resilience to malaria infection). In detail, they collected the longitudinal data from mice that are genetically different and estimated some key biological parameters governing the parasite replication dynamics and immune responses by fitting a mathematical model to the data. By comparing the parameter estimates and a set of cytokine data collected from the same pool of mice, they identified some consistency between model predictions and data and concluded that the longitudinal data contain important information that may facilitate the explanation of the diverse infection outcomes. The methods for model fitting and data analysis are appropriate and rigorous. The conclusion about balanced immune response is partly supported by the data but not clearly associated with model predictions. Some further analyses on fitting results are necessary in order to ensure the reliability of some conclusions.
Reviewer #3:
The first section of the analysis connects parameter estimates with outcome. With such a large model this section needs a thorough analysis of correlations between parameters – the PCA goes a little way towards this, but not enough. Identifying parameter subspaces might also make the interpretation of these fitted values simpler.
The second section – the data are rich, but it feels let down slightly by its qualitative nature. Could the authors connect the cytokine dynamics with the mechanistic model more directly? This seems like a missed opportunity – there are beautiful timecourses here – could the model be reframed to directly include processes reflected by the cytokines?
It would be useful to have a detailed graphical representation of the model
in the main text.
The temporal information in Figure 5 is hard to process. I like the idea of the 'phasespace' view but here I think it would be clearer to have the timecourses of pro and antiinflammatory cytokines stacked on top of each other.
At what time(s) of day were the cytokine measurements made? Were they consistent? Given there is a daily replication/infection cycle, I would guess timing matters.
p9 "We found that reducing the expression of antiinflammatory …" This phrase is misleading as there was no intervention that reduced expression.
https://doi.org/10.7554/eLife.65846.sa1Author response
Essential revisions:
1) Parameter correlations. The results of the manuscript are primarily based on the parameter estimates, which are calculated by the samples of the posterior distributions, so the correlations between model parameters (based on the posterior samples) should be examined. One example for the correlated parameter would be the activation rate and decay rate of immune response in equation 1. Since the immune response N_{i} was not measured and used in model fitting, the two parameters should show a strong positive correlation, which would prevent accurate estimates. A clear view of the parameter correlations is critical for the reliability of the estimates and in turn any conclusion based on the estimates.
We fully reexamined our modelling assumptions based on this request. First, we have eliminated the decay parameters ϕ_{N1} and ϕ_{N2}, which were correlated with the activation parameters ψ_{N1} and ψ_{N2}, as the Editor rightly pointed out. Our previous estimates indicate that the response activity of indiscriminate clearance decays approximately in one day while the targeted response decays with a halflife an order of magnitude longer than the duration of the acute phase infection (Kamiya et al., 2020 PLOS Comp Biol). Based on these estimates, we made the simplifying assumptions that the indiscriminate activity decays with a halflife of one day following the Diracδ distribution and the targeted activity does not decay during the acute phase. Consequently, we reformulated immune regulation in discretetime, assuming that the former response resets daily while the latter accumulates over multiple days without any decay (Figure 3a, Equation 2 and 3). These changes are detailed between lines 247 and 253. We acknowledge that it is potentially problematic to rely on previous modelled inferred parameter estimates where reliance on prior information was not systematically quantified. To demonstrate that our estimates are robust despite incorporating estimates from previous modelling studies (namely, Miller et al., 2010, Mideo et al., 2011, and Kamiya et al., 2020), we conducted a rigorous examination of the posterior distributions (i.e., posterior zscore and posterior contraction) to demonstrate that our inference was not biased and prior information had relatively small influence on the posterior compared to data.
Second, we found that the ratio of merozoite mortality to invasion rate showed a moderately high correlation (>0.65) with burst size and did not show substantial strainspecific variation. Thus we have now fixed the ratio based on a previously published estimate (Miller et al., 2010 PLOS Comp Biol).
We confirm that the new simplified model preserves the goodness of fit of the older model (Appendix 1) while markedly improving the MCMC sampling behaviour. We believe that eliminating ϕ_{N1} and ϕ_{N2} does not impact our inference substantially because we found larger strainspecific variability in the rates of activation (ψ_{N1} and ψ_{N2}) than decay of responses (ϕ_{N1} and ϕ_{N2}) in the previously submitted version. This indicated to us that immunogenic differences between strains that we can infer using our model are largely due to differences in the speed and strength with which responses are turned on, rather than how long those responses last.
Based on this new model, we now provide a pairwise correlation plot, which shows that there are no strong correlations between modelled parameters: the highest correlation (r = 0.54) was found between background RBC mortality rate and densityindependent RBC replenishment rate during infection (Appendix 2 Figure 1).
2) Prior distributions. These were not shown on top of the posteriors such that it is difficult to judge how much information was from the priors and how much was from the data. Since the priors are relatively informative, e.g. 2.5 and sqrt(2.5) (see Table 1), it is also necessary to examine the impact of the choice of the priors on the posteriors by a sensitivity analysis. The choice of priors is not welljustified. How were the hyperparameters for the host response parameters chosen? Is the model sensitive to these?
Thanks to eliminating poorly identifiable parameters, our model now samples well with weaker priors in the scale of exp(𝒩(0,1)) compared to exp(𝒩(0,0.25)) in the previously submitted version. We also corrected an oversight in leaving out the hyperpriors from the parameter table (Table 1), which are now estimated with a prior, exp(𝒩(0,1)). This prior is generic and even less informative than recommended by the Stan Developmental Team (https://github.com/standev/stan/wiki/PriorChoiceRecommendations).
We agree with the Editor that the influence of prior information should be studied and presented. To provide a rigorous assessment of Bayesian inference, we now evaluate properties of posterior and prior distributions by estimating the posterior zscore and posterior contraction (Appendix 2). The former measures how closely the posterior recovers the parameters of the “true” data generating process while the latter quantifies the influence of data in relation to prior information. By plotting the two values against each other (Appendix 2 Figure 2), we can identify problematic posterior behaviours including bad priors, poor identifiability (i.e., strong priors), overfitting (i.e., inability to recover “true” parameters.) (Betancourt 2020 Towards A Principled Bayesian Workflow).
We confirm that we detected no clear sign of problematic posterior behaviours (Appendix 2 Figure 2), though the strainlevel variation, σ_{s} for rho (proportion of anaemia restored per day) showed a slight tendency towards overfitting. Our model successfully recovers “true” parameter values from simulated data (i.e., low absolute posterior zscore, – 2.99 at maximum) and posterior distributions are substantially narrower than the prior distributions (high posterior contraction, >75% contraction for all but one and 64% at minimum).
3) The authors showed that the too strong and too weak immune activation predicted by the model was associated with high host mortality and concluded that a balanced immune response may be a major determinant of host survival. However, a stronger immune activation in the model refers to a higher activation rate (i.e., a higher value of parameter psi in equation 1). This is not necessarily related to a balance immune response, which is a balanced level of expression of proinflammatory and anti inflammatory cytokines and is mathematically interpreted by that a strong/weak activation should be associated with a strong/weak decay such that the net effect (N_{i} in equation 1) is maintained at a healthy level. Further clarification on this point is necessary.
In the previously submitted version, we considered our cytokine results as “confirmatory” of our mathematical modelling. Instead, this comment helped us realise that the cytokine results actually offer “complementary” insights at different scales: the cytokine data provide molecular explanations of the differences in the net effect of host responses we predicted using the mathematical model.
We agree with the editor that a higher immune activation rate does not relate directly to balanced immunity. Our model predicted a robust targeted response for resilient strains while a weak targeted response (CAST/EiJ, NOD/ShiLtJ, PWK/PhJ) or a too strong indiscriminate response (A/J) for poorly resilient strains. Thus, we now interpret the primary motif for resilience as the strength and precision of parasite clearance. Cytokine assays then revealed that a strong and precise net effect of immune response was underpinned by strong and balanced expressions of pro and antiinflammatory cytokines while poorly resilient strains lacked either pro (CAST/EiJ, PWK/PhJ) or antiinflammatory cytokine (A/J). Thus, we now distinguish that our inference about the role of balanced immunity was derived from cytokine assays, and not mathematical modelling. To clarify this nuanced distinction, we changed the phrasing from “balance” to “precision” when discussing our modelling results.
4) Cytokine dynamics. In the results, the authors selected and discussed some model predictions that are consistent with what the cytokine data suggest. However, it lacks a comprehensive comparison between the data and the model prediction. I am wondering if it is possible to create a figure similar to Figure 3b (only psi's and phi's) using the cytokine data and compare with the model prediction. It will tell clearly how well the model predictions are consistent with the data. Could you connect the cytokine dynamics with the mechanistic model more directly? This seems like a missed opportunity – you have beautiful timecourses here – could you reframe the model to directly include processes reflected by the cytokines?
As we mention in the reply above, we realise that our previous assertion that the cytokine data offer direct validation of the model predictions was misleading, and corrected this error. We also agree with the Editor that the temporal aspect of our cytokine data was not communicated well. To improve, we now provide a new figure, Figure 6 to better demonstrate the temporal change in the cytokine expressions. We note that this figure is not meant to offer a direct comparison of ψ_{N1} and ψ_{N2} in Figure 4b as cytokine data offer insights at a different scale (i.e., molecular) from the modelling result (i.e., withinhost ecology).
In part, the model parameters like ψ_{N1} and ψ_{N2} are not directly comparable to observable phenotypes because variation in ψ_{N1} and ψ_{N2} represents the intrinsic differences independent of withinhost cues (e.g., infected red blood cell density). In contrast, cytokine expression is a product of the intrinsic differences and the changing withinhost cues. Withinhost cues are also different among host strains. To facilitate the comparison among strains, we now provide a figure of cytokine expressions scaled by the iRBC density per day (Figure 6c). This new figure demonstrates, for example, that the cytokine intensity of the WSB/EiJ strain is comparable with the other three resilient strains once standardised for its low parasite density.
Even after controlling for variation in withinhost cues, our modelling of withinhost ecology is not designed to correspond directly with cytokine expressions as they are emergent properties of a complex network, of which cytokines take part as regulators (i.e., upstream of effector responses). For example, we observed high proinflammatory cytokine expression in NOD/ShiLtJ, despite our prediction of low immune responses. This discrepancy is likely due to previously documented immunodeficiency downstream at the cellular level in this strain. We believe that this difference highlights a unique merit of Bayesian inference using the withinhost ecology data in revealing complex traits like the strength of response activation, which directly relates to infection dynamics and outcomes, yet it is difficult to measure directly.
We agree that it would be desirable to quantitatively link cytokine expressions to infection dynamics, and ultimately infection outcomes. We recognise two potential approaches. However, we foresee major limitations given our dataset and the current state of knowledge of immune interactions. First, as the editor suggests, it may seem reasonable to correlate the modelling outputs with cytokine data. A key limitation of the correlation approach is that longitudinal data used to fit the model and cytokine data come from different mice as the latter required mouse sacrifice to collect enough blood. As such one would have to reduce the data to the strain level (a maximum of N=8 data points), which severely limits our power to detect meaningful correlations. To clarify this point, we now mention that cytokine data were sampled destructively earlier in the manuscript (L139141).
The second, and a more conceptually pleasing approach would be to directly integrate cytokine interactions in a mechanistic modelling framework. A major conceptual challenge here is that we lack a complete map of molecular and cellular interactions that generates the net effect of host responses that impacts withinhost ecology. Also, we have access to data on only a small number of known molecular components. As such, a priori, we could only construct a partial, sparsely observed network of molecular and cellular interactions that drives the process of parasite clearance. A case study demonstrates that Bayesian estimation of network interactions is plagued by poor identifiability when network regulatory nodes are unobserved (https://www.martinmodrak.cz/2018/05/14/identifyingnonidentifiability/). As such, we believe that neither our dataset nor modelling approach is suitable for modelling a network of molecular interactions directly at this time.
We believe that the combination of modelling and cytokine assays among diverse genetic backgrounds offers unique qualitative insights that narrow the gap between diverse infection dynamics (i.e., withinhost ecology) and their molecular underpinnings.
5) Figures. It would be useful to have a detailed graphical representation of the model
in the main text.
We added a graphical schematic of our mathematical model (Figure 3).
The temporal information in Figure 5 is hard to process.
We now provide a longitudinal representation of this data (new Figure 6a).
https://doi.org/10.7554/eLife.65846.sa2Article and author information
Author details
Funding
Mitacs (Globalink)
 Tsukushi Kamiya
National Science Foundation (IDEAS RCN exchange grant)
 Tsukushi Kamiya
Natural Sciences and Engineering Research Council of Canada (Discovery Grant)
 Nicole Mideo
Compute Canada (Resource Allocation Competition Award)
 Tsukushi Kamiya
 Nicole Mideo
National Institutes of Health (T32 T32 AI00)
 Nicole M Davis
 David Schneider
National Science Foundation (GRFP)
 Nicole M Davis
Defense Advanced Research Projects Agency (W911NF160052)
 David Schneider
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
We thank members of the Mideo group at University of Toronto and the Schneider group at Stanford University for discussions. This work was funded by Mitacs (Globalink Research Exchange Award), IDEAS (RCN Exchange Award), NSERC (Discovery Grant), Compute Canada (Resource Allocation Competition Award), T32 AI007328 and NSF GRFP (NMD), DARPA W911NF16–0052 (DSS). We are also grateful to the senior editor, Prof. Miles Davenport and three anonymous reviewers for their helpful comments.
Ethics
Animal experimentation: Animals were maintained specificpathogen free (SPF) and housed in the Stanford Research Animal Facility according to Stanford University guidelines, accredited by the Association of Assessment and Accreditation of Laboratory Animal Care (AAALAC) International. All mouse experiments were approved by the Stanford Administrative Panel on Laboratory Care (APLAC) under protocol #30923, and every effort was made to minimize animal suffering.
Senior Editor
 Miles P Davenport, University of New South Wales, Australia
Reviewing Editor
 Jennifer Flegg, The University of Melbourne, Australia
Reviewer
 Jennifer Flegg, The University of Melbourne, Australia
Publication history
 Received: December 16, 2020
 Accepted: August 16, 2021
 Version of Record published: October 12, 2021 (version 1)
Copyright
© 2021, Kamiya et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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Further reading

 Computational and Systems Biology
Different strains of a microorganism growing in the same environment display a wide variety of growth rates and growth yields. We developed a coarsegrained model to test the hypothesis that different resource allocation strategies, corresponding to different compositions of the proteome, can account for the observed rateyield variability. The model predictions were verified by means of a database of hundreds of published rateyield and uptakesecretion phenotypes of Escherichia coli strains grown in standard laboratory conditions. We found a very good quantitative agreement between the range of predicted and observed growth rates, growth yields, and glucose uptake and acetate secretion rates. These results support the hypothesis that resource allocation is a major explanatory factor of the observed variability of growth rates and growth yields across different bacterial strains. An interesting prediction of our model, supported by the experimental data, is that high growth rates are not necessarily accompanied by low growth yields. The resource allocation strategies enabling highrate, highyield growth of E. coli lead to a higher saturation of enzymes and ribosomes, and thus to a more efficient utilization of proteomic resources. Our model thus contributes to a fundamental understanding of the quantitative relationship between rate and yield in E. coli and other microorganisms. It may also be useful for the rapid screening of strains in metabolic engineering and synthetic biology.

 Computational and Systems Biology
 Neuroscience
Biological motor control is versatile, efficient, and depends on proprioceptive feedback. Muscles are flexible and undergo continuous changes, requiring distributed adaptive control mechanisms that continuously account for the body's state. The canonical role of proprioception is representing the body state. We hypothesize that the proprioceptive system could also be critical for highlevel tasks such as action recognition. To test this theory, we pursued a taskdriven modeling approach, which allowed us to isolate the study of proprioception. We generated a large synthetic dataset of human arm trajectories tracing characters of the Latin alphabet in 3D space, together with muscle activities obtained from a musculoskeletal model and modelbased muscle spindle activity. Next, we compared two classes of tasks: trajectory decoding and action recognition, which allowed us to train hierarchical models to decode either the position and velocity of the endeffector of one's posture or the character (action) identity from the spindle firing patterns. We found that artificial neural networks could robustly solve both tasks, and the networks'units show tuning properties similar to neurons in the primate somatosensory cortex and the brainstem. Remarkably, we found uniformly distributed directional selective units only with the actionrecognitiontrained models and not the trajectorydecodingtrained models. This suggests that proprioceptive encoding is additionally associated with higherlevel functions such as action recognition and therefore provides new, experimentally testable hypotheses of how proprioception aids in adaptive motor control.