Molecular crypsis by pathogenic fungi using human factor H. A numerical model

Molecular mimicry is the formation of specific molecules by microbial pathogens to avoid recognition and attack by the immune system of the host. Several pathogenic Ascomycota and Zygomycota show such a behaviour by utilizing human complement factor H to hide in the blood stream. We call this type of mimicry molecular crypsis. Such a crypsis can reach a point where the immune system can no longer clearly distinguish between self and non-self cells. Thus, a trade-off between attacking disguised pathogens and erroneously attacking host cells has to be made. Based on signalling theory and protein-interaction modelling, we here present a mathematical model of molecular crypsis of pathogenic fungi using the example of Candida albicans. We tackle the question whether perfect crypsis is feasible, which would imply that protection of human cells by complement factors would be useless. The model identifies pathogen abundance relative to host cell abundance as the predominant factor influencing successful or unsuccessful molecular crypsis. If pathogen cells gain a (locally) quantitative advantage over host cells, even autoreactivity may occur. Our new model enables insights into the mechanisms of candidiasis-induced sepsis and complement-associated autoimmune diseases.


Introduction
Crypsis and mimicry are wide-spread phenomena in biology. They are used to deceive predators, prey, hosts or other interaction partners. Most commonly they are used as defensive strategies like imitation of harmful species by harmless species (Batesian mimicry) or imitation of a dominant element of the environment (cryptic mimesis) [1][2][3]. Aggressive mimicry refers to the observation that some predators use camouflage not to be recognized by their prey. Mimicry can also be useful for reproduction, for example, in egg mimicry realized by some birds [1][2][3]. The effect of mimicry can be understood by analogies from human society. For example, uniforms may be misused by civilians, as was impressively described in the movies "The Sting" and "Catch Me If You Can" and in the theatre play "The Captain of Koepenick" by Carl a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 proteins [14][15][16][17][18][19]. FH recruitment to the pathogen surface could then prevent C3b opsonization, as it does for host cells. The opaque form of C. albicans [27][28][29] is additionally able to degrade C3 and C3b directly using for example the endopeptidase SAP1. This may enhance or enable FH-independent crypsis of the pathogen. In this manuscript, we focus mainly on the white form, which does not express SAP1 [30].
As the difference in C3b deposition on host and pathogen cells decreases, phagocytes may have more trouble to discriminate between self and non-self. In consequence, the host may have to decide on a trade-off between erroneously attacking own cells, possibly inducing autoreactivity, and the effectiveness by which it is able to clear mimetic pathogens from the blood (see Fig 2). This trade-off may not always be easy to find, which might be related to the presence of two different FH alleles in the human population, one leading to a higher risk of autoreactivity than the other [31,32]. This illustrates the role of complement as a 'double-edged sword' [33][34][35] where the immune system may erroneously attack own cells and, on the other hand, pathogens may mimic host cells as an evasion strategy (see [36] for alternative evasion strategies).
One may have different opinions on which phenomenon occurs. Pasteur [2] doubts relationships of molecular mimicry to organismic mimicry because in the former case there is "no On cell surfaces, amplification may occur, depending on whether or not FH can be acquired there. If FH cannot be acquired in sufficient amounts, there is a steady but slow conversion of C3 to C3b initiating amplification. C3b then associates with factor B (FB) to form the C3 proconvertase (C3bB), which is activated by factor D, resulting in the active C3 convertase (C3bBb). C3bBb in turn is able to proteolytically convert C3 into C3b at a high rate, starting the loop again and thus acting as an amplifier of C3b production. The amplification loop can be controlled by FH at two stages. Firstly, FH competes with FB for C3b binding, acting as a cofactor for C3b degradation to iC3b. Secondly FH is able to accelerate the decay of C3bBb, reducing proteolytic C3b generation. sensory perception involved". Nevertheless in the case of molecular mimicry antigens are bound by antibodies which activate Fc-receptors on the surface of phagocytes. Also in the case of (unsuccessful) molecular crypsis, various receptors are activated. Activation of those receptors triggers specific signalling pathways, leading to a specific response of the phagocytes. Thus we adopt the view that a molecular equivalent of sensory perception is involved and molecular mimicry should be analysed like any other mimicry system. Furthermore we consider FH recruitment to prevent C3b opsonization as a new type of molecular mimicry with the difference that instead of mimicking a specific antigen (actively transmit a self signal), pathogens prevent deposition of specific opsonins to camouflage in the environment of the blood-stream (actively inhibit a non-self signal).
Accordingly, we classify this system as molecular cryptic mimesis (molecular crypsis), involving the unusual case that the environment is not neutral to the dupe. This is because the environment is a multicellular organism to which the dupe (phagocytes) belongs. When pathogens use camouflage, they make, for phagocytes, the decision of whether or not to attack cells very hard because it may happen that own cells are attacked.
Host-pathogen interactions have been successfully analysed by mathematical modelling [37][38][39][40] In particular, mathematical models describing mimicry by using, for example, methods from game theory, have been presented earlier for higher organisms, either in a general context [6,7,41] or for specific types of mimicry [42,43]. In contrast, models describing molecular mimicry of bacteria or fungi are rare. Our paper is aimed at filling this gap and extending theoretical mimicry research to pathogenic micro-organisms including fungi. This helps us provide an alternative view on a wide variety of complement-associated autoimmune diseases, including the ambiguous or paradoxical role complement plays in the pathogenesis  of sepsis, which is an exaggerated inflammation and a multifactorial disease. Despite its role in protection, complement can also contribute to the development of severe complications in sepsis, notably injury to host tissues and organs [44].
The question arises whether the pathogen is capable of perfect mimicry (camouflage). In that situation, the human immune system would not be able to distinguish between human cells and pathogen cells. Then, however, protection of human cells by complement factors would be useless. On the other hand, complement factors are successfully used. Our study is aimed at investigating how the immune system can resolve that paradox.

Materials and methods
We start by analysing the attack-decision problem the host faces. Based on a single trait, which is here opsonization with C3b, phagocytes have to decide whether or not to attack a single cell, being either a host cell or a pathogen. As a result of this decision two errors may arise: erroneously attacking a host cell and erroneously not attacking a pathogen. Using an approach to study mimicry in higher organisms [7,41], we can quantify each of the errors, depending on the encounter rate and the pathogenity of the intruder and the aggressiveness of the host's immune system (above which threshold of C3b opsonization the host attacks). Based on those errors, their respective probabilities and their severity, we use a population-based evolutionary model to outline the key parameters which influence molecular crypsis in a general context.
In the second part, we examine opsonization states of an individual infection scenario. This is done by adapting a model of human complement system [45,46] to account for FH recruitment of pathogenic microbes. For the additional parameter values, experiments were performed (S1 and S2 Tables). Our adapted model is used to predict physiological C3b opsonization states for the two microbial species Candida albicans and Escherichia coli, each in comparison to human erythrocytes. It is worth mentioning that E. coli is able to perform molecular crypsis, utilizing for example the C4b-binding protein, which may also have C3bdegrading function [47]. However, to our knowledge, it is not able to utilize FH. We further generate C3b distributions for a single infection with one of the two microbes, which are then used to analyze possible strategies the host could adopt to react to molecular crypsis of pathogens. Thus, we consider E. coli for comparison to C. albicans, to include a cell type that does not perform molecular crypsis using FH. Note that it is not necessary to distinguish between pathogenic and non-pathogenic E. coli.
For simplicity we did not consider infection with multiple pathogens at the same time in this model. This does not severely limit the value of our model because the decision problem for the host remains self versus non-self. Combined infection with multiple pathogens could be modelled by grouping all infecting species into a "non-self" group and the usage of combined (multimodal) distributions for the parameters of this group (with modes centered at the species-specific parameters).
The following modelling approaches share the same basic assumptions about complement dynamics which will be briefly summarized here. C3b attaches equally to all cell surfaces present in the blood. Once bound, it may amplify on the cell surface. This leads to deposition of high amounts of C3b, which acts as a signal for phagocytosis or directly triggers lysis of the cell. Deposition of C3b can be prevented by recruitment of FH to the cell surface. The host can only distinguish between self and non-self if the difference in C3b deposition is sufficiently high. This way, pathogens which are able to acquire FH on their cell surface (by producing the appropriate binding receptors) may be able to camouflage as host cells.

Computing distributions of surface-bound FH
The binding reaction of FH (denoted H here) to a binding site B in a fixed-volume reaction vessel can be described in an approximative way by the following reaction equation: where k + and k − are rate constants. We use this reaction equation to approximate the binding of FH to the host cell surface, although the localisation of the binding sites on the surface induces spatial effects, which we consider later. Let H and HB denote the number of free and bound FH molecules, respectively, and B the number of free binding sites. Then, for the mean dissociation constant K d of all molecules binding FH on the cell surface, we have Defining n as the total number of all binding sites on pathogen surfaces in the medium, i.e.
it follows for the dissociation constant This can be rewritten as where we define the variable p for the fraction of occupied binding sites, or, the probability that a single binding site is occupied. For the immune system, it is of interest how many of all binding sites are occupied. This number is here denoted by k. The binomial distribution applies because that corresponds to the number of k-combinations from a given set of n elements: Note that n − k binding sites are not occupied. The expected value μ and the standard deviation σ of this distribution are Note that we can generally assume large n on the host side, but also on the pathogen side if cell numbers are high enough. For large n the binomial distribution may converge to a Poisson distribution (p ! 0) or a normal distribution, as depicted in Fig 2.

Defining the attack decision
Approximation of the attack decision follows the theoretical work of Holen and Johnstone [7] on mimicry systems and the work of Wiley [48] on signal perception in biological systems. Given a particular C3b opsonization state k, the decision of attacking can be made by the host using any function, for example a sigmoid, where all trait values k above a certain threshold t will be attacked. For simplicity we will assume a step function. This implies that once a cell is opsonized sufficiently, there will be no chance evading phagocytosis or lysis.
Given a threshold t the two errors arising can be quantified using the cumulative distribution functions F H and F P of the C3b distribution of host and pathogens. This distribution depends inversely on the distribution of surface bound FH derived above and is quantified with real-world data in the protein-interaction model later. That is, F H ðtÞ is the probability of erroneously attacking a host cell and 1 À F P ðtÞ is the probability of erroneously not attacking a pathogen.
If there is no further information on the two signal distributions, signalling theory suggests the optimal way of discriminating the signals to be minimisation of a linear combination of these two errors, where the weights depend on the encounter probabilities, which are proportional to the respective cell numbers [7,48]. It should be considered that in the case of infection by a pathogen, the weighting factors may be time-dependent because the cell numbers of pathogens vary. Thus, it would be beneficial to adjust the threshold during inflammation. It has indeed been observed that autoreactivity and inflammation are correlated [49]. We consider this effect in that the optimal threshold depends on pathogen abundance.
Moreover, in systems where the attack decision implies some benefit or cost, depending on whether it was right or erroneous, the probabilities of the two errors should additionally be weighted by the benefit in correctly attacking a pathogen and the costs arising by erroneously attacking an own cell [48]. The more virulent a pathogen is, the higher is the benefit of attack.
Taking all together, in the presented model, it is beneficial for phagocytes to choose a threshold t � , where the difference of the probability of attacking a pathogen, weighted by the benefit B and the probability of attacking an own cell weighted by the cost C is maximized: Note that in this formula, we assume an infection with a single pathogen, so the encounter probability of the pathogen can be approximated by (1 − p) with p being the encounter probability of the host. A more exact model should not only include the frequency-but also the density dependency.
Assuming a normally distributed trait value with standard deviation σ (note that for high receptor numbers the binomial C3b distribution converges to a normal distribution) Holen and Johnstone [7] identified the optimal attack threshold for this type of fitness function to be: In this formula m is the distance in the trait value space of both species. Smaller m values indicate better mimetic resemblance and generally result in a higher (more aggressive) threshold. This is dampened by the mimetic load K, which increases with low probability of encountering a pathogen or low benefit to-cost-ratio of attacking that pathogen (low pathogenity). When m approaches zero, the dupe cannot distinguish between mimic and model and tends to attack both in the same manner (t � ! −1). Transferred to our model, the trait is opsonization with C3b and m is the difference in opsonization of the pathogen compared to the host. As the dupe and the model represent one entity in our model (the host), for m ! 0, autoreactivity may occur.

The population based evolutionary model
For the evolutionary model we assume that the C3b distribution of a given species is defined by the ability of FH surface acquisition alone. This implies that in this case all species below the FH threshold concentration t � H are attacked (because we assume those have a C3b opsonization above the "real" threshold regarding C3b). We next define the maximum possible binding sites the cells can produce (restricted due to limited energy and/or space), called n max . Furthermore we define the relative energetic cost c 2 [0, 1] to produce these binding sites, compared to the total energy available to the pathogen. The overall payoff u of a species S is the product of the energy available for growth and maintenance after receptor production and its survival probability. This energy is, for simplicity's sake, assumed to depend linearly on n S , the number of binding sites species S actually produces: The payoff for the pathogen then reads where we use the abbreviation a P ¼ F P ðtjp P ; n P Þ, which is the pathogen's probability of being attacked, given threshold t.
The payoff of the host H will depend on the attack decision: If we assume that both the pathogen and the host can alter their amount of receptors and their dissociation constant and the host decides which threshold it should assign to a given pathogen (i.e. based on its degree of mimetic resemblance and on the pathogenity in general) strategies of pathogen cells are a pair (n, K d ), with n 2 N = {0, 1, . . ., n max }, while host strategies are a triple (n, t, K d ) with n, t 2 N. Thus, the host has one degree of freedom more than the pathogen. Note that by decreasing the threshold for pathogenic cells which can bind FH very effectively the host can avoid autoimmunity, but has to rely on other mechanisms of immune defense than the complement system. Optimization of the parameters was done using an evolutionary algorithm [50]. Starting from random populations of 50 host individuals and 100 pathogen individuals, it simultaneously evolves both populations by mutation and selection of individuals according to the payoff-functions u S . The opponent is chosen randomly. Pathogens are evaluated first, so they play against the hosts of the last generation. The host population is optimized using a (50 / 2 + 200) evolution strategy [50]. This means that from 50 parent individuals 200 (mutated and recombined) offspring are generated by mating of two random individuals from the parent population. The best 50 individuals are then selected to form the new host population. The pathogen population is optimized using a (100 / 1 + 1000) evolution strategy (cloning and mutation) to simulate the higher mutation rates observed in the pathogens. Termination occurs after 1000 generations.

The protein-interaction model
To investigate C3b opsonization in a single infection scenario, we have adapted a model of human complement system [45,46] to account for FH surface acquisition of pathogens. Our adapted model focuses on the initial part of complement activation. Thus we did not consider stabilization of the C3bBb complex with properdin or formation of the TCC. Our model of initiation of the alternative pathway of complement has 28 species and 38 reactions. The parameter values of FH binding to host and pathogen surfaces were obtained by own experiments. Red blood cell data was taken from the combined NHANES datasets from 2001 to 2014 [51]. Altogether the parameter values were chosen so as to provide realistic estimates of the conditions encountered in the blood stream.
We implemented the model using the COmplex PAthway SImulator, COPASI [52]. Units of the model are micromole for amount of substance, litre for volume and milliseconds for time. C3b amplification may occur in blood plasma, serum or on surfaces. In serum it is activated by a spontaneous tick-over reaction of C3 into C3 H 2 O . As C3 H 2 O has similar properties to C3b, we did not distinguish between C3 H 2 O and C3b in the model, but between fluid and surface bound molecular species (denoted by the prefixes f and b respectively). C3b will then associate with factor B (FB) to form the C3 proconvertase (C3bB), which is activated by factor D, resulting in the active C3 convertase (C3bBb). C3bBb in turn is able to convert more C3 into C3b, starting the loop again (see Fig 1).
In contrast to the spontaneous cleavage, the proteolytic cleavage of C3 into C3b results in a reactive intermediate called nascent C3b (nC3b), which can bind covalently to cell surfaces before it associates with water to form C3b [21]. If nC3b encounters a surface, the same positive feedback as in fluid may occur, where the tick-over (basal rate of spontaneous decay) is replaced by the rate of attachment of nC3b to host or pathogen surfaces (k þ nC3b ). Proteolytic cleavage of C3 by the C3 convertase assembled on surfaces (bC3bBb) will again result in nascent C3b (bnC3b). Although fnC3b and bnC3b are chemically identical, the dynamics of bnC3b surface binding in an ODE system will differ from those of fnC3b, as was already pointed out by Zewde [45]. This is because the local concentration of binding sites surrounding a bC3bBb unit will differ from the global concentration (B C3b ) due to spatial effects which should not be neglected (see S5 Appendix and S7 Fig). This will effectively increase the rate of bnC3b binding to their originating surfaces, compared to other surfaces or fnC3b binding.
A diagram of all reactions, as well as tables of complement concentrations and kinetic rate constants used can be found in the Supporting Information (see S2 Fig, S1 and S2 Tables).
Generally, we distinguish two cases, one where the pathogen species is not able to acquire FH on the cell surface (represented by E. coli) and one where the species is able to acquire FH (represented by C. albicans which can bind FH by the surface molecule Pra1). The host (represented by erythrocytes) can always acquire FH by heparan sulfates (HS) on the cell surface. In the case that FH can be acquired, we assume the same amount of FH binding sites compared to C3b binding sites, which is the whole surface (maximum number). This assumption is supported by the results of population based simulations (see Fig 4c).
As was already mentioned in the section on Dynamics of C3b opsonization, we included some spatial aspect into the ODE model, by assuming a higher affinity of surface derived nascent C3b to the originating surface. Derivation of the scaling factor and simulations not considering this scaling can be found in the Supporting Information (S5 Appendix and S7 Fig).
Complement factors B, D and I were fixed at their physiological concentrations (S1 Table). C3a and iC3b act as a sink in the model and were fixed at zero concentration. C3 and FH were modelled with an explicit inflow and outflow limited only by blood flow. Inflows and outflows were chosen such that the steady state corresponds to the physiological concentrations of C3 and FH. This is to prevent unrealistic behaviour, where especially C3 consumption grows infinitely due to the positive feedback loop in C3b production at high surface concentrations. A detailed description of assumptions regarding C3 and FH dynamics and sensitivity of the model with respect to these assumptions can be found in the Supporting Information (S3, S5 and S6 Figs).
During simulations, host cells were inoculated into the medium first and the system was given enough time to approach a steady state. In the case that no FH and C3 inflow occurs, the system cannot approach the physiological steady state because the initial concentrations of C3 and FH are not sufficient to cover all cells simultaneously. In this case we gave the system several C3 and FH pulses corresponding to physiological concentrations until steady state was reached (see S4 Fig for example runs).
After 1000 seconds, a pathogen is assumed to enter the bloodstream, where the concentrations of pathogenic C3b and FH (if suitable) binding sites were set to the concentrations calculated from the pathogen count defined. Final concentrations of all relevant chemical species were obtained after additional 800 seconds have passed and the system again had enough time to reach steady state (half an hour of simulation in total). Because of the pathogen arrival, we  bnC3b) which is able to covalently attach to surfaces due to an exposed thioester bond. While chemically there is no difference between fnC3b and bnC3b, we distinguish them in the model, because we assume a higher binding rate of bnC3b to the originating surface (see text and S5 Appendix).
https://doi.org/10.1371/journal.pone.0212187.g003 had to use fixed time points for measurements and could not use steady-state information directly.
If distributions were available (in particular, of the erythrocyte count as well as erythrocyte and pathogen surface area, see S1 Fig), 1000 values were sampled from those distributions to obtain C3b distributions for host and pathogen cells at specified pathogen concentrations. The optimal attack threshold in terms of signalling theory was computed for distributions as described in the previous section. If there is no benefit in attacking the pathogen at all (in which case it is no pathogen), no cells are attacked (threshold 0) and no binding sites are produced, yielding the highest payoff (1 for pathogen and host). As soon as there is some benefit in attacking the pathogen, both the pathogen and host will start producing their maximum number of binding sites and minimizing the dissociation constant for FH. In consequence, the FH and therefore the C3b distributions on the cell surface will be almost identical and the host cannot distinguish between self and non-self. This leads to a scenario where a lower number of cells are attacked if the benefit in attacking a pathogen is smaller than the cost of attacking an own cell, but all cells are attacked if it is the other way around. Note that when all cells are attacked, there is a random drift in the dissociation constant and the payoff of the host will increase linearly with r. Hosts save the cost of producing binding sites altogether in this case, while pathogens do not. Fig 5 shows the fitness landscapes of pathogens with fixed host parameters and unlimited (A) or restricted (B) binding site production of the pathogen. In both of the scenarios, there is one local fitness maximum if no binding sites are produced at all and one global maximum at perfect (possible) resemblance. Nevertheless we can observe that the global maximum at the best possible resemblance tends towards the local one as pathogens are less able to obtain perfect resemblance (similar results are observed when restricting for example metabolic energy gain or binding effectiveness).

C. albicans opsonization
In the following, C3b opsonization means all C3b derived chemical species bound on the surface of host or pathogen. This is the sum of bC3b, bC3bB and bC3bBb concentrations. After this point, opsonization of both pathogen and host cells sharply decreases, reaching the opsonization state of the host as if no pathogen were present. The distribution at this point shows that the predicted attack threshold is high, implying very low autoreactivity but effective crypsis of a great deal of pathogens. This behaviour continues until C. albicans reaches nearly Equal pathogen and host concentrations were used. Each blue dot represents one individual with a certain investment into the number of binding sites produced (x-axis location) and a certain fitness (y-axis location, determined by the attack probability solely). The red line represents the goodness of thresholding, which is the relative fitness of a host with attack threshold t, compared to the fitness of a host using the optimal attack threshold t � . Host investment into number of binding sites is arbitrarily fixed to 40% (left) and 60% (right). In the right subfigure, pathogens are restricted to produce a maximum of 90% of host binding sites. https://doi.org/10.1371/journal.pone.0212187.g005 Molecular crypsis by pathogenic fungi using human factor H the same concentration as the erythrocytes (4.64 � 10 12 ). Actually the turning point (the intersection point of the red and blue curves in Fig 6) is slightly less than the mean erythrocyte concentration. Beyond this point, the signals become clearly inseparable and the predicted threshold is very low, meaning the attack of the majority of host and pathogen cells.
For even higher C. albicans concentrations, we observe competition for FH (see S3 Fig) 6).
In the case of molecular crypsis of pathogens, the model suggests three discriminable regimes of opsonization, mainly depending on the concentration of pathogens in the blood (see Fig 7). In the first regime (low pathogen concentration), molecular crypsis is not effective and the host is able to clearly discriminate between self and non-self. In the second regime (medium pathogen concentration), molecular crypsis is successful and opsonization of host Molecular crypsis by pathogenic fungi using human factor H and pathogen is low. In the third regime (high pathogen concentration) complement is hyperactive on self, leading to autoreactivity. High autoreactivity due to high pathogen cell density is doubtless not beneficial for the host but may also not be beneficial for the pathogen [8]. It is an interesting question in this context if there are mechanisms of pathogens to avoid too high concentrations in the host system.

Discussion
Here, we have analysed molecular crypsis by microbial pathogens involving human factor H. In this case, crypsis is defensive and aggressive at the same time because the mimics' intention is invasion of unprotected host cells and mimicry makes them less suspicious, but at the same time other cell lines of the host can be regarded predators of the mimics so they need to hide from them in a defensive manner.
In particular, we established a mathematical model of molecular crypsis by Candida albicans. It integrates methods from signal detection and protein interaction modelling. By choosing appropriate parameter values, it is easily adaptable to be applied to other pathogens. Results were compared to a strain of Escherichia coli that is considered not to perform molecular crypsis using FH.
Our results indicate that if microbes cannot acquire FH on their cell surface, the alternative pathway of complement is active on microbial cells and inactive on host cells, clearly separating self from non-self over a wide range of parameters. We demonstrated that, if pathogens are able to acquire FH on their surface, the effectiveness of complement may depend substantially and highly non-linearly on the quantity (and FH binding quality) of pathogen surface in the blood. This may come as a surprise because intuitively one would expect a linear dependence of opsonization on the target surface present in the medium.
The failure of molecular crypsis to be successful at low pathogen concentrations can be explained by the fact that, due to the positive feedback loop leading to amplification of C3b, there is no pure competition between C3b and FH. Rather, it is important which of the two binds first. This corresponds to the idea that at the very start of an infection, a single molecule  of C3b encountering an unprotected surface may initiate a positive feedback loop of C3b generation (amplification) on that surface that is faster than the degradation or inactivation of C3b by surface-acquired FH. In contrast, C3b encountering a surface protected already by FH may not initiate C3b amplification. For increasing numbers of pathogen cells in the medium (injected simultaneously) the dominance of C3b amplification compared to FH acquisition may vanish as the total amount of C3 required to saturate the pathogen cell surface increases. For (temporarily) decreasing C3 levels, chances reduce that an amplification loop initiates on a pathogen surface and the speed of C3b amplification will slow down due to the short supply. The high binding affinity of FH to the pathogen surface (higher than to the host surface) may then prevent C3b amplification and ultimately lead to decomposition of all the acquired C3b, making the pathogens appear as host cells.
This result is supported by experiments performed by Stone [53]. They infused varying concentrations of 10 5 to 10 19 C. albicans cells per litre at a rate of 1 ml per minute into the portal vein of mongrel dogs and measured the presence of C. albicans in several compartments of the blood system and other tissues after 10 minutes (see Fig 9). Their results correlate with our overall finding that clearance of mimetic pathogens is feasible at a low pathogen concentration,  Fig 8. Data was converted from samples positive for C. albicans to relative effectiveness in clearing C. albicans cells and scaled to units used in our model for better comparability. Top: Samples with no C. albicans cells present after 10 minutes, measured in different compartments of the blood system. For infusion rates below 10 11 cells per litre per minute, C. albicans cells are removed to a great deal by the liver and cannot be detected in subsequent compartments of the blood system. For higher infusion rates C. albicans cells are present in all compartments. Bottom: Comparison to results of our model. The predicted C3b opsonization (red line) correlates to the effectiveness in clearing C. albicans from the blood (black lines). High opsonization means clear identifiability of mimetic pathogens and therefore easy clearance. As the predicted opsonization decreases, also a decrease in clearance of C. albicans cells can be observed, especially in compartments subsequent to the liver (dashed line).
https://doi.org/10.1371/journal.pone.0212187.g009 but not at a high concentration. For C. albicans concentrations below 10 11 cells per litre, pathogen cells could only be detected at the infusion site and the liver. In subsequent parts of the blood system C. albicans was cleared successfully. For higher infusion rates nevertheless C. albicans was detected in all parts of the blood system. These results support our predicted transition point of the regime where molecular crypsis is not successful to the regime where it is successful, although the transition is smoother than predicted by our model and there is no increase in C. albicans clearing effectiveness at around 10 14 cells per litre. This may be an artifact of the fixed C3 and FH inflow (production) rates assumed in the model, which are dynamically regulated in vivo.
Autoreactivity at high infusion rates was mentioned by Stone [53], but not quantified, so the transition point to the regime where autoreactivity may occur could not be compared. Note that we have not assumed, in each simulation, the C. albicans concentration to be timedependent and that we have explicitly modelled the inflow of FH and C3 to regions different from the production site (the liver, see S6 Appendix).
From the above, we can conclude that the positive feedback loop of C3b amplification is beneficial for host cells in several respects. It prevents molecular crypsis from being perfect at low pathogen load and decreases the response time. It allows the host to some extent to shape the environment.
Regarding spatiality, Pangburn [22] showed that cells with activated complement and close proximity tend to "infect" each other with C3b. In our model this effect is only partially treated by increasing the rate of binding to the originating surface, but distributing the remaining increased C3b concentration to all cells in the medium (not only to cells in close proximity). If we neglect the scaling factor, we observe different results, indicating that spatiality really plays an important role (see S7 Fig). We suggest that simulations taking into account the spatial distribution and dynamics of C. albicans cells and erythrocytes (especially in differently sized blood vessels, like veins and capillaries), would be very helpful to fully understand the effects of molecular crypsis by pathogenic microbes. For example, the concentration of the same amount of pathogens invading a capillary will be higher compared to veins, simply because the "local" volume of capillaries is smaller than for veins. Also blood viscosity varies with the radius of the vessel [54]. Besides the different susceptibility of tissues to pathogen invasion, this could help to understand why several autoimmune diseases like age-related macular degeneration (AMD) seem to be tissue-specific.
The effect of increasing opsonization on host cells but not on C. albicans cells at increasing pathogen concentrations can be explained by a higher binding affinity of FH to C. albicans surfaces than to host surfaces. The dissociation constant of FH at C. albicans surfaces is in the nano-molar range, while the dissociation constant of FH at host surfaces is in the micro-molar range. So the binding of FH to C. albicans surfaces is stronger than to host surfaces. This means that if we assume a somehow limited FH production, the pathogen can gain a substantial advantage in FH binding. As a result it is able to sequester dissociating protection from host cells (in addition to newly produced FH molecules), leaving them unprotected regarding C3b opsonization (see S4 and S9 Figs). Of course, the actual values will depend on the maximum expression rates of FH (see S3 and S6 Figs) and its binding sites on host and pathogen surfaces (see S8 Fig).
The question remains whether the host can avoid perfect mimicry as an evasion mechanism of the pathogen. One option for the host would be to change the total amount of complement factors, which cannot be influenced by the pathogen. However, our model shows that changing these amounts does not help because the pathogen can always bind an appropriate fraction of factor H.
To respond to a decrease in the dissociation constant of FH by the pathogen, the host should do the same and could, for example, adapt the speed of C3b amplification while increasing binding affinities to FH, reaching the same balanced system but with an advantage in FH binding. Importantly, the alternative pathway of complement is not the only defence mechanism of the host. We can imagine a scenario following the so-called Red queen hypothesis [55], where (spatially related) hosts and pathogens evolve in a permanent arms race, trying to be always one step ahead of the other [56]. If one of them is not able to adapt to the advance of the other any more, it will lose the race. Maximizing binding affinities to FH nevertheless may become harder and harder in each round and is certainly limited by physical constraints and side-effects to other systems. This means that at some point, it could have been beneficial (or unavoidable) for the host to develop a completely new system of defence against mimicking pathogens.
Besides maximizing binding affinities, another way of gaining an advantage for hosts and pathogens, is the production of additional binding sites. For example C3d, a cleavage product of C3b, can provide new binding sites for FH [57]. While it is generated on the surface of pathogens capable of molecular crypsis as well (since they also utilise FH to cleave C3b), it could be a mechanism of producing binding sites faster than the pathogen, if this cleavage product binds preferentially to host surfaces. Additionally this strategy practically does not involve any costs, since C3b is already needed to ensure complement activity and is cleaved on host cell surfaces anyway. The results of the population based model clearly indicated that both the host and the pathogen adjust the binding affinities and binding sites of factor H ligands to a maximum, although binding site production involves costs.
An important prediction of our model is that that pathogens evolve to produce the same number of binding sites as hosts or none at all. This is in line with results by Holen and Johnstone [7], showing that costs of mimicry can lead to mimetic dimorphism (explaining why not all pathogens do mimicry). We observed two fitness maxima, one local maximum if no binding sites were produced at all and one global at perfect resemblance. This can be explained as follows: The benefit of mimicry is described by a constant basal growth rate minus a linear function of costs plus a saturation function above a threshold. The sum gives a non-linear function with two maxima. It does not pay at all to have a low degree of mimicry. For example, a fly being purely black or purely yellow has achieved half of the colouring of wasps but has (nearly) no mimicry effect. No mimicry is better than low mimicry, because the former is the local maximum at zero investment. The global maximum is achieved at a certain high investment (all or nothing). It is an interesting question how the global maximum could be attained during evolution, because it is difficult to explain by micro-evolution.
One way of avoiding perfect resemblance of pathogens in a population of hosts could be the utilization of polymorphisms of relevant proteins. On the host side there is the Y402H polymorphism in the FH gene and there is also a factor H like protein (FHL-1) which is an alternatively spliced product of the FH gene. It mainly consists of the first seven (of the 20) small consensus repeats (SCRs) responsible for binding. Those polymorphisms make adaptation to specific environments harder, as was already proposed by Damian [8] in its initial study on molecular mimicry due to antigen sharing by hosts and parasites. There he suspected the AB0-polymorphism, defining the human blood groups, to be a consequence of molecular mimicry. On the pathogen side, there are also polymorphisms observable. For example pathogens often have more than one FH acquiring protein (CRASP) with slightly different binding affinity and different expression levels. Those could be adaptations to host polymorphisms, but also provide some robustness of the system if one receptor type gets lost or inefficient somehow. For example, neutrophils are able to detect the C. albicans FH-acquiring protein Pra1 using their α M β 2 receptor [58]. By targeting the pathogen molecule directly, neutrophils can discriminate this specific pathogen, but it is no general mechanism to avoid mimicry, as it has to be learned for each antigen and the pathogen could adapt using another FH-acquiring molecule (i.e. C. albicans might also use Tef1).
It remains to be studied in more detail above which C3b concentration phagocytes decide to attack cells and if this threshold is adaptable. Based on methods of signalling theory, one can calculate optimal ways of discriminating two signals, as was shown in the Methods section. Still, it is hard to quantify the benefit of correctly attacking a pathogen and the cost of erroneously attacking a host cell. Generally we can assume none of the errors to be negligible, as was shown in a previous paper [32]. If we weight both errors equally, as was done in most of the simulations, we see that at high pathogen concentrations there is no optimal decision. The model would predict for this case that all cells present in the medium should be attacked. This is clearly not the optimal response of the host. In theory, this could only be prevented by zero benefits in attacking pathogen cells or equivalently infinite costs of attacking host cells. Such benefits and costs would be, on the other hand, non-optimal for low pathogen concentrations, as they may allow infections.
It would be optimal for the host to adapt its attack threshold based on the pathogen concentration in the blood. The threshold should be shifted towards effectiveness in clearing mimetic pathogens for low pathogen concentrations and shifted towards avoidance of autoreactivity for high pathogen concentrations. Because the pathogens hide, the host can not directly sense the pathogen concentration and adjust thresholds, but it could use other clues, like blood vessel damage, indicating that a high amount of pathogens might enter and adjust the response. This context-dependant decision making could be modeled using the extended model of Holen and Johnstone [59].
In addition to changing the threshold directly by alteration of the phagocyte's predation strategy, also alteration of the binding effectiveness of FH to surfaces may be an option to control general reactiveness of complement. This is because FH removes deposited C3b from cell surfaces and thus higher binding affinity of FH to host surfaces results in better protection (calmed down complement) of host cells. On the other hand this one-sided alteration of protection of host cells will likely result in adaptation of the FH binding effectiveness of the mimic by exactly the same amount, as we showed in the population based model. So basically both distributions would shift by the same amount in the same direction, which is then mathematically equivalent of changing the attack threshold, since it affects both errors equally. So although this may have no effect on seperability of host and pathogen cells it is a way of easily adapting general aggressiveness of complement without the need for alterations in the phagocyte's predation strategy. So in fact the Y402H polymorphism could be an example of adaptation of the attack threshold, where the YY homozygote has a reduced complement reaction, while the HH homozygote may react more aggressively and the YH heterozygote is somewhere in the middle [32].
Regardless of how the attack threshold is fixed exactly, we see that in the case where there is a high concentration of pathogens performing molecular crypsis, host opsonization can reach a point where autoreactivity seems nearly unavoidable. This may have severe consequences especially in situations where the adaptive immune system is suppressed.
In such a scenario the pathogen load can hardly be regulated at all, possibly contributing to severe malfunctions of the immune system, like autoimmune diseases (in which self cells are attacked) and sepsis. Lethal sepsis can indeed occur, for example, as a consequence of a disseminated candidiasis [60]. Depending on the specific microbial environment, the host may accept the cost of a certain degree of autoreactivity in an early stage of infection to avoid pathogen concentration reaching the regime of successful crypsis. As mentioned above, there is a polymorphism Y402H in the human FH gene, where the H variant predisposes individuals to age-related macular degeneration [31]. It is an interesting question arising from the results of our model, if the H variant of the polymorphism on the other hand could act protectively in the context of sepsis. In addition to changes in parameter values by adaptation to the disease, some values such as C3 level are likely change due to the impact of the disease or predispositions such as anemia. This is worth studying in future extensions of the model.
Also the fungus has the capability to adapt its behaviour. For example C. albicans is able to switch between a white and an opaque form [27][28][29]. Although we did not find FH-binding proteins differentially expressed between these forms in the data by [30], there is high overexpression of SAP1 in the opaque form. SAP1 has complement-component degrading function [61] and thus may promote crypsis of the opaque form. As SAP1 also causes epithelial cell damage [62], the opaque form may be especially involved in invasion of the blood or tissue where the C3b degrading function would cause an initial depletion of C3b, increasing the chances of successful invasion. At 37˚C the fungus will switch to the white form [29], which is the predominant form in disseminated candidias [28]. To analyse white-opaque switching in the model, it could be extended to include SAP1 production of C. albicans. The spontaneous type-switching could be modelled using a rate constant on the population level, where SAP1 production would be increased proportional to the amount of opaque cells in the population. The initial ratio of opaque to white cells could be chosen to represent the ratio of the originating compartment (the opaque type is a better colonizer of the skin), where the equilibrium ratio in the blood will be determined by the switching rate at 37˚C, which is around 10 −3 (1000 white on 1 opaque) [29].
Returning to the analogy of the Captain of Koepenick [4], it is worth noting that to distinguish true from false policemen, also other signals are needed, such as correct commands. The deception by the Captain of Koepenick was so perfect because he had learned and used the army commands. To avoid a perfect camouflage, the human adaptive immune system is used in addition to the innate immune system to recognize antigens as further signals for discrimination. For molecular mimicry to be successful, pathogens need to deceive both systems.