Modeling the activation of the alternative complement pathway and its effects on hemolysis in health and disease

Antonello Caruso; Jannik Vollmer; Matthias Machacek; Elod Kortvely

doi:10.1371/journal.pcbi.1008139

Abstract

The complement system is a powerful mechanism of innate immunity poised to eliminate foreign cells and pathogens. It is an intricate network of >35 proteins, which, once activated, leads to the tagging of the surface to be eliminated, produces potent chemoattractants to recruit immune cells, and inserts cytotoxic pores into nearby lipid surfaces. Although it can be triggered via different pathways, its net output is largely based on the direct or indirect activation of the alternative pathway. Complement dysregulation or deficiencies may cause severe pathologies, such as paroxysmal nocturnal hemoglobinuria (PNH), where a lack of complement control proteins leads to hemolysis and life-threatening anemia. The complexity of the system poses a challenge for the interpretation of experimental data and the design of effective pharmacological therapies. To address this issue, we developed a mathematical model of the alternative complement pathway building on previous modelling efforts. The model links complement activation to the hemolytic activity of the terminal alternative pathway, providing an accurate description of pathway activity as observed in vitro and in vivo, in health and disease. Through adjustment of the parameters describing experimental conditions, the model was capable of reproducing the results of an array of standard assays used in complement research. To demonstrate its clinical applicability, we compared model predictions with clinical observations of the recovery of hematological biomarkers in PNH patients treated with the complement inhibiting anti-C5 antibody eculizumab. In conclusion, the model can enhance the understanding of complement biology and its role in disease pathogenesis, help identifying promising targets for pharmacological intervention, and predict the outcome of complement-targeting pharmacological interventions.

Author summary

The complement system as a part of the innate immunity builds a first line of defense of the body against foreign and altered host structures, such as pathogens and damaged cells. In contrast to the adaptive immune system, which takes days to weeks to mount a response, the complement system can react within seconds to minutes and therefore enables the body to quickly react to infections. Dysregulation of the system, however, has been identified as a major driver or significant contributor to several pathologies, ranging from the pathologic lysis of body’s own erythrocytes to the progressive loss of eyesight. The pathway’s highly diverse and interconnected architecture makes it difficult to identify promising points of therapeutic intervention and to predict clinical study outcomes. Building on previous modelling efforts, in this work we developed a mathematical model of the alternative pathway of the complement system that can explain experimental observations acquired over more than 50 years of complement research. These include results acquired in the laboratory as well as in patients. We envision that this work can be used to support the development of novel treatments for complement-mediated diseases at several key points such as target selection or design of clinical studies.

Citation: Caruso A, Vollmer J, Machacek M, Kortvely E (2020) Modeling the activation of the alternative complement pathway and its effects on hemolysis in health and disease. PLoS Comput Biol 16(10): e1008139. https://doi.org/10.1371/journal.pcbi.1008139

Editor: James R. Faeder, University of Pittsburgh, UNITED STATES

Received: November 19, 2019; Accepted: July 9, 2020; Published: October 2, 2020

Copyright: © 2020 Caruso et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the manuscript and its Supporting Information files.

Funding: The authors received no specific funding for this work.

Competing interests: Antonello Caruso and Elod Kortvely are employees and shareholders of F. Hoffmann-La Roche AG. Jannik Vollmer and Matthias Machacek are employees of LYO-X.

Introduction

As a first responder of innate immunity, the complement system, a network of more than 35 proteins, provides a powerful mechanism to mark and eliminate unwanted foreign and self-structures [1]. Complement activation triggers a proteolytic cascade leading to the production of cleavage products with various biological functions [2,3]. Diffusible fragments (such as the anaphylatoxins) trigger degranulation of neighboring endothelial and mast cells, increase the permeability of capillaries and recruit immune cells (chemotaxis) [4]. Furthermore, complement actively tags surfaces destined for elimination by covalent attachment of specific fragments to the activating surface [5]. Finally, it is actively involved in the disruption of pathogens owing to the insertion of pore-forming complement proteins into the membrane of the invading microorganism [6].

Three pathways may activate the complement system, depending on the mechanism triggering response. Antigen-bound antibodies and sugar moieties initiate the classical (CP) and lectin pathway (LP), respectively [7]. The third pathway, the alternative pathway (AP), does not require a specific activator. Instead, it is always active at low level (tick-over) [8–11] and continuously challenges any surface it encounters, including self-cells. Healthy cells can counterbalance this pressure, while damaged cells or pathogens are rapidly eliminated. All three pathways lead to the cleavage of complement component 3 (C3) into C3a and C3b, with the difference that the composition of the enzyme complex catalyzing this reaction is different for the CP/LP and the AP. The generated C3b becomes a part of the AP C3 convertase (C3bBb), responsible for cleaving C3 and thereby providing amplification for the activation of the other pathways. Amplification by AP may actually account for up to 80% of complement effector functions, irrespective of the pathway initially activated [12].

The complement system can eliminate cells within minutes, while the adaptive immune system requires weeks to mount an immune response. Complement can however harm self-structures when its regulation is out of balance, making it a major driver in several pathologies [13]. The rare kidney disease paroxysmal nocturnal hemoglobinuria (PNH) is a prominent example of a complement-driven disease [14]. In PNH, the lack of negative complement regulator proteins at the surface of erythrocytes leaves them unprotected against complement attack [15–17], leading to continual hemolysis and life-threatening anemia. The US Food and Drug Administration (FDA) approved eculizumab (Soliris), a complement-targeting monoclonal antibody for treatment of PNH and later for a similarly rare kidney disease, atypical hemolytic uremic syndrome (aHUS). This success led to a deluge of clinical trials targeting the complement system [18–20]. The genetic architecture of age-related macular degeneration (AMD), the most prevalent blinding disease of the elderly, also suggests that complement plays a pivotal role in the pathogenesis [21]. Protein-changing polymorphisms in several complement genes were found to increase the risk of developing this sight-threatening condition [22–27], albeit the molecular underpinnings of the manifestation of the disease are poorly understood. Strikingly, a coding SNP in the complement protein Factor H (FH) is one of the two major susceptibility loci, and the encoded negative regulator is responsible for holding the AP at bay [28–32]. Besides, polymorphisms affecting the activity of Factor B (FB), a serine protease exclusive to the AP, were found to have a direct impact on disease risk further supporting the pronounced role of this pathway in the etiology of AMD [33].

Nevertheless, due to the cascade nature of the complement system and the numerous proteins, activation products, positive and negative regulators, as well as transient protein complexes involved, finding promising points of intervention and predicting the outcome of clinical trials remains a challenging task [34]. A drug candidate may excel when its efficacy is assessed in an ex vivo assay (e.g. in a blood sample) and fall short in vivo where the constant synthesis and elimination (homeostasis) of complement proteins also influence the outcome, among other factors [34]. This high degree of complexity prevents simple interpretations and renders intuitive predictions of complement activation and regulation unreliable. On the other hand, the growing number of drugs in development pipelines necessitates the organization of all relevant components into a testable framework.

Mathematical models are often employed to facilitate the study of complex systems. Modeling techniques offer a means to formalize the available biological knowledge into a quantitative description of the system dynamics and obtain a reproducible testbed. A few models of the complement system have been developed, aiming to enhance our understanding of homeostasis and regulation [35–41]. Hirayama and co-workers [35] first studied CP activation by linear systems analysis to investigate its stability and controllability. Korotaevskiy and colleagues [36] investigated the bactericidal dynamics of the CP and AP and Liu and colleagues [37] that of the CP and LP. More recently, Sagar and colleagues [38] developed a reduced model for the LP and AP capable of mimicking C3a and C5a kinetics in vitro, but which did not describe the terminal pathway beyond the formation of the C5 convertase. Bakshi and co-workers [41] developed two parsimonious models of the AP that took into account the synthesis and degradation of precursors with the aim to understand the steady-state response of the pathway. A model of the AP and CP that aimed at including all relevant molecules and reactions was presented by Zewde and colleagues [39,40], describing pathway interactions with pathogen surfaces and host cells in an in vitro setting. However, none of the published models studied the role of the AP as a driver of disease while aiming to reproduce experimental observations obtained in vitro and in vivo.

The aim of this work was to develop a mathematical model of the AP that allows to investigate complement activation as measured in vitro in laboratory assays and in vivo via clinical markers of disease. The goal was to develop a model that contains all relevant molecules and reactions as opposed to a reductionist approach. Such a model would allow probing the pathway at different nodes and facilitate the comparison to current and emerging experimental data. Being the most comprehensive model of the AP to date, the work by Zewde and colleagues [39] was used as a basis for this work. This mathematical model was evolved by estimating model parameters to fit an extensive set of experimental observations and by incorporating several disease relevant readouts, such as hemolysis of rabbit and human erythrocytes and hematological biomarkers (hemoglobin, LDH and hematocrit). By doing so, it allowed to compare the model predictions to the results of commonly used experimental setups and in vivo observations. The focus was the AP because of its prominent contribution to the generation of complement activation products, as well as to balance biological completeness and mathematical tractability. An ordinary differential equation (ODE) model was formulated and its predictive accuracy assessed under a range of experimental conditions, representing both the in vitro and in vivo, healthy and diseased setting using all published experimental data that was identified from peer-reviewed publications.

Hemolytic assays quantify the functional activity of the complement system by measuring the degree of erythrocyte lysis upon incubation in serum. They have been the cornerstone of complement research for over 50 years [42]. Therefore, we included complement-mediated hemolysis into the model and assessed the predictions against experimental measurements. The turnover of complement proteins and erythrocytes was also taken into account to mimic the metabolic processes that underpin homeostasis in vivo. As a result, the model can simulate the pathological changes induced by complement dysregulation in conditions like PNH, and how these are reflected in clinical markers of disease, such as hemoglobin concentration and hematocrit levels. With the addition of a mathematical description of eculizumab pharmacokinetics and target engagement, we finally simulated the pharmacological inhibition of complement-mediated hemolysis and compared it to clinical observations of hemoglobin recovery in PNH patients.

In summary, the AP model provides a platform to investigate pathway activity in vitro and in vivo and predict the outcomes of pathway modulation by disease and pharmacological intervention. The model may be useful to identify promising targets for drug development and assist the selection of biomarkers of clinical efficacy. It may also guide the definition of personalized treatments in the future, assuming the model can be adapted to match data from individual patients, and be used to study the optimal administration of drugs.

Methods

Alternative pathway model

Model structure.

We aimed at building a single model of the AP closely representing the biochemical pathway capable of reproducing in vitro and in vivo experimental observations. The structure of the proposed model is depicted in Fig 1. The model describes:

the known complement protein interactions in the fluid phase and their negative regulators (reaction 1–36 in S1 Appendix)
the association of complement proteins to erythrocytes, their interaction on the cell membranes together with their positive and negative regulators (reaction 37–111)
the hemolysis of erythrocytes as a function of AP activation (Eq 1)
the synthesis (reaction 114–129) and degradation (reaction 130–223, Eq 2) of complement proteins and their complexes in the human body
the physiological turnover of human erythrocytes, i.e. their natural production and elimination (reaction 112 and 113, Eq 3)
the kinetics of the hematological biomarkers, hematocrit (Eq 4), hemoglobin (Eq 5) and lactate dehydrogenase (LDH, Eq 6).

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Fig 1. The alternative pathway of the complement system.

The diagram shows the pathway as described in the model with its components (orange boxes), reactions (grey arrows), proactivators (green arrows), regulators (red arrows), convertases (yellow boxes), effectors (red boxes), and the positive regulator properdin (brown triangles). Association/dissociation reactions are displayed with a pointed arrowhead, enzymatic reactions with an oval. Degradation was implemented for all proteins and complexes. Proteins marked with a (*) are synthesized. The blue box indicates the reactions of the fluid phase in absence of erythrocytes. Activation of the AP begins with spontaneous hydrolysis (“tick-over”) of C3 producing C3(H₂O), which in turn can form the initial convertase C3(H₂O)Bb in presence of FB and factor D (FD). The initial convertase can generate more C3b through C3 cleavage. C3b may bind FB, leading to the formation of the fluid-phase convertase C3bBb, or attach to a nearby surface, such as a cell membrane. Surface-attached C3b can combine with FB to form the surface C3 convertase C3bBb, reacting with C3b to generate the C5 convertase, C3bBbC3b. Cleavage of C5 by the C5 convertase activates the terminal pathway. The anaphylatoxin C5a is released while C5b remains attached to the C5 convertase, followed by C6 and C7 binding. The complex C5b-7 is released into the fluid phase, from where it can reinsert into the cell membrane if not sequestered by the regulators vitronectin (Vn) or clusterin (Cn). Upon binding of membrane C5b-7 to C8 and up to 18 C9 molecules [42], the membrane attack complex (MAC) is formed. With sufficient terminal pathway activity, the cell is lysed as a result of the accumulation of MAC complexes, permitting free diffusion of molecules across its membrane. All negative regulators (FH, decay-accelerating factor (DAF), CD59, complement receptor type 1 (CR1), factor I (FI), Vn and Cn) are included in the model, as well as the only known positive regulator, properdin (P), which can stabilize the C3 and C5 convertase.

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These individual modules are described in more detail below.

All simulations presented in this work are based on the same model. For prediction of experimental outcomes, the model parameters related to the experimental conditions were adjusted accordingly, while all the parameters intrinsic to the AP, such as the reactions rates, were kept constant. To simulate the in vitro experiments involving fluid phase proteins only, only module 1) was used. To simulate in vitro experiments involving rabbit or human erythrocytes, modules 1) to 3) were used. Details on each of the experimental setups and the respective implementation are provided below. All model equations and parameters are summarized in S1 and S2 Appendix, respectively. An overview of the literature sources used for model development and verification is provided in S1 Table.

Module 1 and 2 –Modelling of complement protein interactions.

Complement protein interactions were modelled with a set of ODEs according to the law of mass action in case of association or dissociation of proteins, else as Michaelis-Menten kinetics for enzymatic reactions. Michaelis-Menten enzymatic reactions were parameterized with a catalytic rate constant k_cat and a Michaelis-Menten constant K_m (S1 Appendix and S2 Appendix) according to , where [P], [E] and [S] are the concentration of the product, enzyme and substrate, respectively. It was assumed that the fluid phase is a well-mixed compartment without spatial concentration gradients.

The model by Zewde and coworkers [39] was used as a foundation for this work to investigate complement activation as measured in vitro in laboratory assays and in vivo via clinical markers of disease. To this end, the model was modified in several ways. The following reactions, assumed to occur physiologically [40], were included: dissociation of FH from C3bBbH_fluid, C3bBbH_host and C3bBbC3bH_host (where “fluid” and “host” indicate fluid-phase and surface-bound complexes, respectively; reactions 35 and 104 in S1 Appendix); dissociation of CR1 from C3bBbCR1_fluid, C3bBbCR1_host and C3bBbC3bCR1_host (reactions 36, 106 and 107), dissociation of DAF from C3bBbDAF_host and C3bBbC3bDAF_host (reactions 108 and 109), and dissociation of properdin from C3bBbP_host and C3bBbC3bP_host (reactions 110 and 111). Additionally, conversion of C5 by the convertase was modelled as an enzymatic reaction (reactions 55 and 82), with the associated kinetic rate constants (k_{C5,cat,C3bBbC3b(P)} and K_{C5,m,C3bBbC3b(P)}) obtained from the literature [43–45]. Binding kinetic parameters of FH (k_p,C3bH and k_m,C3bH) and CR1 (k_p,C3bCR1 and k_m,C3bCR1) were obtained from [40,46]. Finally, the binding of nascent C3b with its reactive thioester to water forming fluid phase C3b was modelled as a first-order reaction with a half-life of 60 μs [47] (reactions 9 and 10 in S1 Appendix) instead of a second order reaction of binding of H₂0 and nascent C3b. Further steps of model development are described in the following sections.

Module 3 –MAC-mediated lysis.

The cytolytic effects of the MAC were included by modelling the relationship between MAC density and cell lysis (Eq 1). Takeda and colleagues [48] provided data on hemolysis of sheep erythrocytes carrying human C5b-7 complexes that were treated with fixed amounts of C8 and excess of C9. Based on this data, a dose-response curve for MAC-mediated lysis was obtained (Fig 2, S3 Appendix). We found that the percentage of cells lysed can be described by a sigmoid function of the number of MAC pores per cell, according to Eq 1 (Hill equation): (1) where H is the percent hemolysis, MAC the number of MACs per cell, MAC50 the number resulting in 50% hemolysis, and γ the Hill coefficient determining the steepness of the response. While there is experimental evidence that MAC complexes with less than 18 C9 molecules can lead to cell lysis [49], smaller MAC pores are likely less lytic than bigger MAC pores. Since there was insufficient experimental data to parameterize this relationship, it was assumed that only MACs complete with 18 C9 molecules [50] are functional structures capable of lysis.

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Fig 2. MAC-mediated hemolysis.

Percent hemolysis displayed against the number of MACs per cell (module 3). Data (circles) of human complement-mediated lysis experiments on sheep erythrocytes were obtained from [48] (S3 Appendix). The data was fitted with the sigmoid model of Eq 1 (line), with γ = 1.60 and MAC50 = 1.15 MAC/cell.

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Module 4 and 5 –Complement protein and erythrocyte turnover.

In order to reproduce pathway homeostasis in the in vivo setting, the non-AP specific production and elimination of complement proteins was incorporated in the model such that the reported in vivo steady state concentrations of all AP molecules would be obtained at homeostasis. This included the synthesis of all components that are not a result of the AP reactions such as C3, FH, FB, FD, as well as the elimination through processes such as renal filtration, receptor mediated uptake or metabolism that are not part of the AP. These processes were not mechanistically modelled, instead production rates and first order elimination rates were used. The elimination rate constants of C3a, C5a, and C3dg were obtained from literature reports (S4 Table). Due to the limited availability of data, the clearance of the other complement proteins was estimated based on their molecular weight, assuming a sigmoidal relationship between the log-transformed hydrodynamic radius and the elimination rate constant (S4 Appendix). To constrain the number of independent parameters, fluid-phase C3 and C3b- and C5b-complexes were assumed to disappear at the same rate as C3 and C5, respectively. Considering that mature erythrocytes do not perform endocytosis [51], the elimination of surface-bound proteins and complexes was instead assumed to occur at the same rate as the cell itself, having a half-life in circulation of 60 days.

Rates of protein synthesis were estimated under the assumption that the AP of healthy individuals produces negligible complement-mediated hemolysis. The rate constants were calculated in order to match protein steady state levels (S2 Table) and elimination half-lives (S4 Table). The parameters controlling C3(H₂O) hydrolysis () and the association of C3b to the surface (k_{p,C3b,surface}) were decreased 10²- and 10⁵-fold relative to the in vitro parameterization, respectively, to compensate the increased pathway reactivity in vitro [52] and attain minimal terminal pathway activity in the absence of a trigger.

The elimination rate constant of cells and surface proteins, k_el,S, was calculated according to Eq 2: (2)

Here k_s describes the physiological turnover of erythrocytes and k_H the degradation due to hemolysis, as per Eq 3: (3)

H being the percent hemolysis (Eq 1) and τ a scaling coefficient, which was estimated by fitting the hemoglobin levels measured in PNH type 3 patients as described below.

Module 6 –Hematological biomarkers.

Several hematological markers are used clinically in the diagnosis and management of PNH disease. Hematocrit, hemoglobin and LDH levels provide relevant information for the monitoring of complement-mediated hemolysis and response to treatment. These markers were therefore included as part of the AP model. Hematocrit (Hct, %) was calculated according to Eq 4: (4) where C_e is the concentration of erythrocytes (5*10⁶ cells μL^-1) and V_e the mean corpuscular volume (90 fL). Hemoglobin levels (C_h, g dL^-1) were calculated using Eq 5: (5) where N_A the Avogadro constant, N_h,e the number of hemoglobin molecules per erythrocyte (assumed as 270*10⁶ cell^-1 [53]) and MW_h their molecular weight (64.5 kDa [54]). Finally, LDH levels (C_LDH, U L^-1) were assumed to be a function of hemoglobin concentration according to Eq 6: (6) where LDH_max, H_LDH50, and LDH₀ are parameters fitted to experimental data in PNH patients. While Eq 6 neglects the delay of LDH removal from circulation following cell lysis, it was found to describe the observations reasonably well (S1 Fig).

Parameter estimation

Data from fluid phase AP activation experiments (Fig 3) and rabbit (Fig 4) and human hemolytic assays (Fig 5) were split into an estimation and validation set. Model parameters were fitted by minimizing the sum of squared residuals (SSR) between simulated and experimental results of rabbit and human hemolytic assays [55–59] (Fig 4A–4D and Fig 5A). While data from the fluid phase AP activation experiments (Fig 3) and observations from Thanassi et al. [60] (Fig 4E and 4F) were used as an independent data set for model verification. A detailed list of the experiments and their use in the parameter estimation is given in S1 Table. For each experiment j, the SSR was calculated according to Eq 7: (7) where y_i is the i^th observation, m_i(x) the model-predicted value with parameter value x, and n the number of data points. Normalization by the number of observations achieved equal weighing among studies. The objective function was defined as the sum of SSR for the five data sets: (8) for S = {[55–59]}

SSR_Sum(x) was minimized with respect to x using the fminbnd function in MATLAB R2018b, with options TolX and TolFun set to 0.01 and 0.5, respectively. Due to computational complexity, the parameters were optimized with an iterative approach. At each iteration, all parameters were individually fitted in a space ±7 orders of magnitude around the initial value; the parameter producing the lowest objective function value was updated in the model. The procedure was repeated until improvement in the objective function was below 1%. The choice of this approach is discussed below.

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Fig 3. Fluid-phase alternative pathway activation.

Observed (symbols) and simulated (lines, module 1) complement factor levels in experiments of fluid-phase AP activation in in vitro samples. (A) Inactivation of C3 in absence or presence of FB, FD, FH, and FI. (B) Formation of C3b by preassembled C3 convertases. (C) Conversion of C3b to iC3b as a function of FH concentration (790 nM (a), 393 nM (b), 197 nM (c), 98.5 nM (d), 49.3 nM (e), 12.4 nM (f), 4.50 nM (g). Dashed lines: experimental data, solid lines: model simulation). (D-F) Formation of activation markers C3a (D), Bb (E), and C3dg (F) due to spontaneous activation of AP in human serum samples. Data was obtained from [10] (A), [62] (B), [61] (C), [38,64] (D), and [63] (E,F). C3a levels from [38,64] were baseline corrected. Bb and C3dg levels from [63] were converted to molar concentrations for comparison to model results, assuming a molecular weight of 63 kDa and 38 kDa, respectively.

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Fig 4. In vitro hemolytic experiments with rabbit erythrocytes.

Experimental data (symbols) and model simulations (lines module 1–3) in hemolytic assays with rabbit erythrocytes. (A) Time course of hemolysis at 400 ng/mL FD. (B-E) Dose-response curves for hemolysis of rabbit erythrocytes. (B,C) Hemolysis at different serum dilutions (B) or when using a mix of normal human serum and FD- (black) or C3-depleted (red) serum (C). (D) Hemolysis as a function of FD serum concentration at 5 (red) and 30 minutes (black). (E) Hemolysis at different concentrations of C5 (green), FB (blue) and FD (black). For titration of C5 and FD, data from two individual experiments are shown. (F) Dose-response curve for activation markers Bb and C5a as a function of FD concentration. Hemolysis was normalized to lysis in water (A-D) or to the activity of NHS assayed in parallel at the same dilution (E). The experimental data was obtained from [56] (A,C,D), [55] (B), and [60] (E,F).

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Fig 5. Hemolysis of human erythrocytes under partially disabled surface regulation.

(A) Observed versus predicted (module 1–3) hemolysis of human erythrocytes. Observations are from [59] (triangles), [57] (circles) and [58] (squares). Inhibited regulators are indicated. Suppression of FH regulation was assumed to be restricted to the surface. (B) Model predicted (module 1–3) hemolysis for pairwise suppressed regulators. Suppressed regulators are indicated on the axes (FH surf: abolished FH surface regulation, FH tot: complete absence of FH regulation). Predicted hemolysis is shown as heat map and with percent values.

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Simulations

Fluid-phase alternative pathway activation.

A fundamental assay of complement research involves the study of AP activation in the fluid phase by the spontaneous hydrolysis of C3. Various experimental setups have been described in the literature [10,38,61–64]. For simulation of these experiments, all surface-related species and reactions were excluded from the model (reaction 37–111 in S1 Appendix). For those experiments involving a subset of (recombinant) complement proteins, the initial concentration of fluid-phase species was set to the assay conditions reported in the respective publication. For simulations of AP activation in whole human serum samples, the initial conditions presented in S2 Table were used.

Rabbit erythrocyte lysis.

Rabbit erythrocytes lack surface regulators of the human AP and rapidly undergo lysis when exposed to normal human serum. They are therefore often employed to perform in vitro studies of pathway activation against foreign cells. For instance, Pangburn and colleagues [55] investigated the lysis of rabbit erythrocytes at different serum dilutions. Wu et al. [56] and Thanassi et al. [60] reported dose-response curves of hemolysis for different levels of complement proteins C3, C5, FB, and FD. In order to achieve an accurate description of AP activity in vitro, we performed model simulations that reproduced these hemolytic assays in silico. In the simulations, rabbit erythrocytes were assumed to lack regulation by CR1, DAF, CD59, and FH [57,65–67]. The concentration of CR1, DAF, and CD59 were therefore set to 0, as well as the association rate constant for the binding of FH to surface-bound proteins (k_p,C3bH,surf). Initial cell concentration, serum dilution and readout time were set to match the reported experimental conditions (S3 Table). Serum dilutions were modelled with equivalent reductions of the initial concentrations in 100% serum (S2 Table).

Human erythrocyte lysis.

Human erythrocytes are protected against autologous complement-mediated lysis, nevertheless they may become susceptible if regulation is deficient, as is the case in PNH disease. To study the contribution of complement regulators, we collected literature data of hemolysis experiments performed with inhibitors [57–59] which we replicated in silico. In those studies, CR1, DAF, and CD59 were inhibited with antibodies. FH was blocked by FH19-20, a recombinant protein that inhibits FH cell-surface regulatory functions while preserving fluid-phase control of complement [57]. Due to limited information on the dose-response of inhibition, complete blockage was assumed in the simulations and the results were compared to the observations at the highest inhibitor concentration. The initial cell concentration and serum dilution used in the simulations matched the experimental conditions (S3 Table). The readout time was set to 30 minutes, a typical duration used in hemolytic experiments to allow sufficient pathway activation.

Alternative pathway in PNH.

PNH is a disease resulting from impaired cell surface complement regulation. Deficiency of the regulatory proteins DAF and CD59 on erythrocyte membranes accounts for the intravascular hemolysis that is the clinical hallmark of the disease [15–17]. In order to study the behavior of the pathway in PNH patients, surface dysregulation was implemented in a model of disease. CD59 and DAF were assumed to be entirely absent from erythrocyte membranes in PNH type 3 patients. For type 2 patients, a 10-fold reduction in their synthesis rate was assumed, mimicking the subtotal deficiency (~10% of normal expression) that was determined in flow cytometric analyses [68]. The PNH clone size, namely the fraction of affected erythrocytes, was assumed to be 100%.

Eculizumab target inhibition and pharmacokinetics.

To simulate the effects of eculizumab on hemolysis, a description of target engagement was included in the model (reaction 224 in S1 Appendix). The binding of eculizumab to its target C5 was described according to the law of mass action as: (9) where C_E, C_C5, and C_EC_C5 are the eculizumab, C5, and drug-target complex concentration, respectively. The off-rate was calculated from the reported K_D value of 120 pM [69] and an assumed on-rate of 20 nM^-1 day^-1 based on the on-rates previously determined for 1500 antibodies [70,71]. Drug-bound C5 was assumed to be completely inhibited and functionally inactive.

In simulations of the inhibition of in vitro hemolytic activity, the initial cell concentration in the incubation, serum dilution, assay readout time, and eculizumab concentration were set to match the reported experimental conditions (S3 Table). Although PNH phenotype is known to differ among patients [68,72], type 2 erythrocytes were assumed due to the lack of donor information.

For the simulation of eculizumab treatment in PNH patients, the systemic pharmacokinetics were described with a one-compartmental model, assuming linear clearance (reaction 225 and 226 in S1 Appendix), an elimination half-life of 14.3 days, a molecular weight of 148 kDa, and a volume of distribution of 6.5 L [73]. Eculizumab dosing was recapitulated from [74] with eculizumab administered once weekly as an intravenous infusion of 600 mg for the first 4 weeks and 900 mg once every 2 weeks from week 5. Pre-treatment hemoglobin levels of the clinical study population [74] were matched adjusting the CD59 and DAF synthesis rates. The impact of transfusions was neglected.

Software.

Data from publications was digitized with the DigitizeIt software (I. Bormann, DigitizeIt version 2.3.3, 2016. Retrieved from http://www.digitizeit.de/). ODEs were implemented and simulated in the SimBiology toolbox of MATLAB software version R2018b (MATLAB R2018b, The MathWorks, Inc., Natick, Massachusetts, United States). Unless stated otherwise, parameters were fitted using the lsqcurvefit function with default options as implemented in MATLAB R2018b.