Modeling the Molecular Impact of SARS-CoV-2 Infection on the Renin-Angiotensin System

SARS-CoV-2 infection is mediated by the binding of its spike protein to the angiotensin-converting enzyme 2 (ACE2), which plays a pivotal role in the renin-angiotensin system (RAS). The study of RAS dysregulation due to SARS-CoV-2 infection is fundamentally important for a better understanding of the pathogenic mechanisms and risk factors associated with COVID-19 coronavirus disease and to design effective therapeutic strategies. In this context, we developed a mathematical model of RAS based on data regarding protein and peptide concentrations; the model was tested on clinical data from healthy normotensive and hypertensive individuals. We used our model to analyze the impact of SARS-CoV-2 infection on RAS, which we modeled through a downregulation of ACE2 as a function of viral load. We also used it to predict the effect of RAS-targeting drugs, such as RAS-blockers, human recombinant ACE2, and angiotensin 1–7 peptide, on COVID-19 patients; the model predicted an improvement of the clinical outcome for some drugs and a worsening for others. Our model and its predictions constitute a valuable framework for in silico testing of hypotheses about the COVID-19 pathogenic mechanisms and the effect of drugs aiming to restore RAS functionality.


Introduction
Since December 2019, the world has been facing a global viral pandemic of the novel severe acute respiratory syndrome coronavirus 2, "SARS-CoV-2"; this pandemic has, to date, caused millions of people to be infected and hundreds of thousands to die [1]. First detected in the city of Wuhan (China) [2][3][4][5], SARS-CoV-2 spread rapidly throughout the world. The coronavirus family, to which SARS-CoV-2 belongs, includes a number of viruses, such as SARS-CoV and MERS-CoV, which have been implicated in serious epidemics that cause acute respiratory distress syndrome (ARDS). There is not yet consensus on the origin of SARS-CoV-2 [6][7][8][9], but the primary hypothesis is that it originated from bat (Rhinolophus affisor) or pangolin (Manis javanica), since the genomes of these two viral species share high sequence identity with SARS-CoV-2.
Coronaviral genomes encode a series of structural proteins, one of which is the spike glycoprotein or S-protein that protrudes from the membrane surface [9]. Similar to the SARS-CoV virus that was identified in 2003, the S-protein of SARS-CoV-2 has been shown to bind to the angiotensin-converting enzyme 2 (ACE2), so that it can be used as an entry receptor to the cell [9][10][11][12][13]. This protein plays a pivotal role in the renin-angiotensin system (RAS) signaling pathway [14] by cleaving angiotensin Figure 1. Schematic representation of RAS. In the unperturbed system, soluble proteins that are explicitly considered in the model are in blue grey, the peptides in light blue, and the peptide-bound membrane proteins in medium blue. The activities and enzymes considered only through reaction rates are in green. The feedback loop is indicated in blue. In the perturbed system, the drugs are in orange and SARS-CoV-2 in dark red. Despite these interesting findings, there is not yet a detailed understanding of how SARS-CoV-2 infection leads to a dysregulation of RAS and, in severe cases, to ARDS. It is of fundamental importance that we gain better insights into the perturbed RAS in order to properly elucidate the pathogenic mechanisms and associated risk factors of SARS-CoV-2 infection; this, in turn, will enable novel therapeutic strategies to be designed and tested so that disease progression can be inhibited.

Modeling the Renin-Angiotensin System
RAS has been widely studied both experimentally [32][33][34] and computationally [35][36][37][38]. It plays a key role in the regulation of a large series of physiological systems including the renal, lung, and cardiovascular systems. Consequently, its dysregulation is related to multiple pathological conditions such as hypertension and ARDS, just to mention a few [39][40][41][42][43].
There are two different types of RAS: the circulating RAS that is localized in the plasma and is involved in the regulation of the cardiovascular system and the tissue-localized systems that act intracellularly or interstitially within different organs in association with the systemic RAS or independently of it. Here, we focus on the local RAS within the pulmonary circulation and model its network of biochemical reactions, as schematically depicted in Figure 1.
When the blood pressure decreases, the juxtaglomerular kidney cells that sense changes in renal perfusion pressure secret an aspartic protease protein called renin (RE, EC 3.4.23. 15). The activity of this enzyme, called plasma renin activity (PRA), is the common measure used in clinical practice to set up the diagnosis and treatment design of essential hypertension.
The dynamics of the renin concentration can be modeled as: where h re is renin's half-life and β its production rate. The latter is not constant, but depends on other elements of RAS, which we will discuss later in the section. The role of renin is to cleave the N-terminus of a protein from the serine protease inhibitor family called angiotensinogen (AGT) to form the decapeptide hormone angiotensin I (AngI). The dynamics of the angiotensinogen can be written as: where the reaction rate c re relates the renin concentration to its activity, k agt is AGT's production rate, and h agt its half-life. The AngI peptide is further cleaved by different enzymes: • The angiotensin-converting enzyme (ACE, EC3.4.15.1) is a zinc metalloproteinase located mainly in the capillaries of lung and in the endothelial cells. It catalyzes the transformation of AngI into the octapeptide angiotensin II (AngII). • Chymase (CHY, EC 3.4.21.39), a serine protease that is mainly localized in blood vessels and heart, also catalyzes the transformation of AngI into AngII. • Neprilysin (NEP, EC3.4.24.11), another zinc metalloproteinase that is expressed in a wide variety of tissues, catalyzes the transformation of AngI into the heptapeptide hormone angiotensin-(1-7) (Ang1-7).
The dynamics of AngI can thus be described as: Viruses 2020, 12, 1367 4 of 20 where c ace , c chy , and c nep are the reaction rates associated with the corresponding enzymatic reactions. The role of AngII in RAS is central since it has a vasoconstriction effect, enhances blood pressure, and triggers inflammatory processes and fibrosis. In lung, the capillary blood vessels are among the sites that have the highest ACE expression and production of AngII. Its dysregulation has frequently been related to a wide series of chronic and acute diseases such as pulmonary fibrosis and ARDS.
AngII's effects are mediated by two G-protein coupled receptors (GPCR) called angiotensin II type 1 (AT1R) and type 2 (AT2R). In addition, it can be cleaved by different enzymes. For example, ACE2 generates Ang1-7 peptides, and aminopeptidase A (APA, EC 3.4. 11.7) generates other peptides such as angiotensin III (AngIII), which is further cleaved to AngIV. In our model, we skipped all the details about the enzymatic reactions AngII-AngIII-AngIV and kept only a single equation for their transformation. The dynamics of AngII and AngIV can thus be written as: where h angI I and h angIV are the half-lives of the peptides and c ace2 , c angIV , c at1r , and c at2r the rates of the enzymatic reactions. The dynamics of the peptide-bound form of the GPCRs are expressed as: where [AT1R-AngII] and [AT1R-AngII] are the concentrations of the bound forms of the receptors and h at1r and h at2r their half-lives. Until now, we have modeled the ACE/AngII/AT1R regulatory axis of RAS. Since the last two decades, it became clear that there is another RAS axis that acts as a counterregulator of the first axis [44]. The key role of this second axis is played by the Ang1-7 peptide that is mainly produced from AngII by the ACE2 enzyme and binds to the transmembrane GPCR called MAS. However, Ang1-7 can also be obtained as an enzymatic product from AngI via the catalytic activity of NEP and, to a lesser extent, from Ang1-9 via ACE and NEP. We overlooked the Ang1-9-related enzymatic reactions in our model, as they contribute less to Ang1-7 formation [33,34]. The dynamical equations for the Ang1-7 peptide and the MAS-bound receptor are as follows: Let us now go back to Equation (1) in which we simply expressed the renin production as a baseline term β. To describe the autoregulatory nature of RAS, this term has to depend on the production of other species, thus introducing a feedback regulation. It is known that this feedback depends on AT1R bound to AngII. Following other models [37,38], we express β as:: where β 0 is a constant parameter to be identified and [AT1R-AngII] N 0 the equilibrium concentration for healthy normotensive humans. δ is a positive number that we fixed to 0.8 [37].
Technical details on the procedure used to solve the model and on model stability are given in Sections 2.6 and 2.7.

Modeling Blood Pressure
Blood pressure is well known to be increased by the concentration of AngII bound to AT1R. It has also been described to be decreased by the concentration of MAS bound to Ang1-7 and of AT2R bound to AngII, but the precise mechanism is not yet known [45][46][47]. Therefore, we did not introduce in our model a feedback between these concentrations and renin production, as we did for AT1R-AngII, and modeled the diastolic blood pressure (DBP) simply from the AT1R-AngII concentration: We identified the two parameters P 0 and P 1 by fixing DBP equal to 80 mmHg for normotensive individuals and to 110 mmHg for hypertensive individuals. Hence, P 0 + P 1 [AT1R-AngII] N 0 = 80 mmHg and P 0 + P 1 [AT1R-AngII] H 0 = 110 mmHg, where the N and H superscripts denote the concentration in normotensive and hypertensive individuals and the 0 subscript the equilibrium concentrations.

Modeling RAS-Blocker Effects
Since dysregulated RAS with high levels of AngII is related to essential hypertension, a wide range of RAS-targeting drugs have been developed in the last forty years [48]. They can be classified into three different categories based on their pharmacological target [49]: • Angiotensin-converting enzyme inhibitors (ACE-I) that bind to ACE and thus inhibit the formation of angiotensin II and the associated vasoconstriction and inflammatory cascades. Examples of this type of drug are enalapril, lisinopril, and captopril.

•
Angiotensin receptor blockers (ARB) that block the binding of AngII to AT1R and thus act in antagonism with AngII. Examples are candesartan, losartan, and valsartan.

•
Direct renin inhibitors (DRI) that act on renin and thus inhibit the conversion of AGT to AngI. Examples are aliskiren, enalkiren, and remikiren.
We modeled the action of these three types of drugs by modifying the reaction rates associated with their targets as: where γ ACE-I , γ ARB , and γ DRI are parameters describing the drug activity.

Modeling SARS-CoV-2 Infection
Since ACE2 is the entry point of SARS-CoV-2 [19], it is downregulated upon infection, and this impacts substantially the local and systemic RASs. In order to model the downregulation effect due to the virus, we modified the ACE2 reaction rate with the function γ CoV : We chose γ CoV to be a sigmoid function of the cycle threshold value C t , which is inversely related to the viral load [50]: Viruses 2020, 12, 1367 6 of 20 where a and b are positive real numbers. C t values of 31.5, 27.6, and 23.8 correspond to mild, moderate, and severe disease, respectively, and C t > 40 to undetected viral infection [51]. We thus chose the inflection point of the sigmoid at C t = 31.5 and imposed γ CoV > 0.99 for C t > 40. Using these relations, we identified the values of the parameters a and b. They are reported in Table 1.

Modeling ARDS Severity
To model ARDS severity and how the lungs of SARS-CoV-2 patients evolve in response to RAS dysregulation, we introduced a phenomenological relation to estimate the PAO2/FIO2 ratio, defined as the ratio between the partial pressure of arterial oxygen (PAO2) and the fraction of inspired oxygen (FIO2). This quantity plays a key role in the assessment of ARDS patients [52,53]. The normal range of PAO2/FIO2 is between 400 and 500 mmHg. Mild  We predicted the PAO2/FIO2 ratio as a function of the AngII and Ang1-7 concentrations: where A 0 and A 1 are two parameters that we identified on the basis of our model by comparing the baseline RAS with the same system in which ACE2 is knocked out. In the former case, we fixed PAO2/FIO2 = 450 mmHg and, in the latter, PAO2/FIO2 = 50 mmHg.

Solving the RAS Model
The mathematical model of RAS described in Equations (1)-(10) is a system of ordinary differential equations (ODEs), which are linear except for the feedback loop of Equation (10).
We collected from the literature the values of the equilibrium concentrations of all proteins and peptides except renin and MAS bound to Ang1-7, for normotensive and hypertensive humans ( Table 2). From these values, we fixed the parameters that appear in the phenomenological relations (11) and (15) for DBP and PAO2/FIO2 (Table 1). We also collected the values of the half-life of all proteins and peptides but MAS; we assumed the latter to be equal to that of the other membrane receptors (Table 1). Moreover, we estimated the value of reaction rate c re from [36,54].
Using these concentration and parameter values, we solved the system of nine ODEs (Equations (1)-(9)) in the stationary state to identify the unknown parameters and concentrations. However, these equations have 12 unknowns: k agt , β 0 , c ace , c ace2 , c angIV , c at1r , c at2r , c mas , c chy , c nep , [RE] 0 , and [MAS-Ang1-7] 0 . We had thus to assume three additional relations, which are: Since no quantitative data related to the MAS receptor can be found in the literature, we hypothesized the first relation assuming MAS and AT2R to be equally expressed and the affinity of Ang1-7 for MAS to be similar to the affinity of AngII for AT2R [46]. Moreover, we assumed c chy = 0 and c nep = 0, but discuss the effect of non-vanishing values in Section 4. By imposing these three additional relations, we solved the system of nine ODEs in the stationary state. The values obtained for [RE] 0 , [MAS-Ang1-7] 0 , k agt , β 0 , c ace , c ace2 , c angIV , c at1r , and c at2r for normotensive and hypertensive humans, are given in Table 2.

Stability of the RAS Model
The system of nine ODEs (Equations (1)-(9)) can be summarized in the form: where x(t) is the vector containing the nine state variables, i.e., the concentrations of all proteins and peptides at time t, θ is the vector with all the production, kinetic, and half-life parameters, and f represents the vector that corresponds to the right-hand sides of Equations (1)- (9). In order to analyze the stability of the two steady states x N 0 and x H 0 for normotensive and hypertensive individuals, respectively, we computed the eigenvalues of the Jacobian matrix: where x 0 stands for either x N 0 or x H 0 . In both the normotensive and hypertensive cases, seven strictly negative real values were obtained, together with two complex conjugate eigenvalues with strictly negative real parts. Both steady states x N 0 and x H 0 are therefore stable. The nonzero imaginary parts of the two complex conjugate eigenvalues are responsible for some damped oscillations in transient responses to parameter changes, but the overshoots are limited. It is interesting to note that the imaginary part is more than three times lower in the hypertensive case, hence leading to more damped responses in comparison with the normotensive case.
To quantify the state variable transients and the aforementioned overshoots, we simulated step responses corresponding to a 10% increase in the normal baseline for renin production β 0 . We observed some damped oscillations during the transient phase of the normotensive case, with very limited overshoots, e.g., 1.3% for the RE concentration. In the hypertensive case, the imaginary part of the complex conjugate eigenvalues is so low that the overshoots become almost undetectable (0.025%).

Results
The main objective of this paper is to investigate the effect of RAS-targeting drugs and SARS-CoV-2 infection, both individually and in combination, on the RAS of normotensive and hypertensive individuals. The robustness and predictive power of our model was first assessed by investigating the effects on RAS of three types of antihypertensive drugs: (i) ACE-I, (ii) ARB, and (iii) DRI (described in Section 2.3). This assessment included a comparison of model simulations with patient clinical data. Following the confirmation of model robustness and accuracy, ACE2 downregulation due to viral infection was introduced into the model to quantitatively predict how RAS is perturbed in COVID-19.

Model Predictions and Clinical Data on RAS-Blocker Drugs
The effect of enalapril, an ACE-I type drug, on plasma ACE activity and on plasma levels of AngI and AngII has been measured in normotensive individuals who received a single oral dose of 20 mg [60]. To compare these data with model predictions, we first fitted the γ ACE-I parameter introduced in Equation (12) to the ACE activity values during enalapril administration divided by the pre-treatment activity (measured by an antibody-trapping assay). Once γ ACE-I was set, we used our model to predict the dynamical response of RAS to this inhibitor drug. The time-dependent values of the AngI and AngII concentrations, normalized by their concentration at time 0, are shown in Figure 2a,b, both for our model predictions and experimental enalapril data; there is very good agreement between the two curves, without any further parameter fitting. The excellent correspondence between model prediction and experimental data is also clear from the root mean squared deviation (rmsd) between model prediction and experimental data on all time points following drug administration, as shown in Table 3.
Our model, thus, captures the known dynamics of ACE inhibition, (i.e., increased AngI levels and decreased AngII levels); this has the effect of lowering the concentration of AngII bound to AT1R and, thus, also lowers the blood pressure (Equation (11)).
To study the effect of ARB antihypertensive drugs on RAS, we considered data from [61], which measured the effects of different types of AT1R blocking molecules on the plasma levels of AngII in normotensive individuals. Specifically, the study participants received a single 50 mg dose of losartan, 80 mg of valsartan, or 150 mg of irbesartan. First, we fitted the γ ARB parameter (defined in Equation (12)) to the in vitro ability of the administered drug to induce the AngII receptor blockade, as measured by an AT1R radioreceptor binding assay [61]. We then used our model to predict the time-dependent AngI level, which was normalized by its concentration prior to drug administration. The results were evaluated through the rmsd between experimental and predicted values of AngI/AngI 0 at different time points after drug administration. The results, which are detailed in Table 3, clearly show that our model accurately predicts the RAS response to ARBs. We also studied the effect of DRI-type drugs using experimental data that describe PRA activity and RE, AngI, and AngII concentrations, when different doses of aliskiren were administered orally to normotensive individuals [62]. We used the PRA activity data to fit the γ DRI parameter (introduced in Equation (12)), and we used our model to calculate the normalized AngI and AngII levels as a function of time. Here also, the results from our model and the experimental concentration data agree very well, as shown in Table 3.
In summary, the rmsd between predicted and experimental values of normalized AngI and AngII levels, averaged over all tested drugs, dosages, and a total of 38 time points, is 0.57 and 0.18, respectively (Table 3). These values should be compared with average experimental values of 1.7 and 0.5, respectively, demonstrating excellent agreement between experimental data and model predictions.
It should be noted that all reported experimental data were obtained after administration of single doses of RAS-targeting drugs. However, for hypertensive patients receiving long-term treatment, the expression of some enzymes involved in RAS could be either up-or down-regulated; we will return to this point in Section 4. Finally, we compared model predictions against clinical data from large cohorts of patients describing the effect of ACE-I and ARB drug administration on blood pressure [63,64]. We first analyzed the response to ACE-I drugs alone. We plot measured DBP values averaged over more than ten ACE-I drug types as a function of the normalized dosage ξ ACE-I [63] in Figure 2c, as well as predicted DBP values. We first considered a linear relation between γ ACE-I and the normalized drug dosage (γ ACE-I = 0.5 ξ ACE-I ). Despite this simplification, chosen to limit the number of parameters and thus overfitting issues, the curve reproduces the experimental data reasonably well. We also defined a non-linear relationship between these two quantities by introducing additional parameters: ACE-I . We thus obtained a better fit as shown in Figure 2c. We then studied the effect of the combined administration of the two drugs, ARB and ACE-I, on blood pressure, plotting the predicted DBP values as a function of both ξ ACE-I and ξ ARB (see Figure 2d). We found that combined administration of ARB and ACE-I reduces DBP by 4 mmHg when compared with ARB monotherapy and by 12 mmHg when compared with ACE-I monotherapy. These predictions should be compared with clinical DBP values of 3 mmHg for combined administration compared to either monotherapy [64]. Thus, our model again provides an excellent prediction of experimental clinical data; further improvements to the model's predictive strength are possible by fixing the γ ARB value at the maximum dose to be slightly lower than the corresponding γ ACE-I value.

RAS in COVID-19
It is known that ACE2 is the cellular receptor of the spike glycoprotein of SARS-CoV-2 [9][10][11][12][13] and that it triggers the entry of SARS-COV-2 into the host cell. Although ACE2 is expressed in a variety of tissues [65][66][67], it is expressed mainly in the alveolar epithelial cells of lung, in the gastrointestinal tract, and in the kidney proximal tubular cells.
Here, we used our model to predict how RAS is perturbed by the SARS-CoV-2 virus. Simulation results of AngII and Ang1-7 concentrations and of the physiological value of PAO2/FIO2 as a function of SARS-CoV-2 viral load are presented in Figure 3 and in Table 4. We observe that the AngII level increases with increasing viral load, with a much stronger increase for hypertensive than for normotensive patients. The AngII level is predicted to increase by approximately 15% for patients with moderate and severe COVID-19 (Table 4); this prediction is in very good agreement with the experimental value of 16% found in [68], but in poorer agreement with the value of 35% resulting from a study of only 12 patients [69].
We also observe that our model predicts a severe reduction of the Ang1-7 level, due to ACE2 downregulation; this reduction is the same for hypertensive and normotensive patients.
The overall result of the model is that RAS becomes imbalanced upon SARS-CoV-2 infection, with the harmful AngII axis upregulated and the counteracting Ang1-7 axis severely downregulated. This imbalance can be related to multiple clinical manifestations of COVID-19. More specifically, increased AngII levels cause hyperinflammation, which, in turn, increases plasma proinflammatory cytokine levels (in particular, IL-6) [70,71]. In addition, thrombotic events are observed, since AngII promotes the expression of plasminogen activator inhibitor-1 (PAI-1) and tissue-factors (TFs) [72,73]. Ang1-7, which normally counteracts these various effects [44], is downregulated by SARS-CoV-2 infection, such that COVID-19 clinical manifestations become increasingly severe as the disease develops. Moreover, our model predicts severe ARDS with PAO2/FIO2 < 100 mmHg for normotensive and hypertensive patients whose C t values are smaller than 24.1 and 27.0, respectively. Our model predicts moderate ARDS, characterized by a PAO2/FIO2 ratio in the range of 100-200 mmHg, for normotensive and hypertensive patients having 24.1 < C t < 29.3 and 27.0 < C t < 29.7, respectively, and mild ARDS, characterized by a PAO2/FIO2 ratio in the range of 200-300 mmHg for normotensive and hypertensive patients having 29.3 < C t < 31.4 and 29.7 < C t < 31.6, respectively.
Our modeling approach suggests a weak relationship between hypertension and ARDS severity resulting from SARS-CoV-2 infection. The mean value of the PAO2/FIO2 ratio over the entire C t range is approximately 20 mmHg lower for hypertensive than for normotensive patients. Indeed, the large difference in AngII levels between normotensive and hypertensive patients is partially compensated by the absence of any difference in Ang1-7 levels. We predicted the PaO2/FiO2 ratio as a function of the AngII and Ang1-7 concen where A 0 and A 1 are two parameters that we identified from our model by comp baseline RAS with the same system in which ACE2 is knocked out. In the former PaO2/FiO2= 450 mmHg and in the latter PaO2/FiO2= 50 mmHg.
Solving the RAS model xx should this part not be in the Results section? With the paragraph of Philippe The mathematical model of the RAS system described in Eqs (1)-(11) is a s ordinary di↵erential equations (ODEs), which are linear except for the feedback lo (11). We collected from the literature the values of the equilibrium concentratio proteins and peptides for both normotensive and hypertensive humans (Table 1 renin and MAS bound to Ang1-7. From these values, we fixed the parameters that the phenomenological relations (12) and (15) for DBP and PaO2/FiO2 (2).
We also got the values of the half-life of all proteins and peptides but MAS; we ass latter to be equal to that of the other membrane receptors (Table 2). Moreover, we e the value of reaction rate c re from [32,33].
Using these concentration and parameter values, we solved the system of 9 ODE at the stationary state to identify the unknown parameters and concentrations. these equations have 12 unknowns: We had thus to assume three additional relations to be abl the system. These are:

Impact of RAS-Modulating Drugs on COVID-19 Severity
We analyzed the effect of administering a selection of drugs to normotensive and hypertensive patients who were infected with SARS-CoV-2. More specifically, we considered RAS-blocking drugs that are already commonly used to treat hypertension, as well as drugs that are currently undergoing clinical trials in the context of COVID-19, such as rhACE2 and Ang1-7.

•
Antihypertensive RAS-blocking drugs: We combined the effect of each of the three RAS-blocking ACE-I, ARB, and DRI drugs, which were modeled by the enzyme-inhibiting γ functions (introduced in Equation (12)), with the ACE2-inhibiting C t -dependent γ CoV function (defined in Equation (14)), which mimics SARS-CoV-2 infection. The PAO2/FIO2 values predicted by our model are presented in Figure 4.
Our model predicts that administration of ACE-I and DRI drugs protect from the adverse effects of ARDS, especially for hypertensive patients, while ARB drugs are predicted to worsen ARDS severity, especially for normotensive patients.
Model predictions for ACE inhibitors are in agreement with clinical data, which indicate that treatment with ACE inhibitors is associated with better survival among COVID-19 patients [31,74]. Indeed, only 3% of non-surviving COVID-19 patients that were monitored were treated with ACE-I drugs compared to 9% of surviving COVID-19 patients [31]. Moreover, in a meta-analysis [74], hypertensive patients treated with ACE-I drugs were associated with a reduced mortality of 35% when compared to patients who were not treated with ACE-I drugs. In another clinical analysis [75], older patients who were treated with ACE-I drugs had a 40% lower risk of hospitalization than those who were not treated with ACE-I drugs.
No data are currently available to validate our model prediction that COVID-19 attenuation due to ACE-I drug treatment is stronger in hypertensive than in normotensive patients. Furthermore, no data are currently available to validate our model prediction that DRI and ACE-I drug treatments cause similar levels of COVID-19 disease attenuation.
In contrast to DRI and ACE-I drugs, our model predicts that ARB drug treatment worsens COVID-19 severity, with the effect being stronger for normotensive compared to hypertensive patients. Here, the agreement between model predictions and clinical data is less clear, with some clinical data in agreement with our model prediction [31,75], while other clinical data suggest that ARB drug treatment does not affect hospitalization risk [75] or mortality [74,76]. This lack of agreement must be further investigated with additional clinical data.
Moreover, we performed a quantitative prediction of the drug effects on disease severity by calculating the RAS peptide concentrations, PAO2/FIO2 values, and DPB for moderate COVID-19 patients. The results are presented in Table 5. Administration of ACE-I drugs, modeled by γ ACE−I = 0.5, increases the PAO2/FIO2 value by approximately 50 and 70 mmHg for normotensive and hypertensive patients, respectively. An equivalent administration of DRI drugs increases this ratio even more, by 70 and 150 mmHg, while ARB administration decreases it by 140 and 30 mmHg for normotensive and hypertensive patients, respectively.
The opposite effect of ARBs administration compared to ACE-I and DRI drugs can be attributed to the substantial increase in AngII concentration, which is only partially balanced by a relatively small increase in Ang1-7 concentration, given that ACE2 is downregulated in SARS-CoV-2 infection.
Note that a number of ARB drugs, including valsartan and losartan, are currently being tested in clinical trials, with the hope that they will rescue RAS in COVID-19 patients [25][26][27]. Our model predicts that this will not be the case.
Finally, as shown in Table 5, the blood pressure is predicted to be unaffected by the administration of either ACE-I, ARB, or DRI to normotensive COVID-19 patients, but to be reduced by approximately 10-20 mmHg by administration to hypertensive patients. • Angiotensin receptor blockers (ARB) that block the binding of AngII to AT1R and thus act in antagonism with AngII. Examples are candesartan, losartan and valsartan. • Direct renin inhibitors (DRI) that act on renin and thus inihibit the conversion of AGT to AngI. Examples are aliskiren, enalkiren and remikiren. We modeled the action of these three types of drugs by modifying the reaction rates associated to their targets as: where ACE-I , ARB and DRI are parameters describing the drug activity.

Modeling CoViD-19 infection
Since ACE2 is the entry point of SARS-CoV-2 [19], it is downregulated upon infection, and this impacts substantially on the local and systemic RAS systems. In order to model the downregulation e↵ect due to the virus, we modified the ACE2 rate with the function CoV as: This function depends on the virus cycle threshold value Ct, which is inversely related to the viral load [69].
Monitoring the acute respiratory distress syndrome and its severity To model how the lungs of the infected patients evolve in response to the modulation of the RAS system, we introduced a phenomenological relation to estimate the PaO2/FiO2 ratio, defined as the ratio between the partial pressure of arterial oxygen (PaO2) and the fraction of inspired oxygen. This quantity plays a key role in the assessment of ARDS patients [34,35]. The normal range of PaO2/FiO2 is between 400 and 500 mmHg. We predicted the PaO2/FiO2 ratio as a function of the AngII and Ang1-7 concentrations: where A0 and A1 are two parameters that we identified from our model by comparing the baseline RAS with the same system in which ACE2 is knocked out. In the former we fixed PaO2/FiO2= 450 mmHg and in the latter PaO2/FiO2= 50 mmHg.

(b) (c)
Monitoring the acute respiratory distress syndrome and its severity To model how the lungs of the infected patients evolve in response to the modulation of the RAS system, we introduced a phenomenological relation to estimate the PaO2/FiO2 ratio, defined as the ratio between the partial pressure of arterial oxygen (PaO2) and the fraction of inspired oxygen. This quantity plays a key role in the assessment of ARDS patients [34,35]. We predicted the PaO2/FiO2 ratio as a function of the AngII and Ang1-7 concentrations: where A0 and A1 are two parameters that we identified from our model by comparing the baseline RAS with the same system in which ACE2 is knocked out. In the former we fixed PaO2/FiO2= 450 mmHg and in the latter PaO2/FiO2= 50 mmHg.
Solving the RAS model xx should this part not be in the Results section? With the paragraph of Philippe ? xx The mathematical model of the RAS system described in Eqs (1)-(11) is a system of ordinary di↵erential equations (ODEs), which are linear except for the feedback loop of Eq. (11). We collected from the literature the values of the equilibrium concentrations of all proteins and peptides for both normotensive and hypertensive humans (Table 1), except renin and MAS bound to Ang1-7. From these values, we fixed the parameters that appear in the phenomenological relations (12) and (15) for DBP and PaO2/FiO2 (2).
We also got the values of the half-life of all proteins and peptides but MAS; we assumed the latter to be equal to that of the other membrane receptors (Table 2). Moreover, we estimated the value of reaction rate cre from [32,33].
Using these concentration and parameter values, we solved the system of 9 ODEs (1)-(11) at the stationary state to identify the unknown parameters and concentrations. However, these equations have 12 unknowns: kagt, 0, cace, cace2, cangIV , cat1r, cat2r, cmas, cchy, cnep, [RE] and [MAS-Ang1-7]. We had thus to assume three additional relations to be able to solve the system. These are: Since no quantitative data related to the MAS receptor can be found in the literature, we hypothesized the first relation assuming MAS and AT2R to be equally expressed and the a nity of Ang1-7 for MAS to be similar to the a nity of AngII for AT2R [46]. Moreover, we assumed cchy = 0 and cnep = 0, but carefully discussed the e↵ect of non-vanishing values in the Discussion section. Imposing these three additional relations, we solved the system of 9 ODEs (1)-(11) at the stationary state. The values obtained for [RE] and [MAS-Ang1-7], kagt, 0, cace, cace2, cangIV , cat1r and cat2r for normotensive and hypertensive humans are given in Table 1. 7

Monitoring the acute respiratory distress syndrome and its severity
To model how the lungs of the infected patients evolve in response to the modulation of the RAS system, we introduced a phenomenological relation to estimate the PaO2/FiO2 ratio, defined as the ratio between the partial pressure of arterial oxygen (PaO2) and the fraction of inspired oxygen. This quantity plays a key role in the assessment of ARDS patients [34,35]. The normal range of PaO2/FiO2 is between 400 and 500 mmHg. Mild and moderate ARDS are characterized by PaO2/FiO2 values in the range [200-300] mmHg and [100-200] mmHg, respectively. ARDS is severe for values below 100 mmHg.
We predicted the PaO2/FiO2 ratio as a function of the AngII and Ang1-7 concentrations: where A0 and A1 are two parameters that we identified from our model by comparing the baseline RAS with the same system in which ACE2 is knocked out. In the former we fixed PaO2/FiO2= 450 mmHg and in the latter PaO2/FiO2= 50 mmHg.
Solving the RAS model xx should this part not be in the Results section? With the paragraph of Philippe ? xx The mathematical model of the RAS system described in Eqs (1)-(11) is a system of ordinary di↵erential equations (ODEs), which are linear except for the feedback loop of Eq. (11). We collected from the literature the values of the equilibrium concentrations of all proteins and peptides for both normotensive and hypertensive humans (Table 1), except renin and MAS bound to Ang1-7. From these values, we fixed the parameters that appear in the phenomenological relations (12) and (15) for DBP and PaO2/FiO2 (2).
We also got the values of the half-life of all proteins and peptides but MAS; we assumed the latter to be equal to that of the other membrane receptors (Table 2). Moreover, we estimated the value of reaction rate cre from [32,33].
Using these concentration and parameter values, we solved the system of 9 ODEs (1)-(11) at the stationary state to identify the unknown parameters and concentrations. However, these equations have 12 unknowns: kagt, 0, cace, cace2, cangIV , cat1r, cat2r, cmas, cchy, cnep, [RE] and [MAS-Ang1-7]. We had thus to assume three additional relations to be able to solve the system. These are: Since no quantitative data related to the MAS receptor can be found in the literature, we hypothesized the first relation assuming MAS and AT2R to be equally expressed and the a nity of Ang1-7 for MAS to be similar to the a nity of AngII for AT2R [46]. Moreover, we assumed cchy = 0 and cnep = 0, but carefully discussed the e↵ect of non-vanishing values in the Discussion section. Imposing these three additional relations, we solved the system of 9 ODEs (1)- (11)  • Angiotensin receptor blockers (ARB) that block the binding of AngII to AT1R and thus act in antagonism with AngII. Examples are candesartan, losartan and valsartan. • Direct renin inhibitors (DRI) that act on renin and thus inihibit the conversion of AGT to AngI. Examples are aliskiren, enalkiren and remikiren. We modeled the action of these three types of drugs by modifying the reaction rates associated to their targets as: where ACE-I , ARB and DRI are parameters describing the drug activity.

Modeling CoViD-19 infection
Since ACE2 is the entry point of SARS-CoV-2 [19], it is downregulated upon infection, and this impacts substantially on the local and systemic RAS systems. In order to model the downregulation e↵ect due to the virus, we modified the ACE2 rate with the function CoV as: This function depends on the virus cycle threshold value Ct, which is inversely related to the viral load [69].
Monitoring the acute respiratory distress syndrome and its severity To model how the lungs of the infected patients evolve in response to the modulation of the RAS system, we introduced a phenomenological relation to estimate the PaO2/FiO2 ratio, defined as the ratio between the partial pressure of arterial oxygen (PaO2) and the fraction of inspired oxygen. This quantity plays a key role in the assessment of ARDS patients [34,35]. We predicted the PaO2/FiO2 ratio as a function of the AngII and Ang1-7 concentrations: where A0 and A1 are two parameters that we identified from our model by comparing the baseline RAS with the same system in which ACE2 is knocked out. In the former we fixed PaO2/FiO2= 450 mmHg and in the latter PaO2/FiO2= 50 mmHg.

Monitoring the acute respiratory distress syndrome and its severity
To model how the lungs of the infected patients evolve in response to the modulation of the RAS system, we introduced a phenomenological relation to estimate the PaO2/FiO2 ratio, defined as the ratio between the partial pressure of arterial oxygen (PaO2) and the fraction of inspired oxygen. This quantity plays a key role in the assessment of ARDS patients [34,35]. We predicted the PaO2/FiO2 ratio as a function of the AngII and Ang1-7 concentrations: where A0 and A1 are two parameters that we identified from our model by comparing the baseline RAS with the same system in which ACE2 is knocked out. In the former we fixed PaO2/FiO2= 450 mmHg and in the latter PaO2/FiO2= 50 mmHg.
Solving the RAS model xx should this part not be in the Results section? With the paragraph of Philippe ? xx The mathematical model of the RAS system described in Eqs (1)-(11) is a system of ordinary di↵erential equations (ODEs), which are linear except for the feedback loop of Eq. (11). We collected from the literature the values of the equilibrium concentrations of all proteins and peptides for both normotensive and hypertensive humans (Table 1), except renin and MAS bound to Ang1-7. From these values, we fixed the parameters that appear in the phenomenological relations (12) and (15) for DBP and PaO2/FiO2 (2).
We also got the values of the half-life of all proteins and peptides but MAS; we assumed the latter to be equal to that of the other membrane receptors (Table 2). Moreover, we estimated the value of reaction rate cre from [32,33].
Using these concentration and parameter values, we solved the system of 9 ODEs (1)-(11) at the stationary state to identify the unknown parameters and concentrations. However, these equations have 12 unknowns: kagt, 0, cace, cace2, cangIV , cat1r, cat2r, cmas, cchy, cnep, [RE] and [MAS-Ang1-7]. We had thus to assume three additional relations to be able to solve the system. These are: Since no quantitative data related to the MAS receptor can be found in the literature, we hypothesized the first relation assuming MAS and AT2R to be equally expressed and the a nity of Ang1-7 for MAS to be similar to the a nity of AngII for AT2R [46]. Moreover, we assumed cchy = 0 and cnep = 0, but carefully discussed the e↵ect of non-vanishing values in the Discussion section. Imposing these three additional relations, we solved the system of 9 ODEs (1)- (11)  • Angiotensin receptor blockers (ARB) that block the binding of AngII to AT1R and thus act in antagonism with AngII. Examples are candesartan, losartan and valsartan. • Direct renin inhibitors (DRI) that act on renin and thus inihibit the conversion of AGT to AngI. Examples are aliskiren, enalkiren and remikiren. We modeled the action of these three types of drugs by modifying the reaction rates associated to their targets as: where ACE-I , ARB and DRI are parameters describing the drug activity.

Modeling CoViD-19 infection
Since ACE2 is the entry point of SARS-CoV-2 [19], it is downregulated upon infection, and this impacts substantially on the local and systemic RAS systems. In order to model the downregulation e↵ect due to the virus, we modified the ACE2 rate with the function CoV as: This function depends on the virus cycle threshold value C t , which is inversely related to the viral load [69].
Monitoring the acute respiratory distress syndrome and its severity To model how the lungs of the infected patients evolve in response to the modulation of the RAS system, we introduced a phenomenological relation to estimate the PaO2/FiO2 ratio, defined as the ratio between the partial pressure of arterial oxygen (PaO2) and the fraction of inspired oxygen. This quantity plays a key role in the assessment of ARDS patients [34,35]. The normal range of PaO2/FiO2 is between 400 and 500 mmHg. Mild and moderate ARDS are characterized by PaO2/FiO2 values in the range [200-300] mmHg and [100-200] mmHg, respectively. ARDS is severe for values below 100 mmHg. We predicted the PaO2/FiO2 ratio as a function of the AngII and Ang1-7 concentrations: where A 0 and A 1 are two parameters that we identified from our model by comparing the baseline RAS with the same system in which ACE2 is knocked out. In the former we fixed PaO2/FiO2= 450 mmHg and in the latter PaO2/FiO2= 50 mmHg.  Table 5. Predicted effects on AngII and Ang1-7 levels, PAO2/FIO2, and DBP upon drug administration to normotensive and hypertensive COVID-19 patients. The drug administrations are modeled by γ ACE−I , γ ARB , γ DRI , γ rhACE2 = 0.5, and η Ang17 = 25 fmol/(mL min) and moderate SARS-CoV-2 infection by γ CoV = 27.6.

Drugs No Drugs ACE-I ARB DRI rhACE2 Ang1-7
Normotensive-Moderate Infection • Other RAS-targeting drugs: We used our model to test the potential of other drugs that are currently in clinical trials to restore the functional activity of the perturbed RAS upon viral infection. First, we modeled how the administration of an exogenous supplement of rhACE2 (GSK2586881) affects RAS by modifying the reaction rate c ace2 defined in Equation (13). This rate already includes the function γ CoV that mimics SARS-CoV-2 infection, and we simply added a second function γ rhACE2 associated with the effects of rhACE2 administration: Our model predicts an increase in PAO2/FIO2 following the administration of exogenous rhACE2, thus predicting an alleviation of disease severity, as shown in Figure 5 and Table 5. Specifically, PAO2/FIO2 is predicted to increase by approximately 200 mmHg when γ rhACE2 is fixed at 0.5. Our model also predicts, as expected, a reduction in AngII level and an increase in Ang1-7 level.
These predictions are in agreement with both animal and in vitro studies [18,22], whereby rhACE2 is observed to alleviate virus-related ARDS severity through a double action. First, by rhACE2 binding to the virus spike protein, interaction with endogenous ACE2 is prevented, and infection is slowed down. Second, rhACE2 administration increases ACE2 activity, thus causing a reduction in AngII level and an increase in Ang1-7 level; this protects lung against severe failure.
Current clinical trial data concerning the administration of different doses of rhACE2 (0.1, 0.2, 0.4, and 0.8 mg/kg) to SARS-CoV patients at different time intervals (2, 4, and 18 h) are only in partial agreement with our model predictions [20]. Specifically, while clinical data followed the predicted decrease in [AngII] and the predicted increase in [Ang1-7], there was no sustained increase in PAO2/FIO2 compared with placebo. It has been suggested that the drug concentrations used in these clinical trials were too low to have a measurable effect on the respiratory system and that drug administration via infusion would have been more sustained [20]. More experimental and clinical data are clearly needed to further investigate the effect of rhACE2 on coronavirus-related ARDS.
Another method of boosting the second RAS axis, ACE2/Ang1-7/MAS, which is downregulated by SARS-CoV-2 infection, is to administer Ang1-7 peptides as a means of triggering anti-inflammatory and anti-fibrotic mechanisms. We modeled Ang1-7 peptide administration by introducing a new parameter, the production rate η Ang17 , to the dynamical Equation (8) of [Ang1 -7]; this allows the model to describe the exogenous Ang1-7 level, which is added to the endogenous Ang1-7 baseline. As shown in Figure 5b and Table 5, our model predicts a clear alleviation of COVID-19 severity, with PAO2/FIO2 increasing by 50 and 130 mmHg for hypertensive and normotensive patients, respectively, upon administration of η Ang17 = 25 fmol/(mL min) Ang1-7 in infusion. Note that COVID-19 alleviation is significantly stronger in normotensive compared to hypertensive patients for the same drug concentrations; a slightly stronger concentration of Ang1-7 must be administered to hypertensive patients for an equivalent effect.
Our model predicts a quantitative reduction in ARDS severity in COVID-19 patients, in agreement with the known anti-inflammation and anti-fibrosis nature of Ang1-7. Model predictions nicely agree with data from animal studies without the need for any additional fitting. For example, administration of Ang1-7 by infusion to acid-injured rats suffering from ARDS increases the baseline Ang1-7 level by a factor 2.5, leading to an increase in PAO2/FIO2 of approximately 70 mmHg [77]. However, while the PAO2/FIO2 value increases linearly in our model as a function of Ang1-7 concentration, it reaches a plateau in rats; this suggests that our model is probably oversimplified, since PAO2/FIO2 is not a linear function of Ang1-7 concentration. Further work on this aspect of our model will be possible when more data become available. Monitoring the acute respiratory distress syndrome and its severity To model how the lungs of the infected patients evolve in response to the modulation of the RAS system, we introduced a phenomenological relation to estimate the PaO2/FiO2 ratio, defined as the ratio between the partial pressure of arterial oxygen (PaO2) and the fraction of inspired oxygen. This quantity plays a key role in the assessment of ARDS patients [34,35]. The normal range of PaO2/FiO2 is between 400 and 500 mmHg. Mild and moderate ARDS are characterized by PaO2/FiO2 values in the range [200-300] mmHg and [100-200] mmHg, respectively. ARDS is severe for values below 100 mmHg.
We predicted the PaO2/FiO2 ratio as a function of the AngII and Ang1-7 concentrations: where A 0 and A 1 are two parameters that we identified from our model by comparing the baseline RAS with the same system in which ACE2 is knocked out. In the former we fixed PaO2/FiO2= 450 mmHg and in the latter PaO2/FiO2= 50 mmHg.
Solving the RAS model xx should this part not be in the Results section? With the paragraph of Philippe ? xx The mathematical model of the RAS system described in Eqs (1)-(11) is a system of ordinary di↵erential equations (ODEs), which are linear except for the feedback loop of Eq. (11). We collected from the literature the values of the equilibrium concentrations of all proteins and peptides for both normotensive and hypertensive humans (Table 1), except renin and MAS bound to Ang1-7. From these values, we fixed the parameters that appear in the phenomenological relations (12) and (15) for DBP and PaO2/FiO2 (2).
We also got the values of the half-life of all proteins and peptides but MAS; we assumed the latter to be equal to that of the other membrane receptors (Table 2). Moreover, we estimated the value of reaction rate c re from [32,33].
Using these concentration and parameter values, we solved the system of 9 ODEs (1)-(11) at the stationary state to identify the unknown parameters and concentrations. However, these equations have 12 unknowns: k agt , 0 , c ace , c ace2 , c angIV , c at1r , c at2r , c mas , c chy , c nep , [RE] and [MAS-Ang1-7]. We had thus to assume three additional relations to be able to solve the system. These are: c chy = 0 (17) Since no quantitative data related to the MAS receptor can be found in the literature, we hypothesized the first relation assuming MAS and AT2R to be equally expressed and the a nity of Ang1-7 for MAS to be similar to the a nity of AngII for AT2R [46]. Moreover, we assumed c chy = 0 and c nep = 0, but carefully discussed the e↵ect of non-vanishing values in the Discussion section. Imposing these three additional relations, we solved the system of 9 ODEs (1)-(11) at the stationary state. The values obtained for [RE] and [MAS-Ang1-7], k agt , 0 , c ace , c ace2 , c angIV , c at1r and c at2r for normotensive and hypertensive humans are given in Table 1.   7 Monitoring the acute respiratory distress syndr To model how the lungs of the infected patients evolve in RAS system, we introduced a phenomenological relation to defined as the ratio between the partial pressure of arterial inspired oxygen. This quantity plays a key role in the asse The normal range of PaO2/FiO2 is between 400 and 500 m are characterized by PaO2/FiO2 values in the range [200respectively. ARDS is severe for values below 100 mmHg.
We predicted the PaO2/FiO2 ratio as a function of the [AngII] 0 + where A 0 and A 1 are two parameters that we identified f baseline RAS with the same system in which ACE2 is kno PaO2/FiO2= 450 mmHg and in the latter PaO2/FiO2= Solving the RAS model xx should this part not be in the Results section? With th The mathematical model of the RAS system describe ordinary di↵erential equations (ODEs), which are linear ex (11). We collected from the literature the values of the proteins and peptides for both normotensive and hypert renin and MAS bound to Ang1-7. From these values, we fix the phenomenological relations (12) and (15) for DBP and We also got the values of the half-life of all proteins and latter to be equal to that of the other membrane receptors ( the value of reaction rate c re from [32,33]. Using these concentration and parameter values, we sol at the stationary state to identify the unknown paramete these equations have 12 unknowns: k agt , 0 , c ace , c ace2 , c a rhACE2 (GSK2586881). We modeled its e↵ect on the RAS system by modifying the c ace2 coe cient defined in Eq. (14) which already mimics the SARS-CoV-2 infection, as: We thus introduced a new gamma function rhACE2 associated to the e↵ect of rhACE2 administration.
The predictions of our model are shown in Fig. 5 and Table 5. We observe an increase of the PaO2/FiO2 value upon intake of exogenous rhACE2, and thus a weakening of the disease severity. The increase of PaO2/FiO2 is of about 200 mmHg when rhACE2 varies in the interval [0-0.5]. We also observe a reduction of the AngII level and an increase of the Ang1-7 level.
These predictions are in agreement with animal and in vitro studies [18,27], where rhACE2 administration has led to an improvement of the disease condition through a double action. First, its binding to the S-protein of the virus xx S or S1 ? xx prevents interaction with endogenous ACE2 and slows down the viral infection. Second, rhACE2 administration increases the ACE2 activity, thus causing a reduction of the AngII level and an increase of the Ang1-7 level, which results in the protection of the lung from severe failure.
However, our predictions and the data described above do not agree with clinical trials clinical data [20] regarding the administration of rhACE2 at di↵erent doses (0.1 mg/kg, 0.2 mg/kg, 0.4 mg/kg and 0.8 mg/kg) and intervals (2, 4, and 18 h) to CoViD patients are less positive. A drop in [ANGII] and an increase in [ANG1 -7] was found, similar to what we found, but no sustained increase of PaO2/FiO2 was observed for these patients compared with placebo, in contrast with what happens from animal model, . However there is the possibility that drug concentrations were not enough substained and that maybe only trough its continuous infusions could reach a more e↵ective result [20]. More experimental data are needed to further investigate e↵ect of rhACE2 on ARDS and preturbed RAS system due to the SARS-CoV-2 infection.
Another way to maintain an high level of the ACE2/Ang1-7/MasR negative counter-regulation in CoViD is to administrate the Ang1-7 peptide to trigger antiinflammatory and antifibrotic mechanisms. In our computation we model this administration introducing the parameter ⌘ describing the endogenously Ang1-7 quantity that is added to the normal baseline quantity. We can see the results of the computation in Fig 5.b and in Table XXX where a clear improvement of the disease severity is observed with an increase PaO2/FiO2 between 70 and 140 mmHg for hypertensive and normotensve patients respectivley and for administration in infusion of 25 fmol/ml that means almost doubling the control values Ang1-7 0 Note that the improvement is significantly more pronounced in normotensive patients than in hypertensive ones for equal drug concentrations and to reach the same effects the Ang1-7 concentration administrated to hypertensive patients have to be slightly increased.
In agreement with clinical data on human ARDS and with the anti-inflammation and anti-fibrosis nature of Ang1-7, our results predict quantitatively an improvenent Current clinical trial data concerning the administration of di↵erent doses of rhACE (0.1, 0.2, 0.4 and 0.8 mg/kg) to SARS-CoV infected patients at di↵erent time interva (2, 4, and 18 h), are only in partial agreement with our model predictions [20]. Speci ically, while clinical data followed the predicted decrease in [AngII] and the predicte increase in [Ang1 -7], there was no sustained increase in PaO2/FiO2 compared wit placebo. It has been suggested that the drug concentrations used in these clinical tr als were too low to have a measurable e↵ect on the respiratory system and that dru administration via infusion would have been more sustained [20]. More experimen tal and clinical data are clearly needed to further investigate the e↵ect of rhACE2 o coronavirus-related ARDS.
Another method of boosting the second RAS axis, ACE2/Ang1-7/MAS, which downregulated by SARS-CoV-2 infection, is to administer Ang1-7 peptides as a mean of triggering anti-inflammatory and anti-fibrotic mechanisms. We modeled Ang1peptide administration by introducing a new parameter, ⌘ Ang17 , to dynamical equatio (9) of [Ang1 -7]; this allowed the model to describe the exogenous Ang1-7 level, which added to the endogenous Ang1-7 baseline. As shown in Fig. 5.b and Table 5, our mod predicts a clear alleviation of COVID-19 severity, with PaO2/FiO2 increasing by 5 and 130 mmHg for hypertensive and normotensive patients, respectively, upon admin istration of ⌘ Ang17 =25 fmol/(ml min) Ang1-7 in infusion. Note that the COVID-1 alleviation is significantly stronger in normotensive compared to hypertensive patien for the same drug concentrations; a slightly stronger concentration of Ang1-7 must b administered to hypertensive patients for an equivalent e↵ect.
Our model predicts a quantitative reduction in ARDS severity in COVID-19 pa tients, in agreement with the known anti-inflammation and anti-fibrosis nature Ang1-7. Model predictions nicely agree with data from animal studies without th need of any additional fitting. For example, administration of Ang1-7 by infusio to acid-injured rats su↵ering from ARDS increases baseline Ang1-7 level by a facto 2.5, leading to an increase in PaO2/FiO2 of approximately 70 mmHg [77]. Howeve while the PaO2/FiO2 value increases linearly in our model as a function of Ang1concentration, it reaches a plateau in rats; this suggests that our model is probab oversimplified, since PaO2/FiO2 is not a linear function of Ang1-7 concentration Further work on this aspect of our model will be possible when more data becom available.

Conclusion and Perspectives
The spike protein of SARS-CoV-2 interferes with the RAS system by binding to th ACE2 receptor, a key element of RAS. Despite recent progress in understanding th COVID-perturbed RAS system and how its functionality can be restored, more wor is urgently needed in the context of the current COVID-19 pandemic.
We here present a simple computational approach to modeling RAS system evolu tion in the context of SARS-CoV-2 infection. Inspired by a number of existing RA models [41,42,52,59], we searched the literature for measured half-lives and con centrations of angiotensin peptides and their receptors in healthy normotensive an hypertensive individuals, and then identified the unknown production and reactio rate parameters from the model. As an initial test of our model, we compared i

Discussion
The spike protein of SARS-CoV-2 interferes with RAS by binding to the ACE2 receptor, a key element of RAS. Despite recent progress in understanding COVID-perturbed RAS and how its functionality can be restored, more work is urgently needed in the context of the current COVID-19 pandemic.
We here present a simple computational approach to modeling RAS evolution in the context of SARS-CoV-2 infection. Inspired by a number of existing RAS models [35][36][37][38], we searched the literature for measured half-lives and concentrations of angiotensin peptides and their receptors in healthy normotensive and hypertensive individuals and then identified the unknown production and reaction rate parameters from the model. As an initial test, we compared our model predictions of how the administration of RAS-blocking drugs would affect Ang peptide concentrations and blood pressure with relevant experimental data; we found good quantitative agreement between our model and experimental data, without the need for further parameter fitting. We then modeled the effect of SARS-CoV-2 infection on RAS through the downregulation of ACE2, which we related to the SARS-CoV-2 viral load.
A focal point of our work was to investigate how a series of RAS-targeting drugs affected COVID-19 patients. We found that the administration of two antihypertensive drugs, ACE-I and DRI, tended to reduce the severity of COVID-19, while ARB drugs worsened it. Clinical data generally support the model's predictions for the administration of ACE-I drugs, but they are either absent or partially contradict the model's prediction for DRI and ARB administration. Additionally, we modeled a potential treatment that is currently under clinical trial in COVID-19 patients: administration of rhACE2 or Ang1-7 by drug infusion. Our model predicts improved clinical outcomes in these cases, in agreement with a series of experimental data on animal models.
It is important to note that, despite its simplicity, our model has excellent accuracy in reproducing clinical and experimental data on the perturbed RAS. Furthermore, the model's predictions of changes in COVID-19 severity due to drug administration are blind predictions, without the fitting of any additional parameters.
Many challenges remain in our current understanding of RAS perturbation in COVID-19 patients. Importantly, more data regarding angiotensin peptide concentrations upon SARS-CoV-2 infection are urgently needed, since currently available data are often inconsistent or conflicting so that reliable comparisons between model predictions and experimental data cannot be made. Even in healthy individuals, angiotensin peptide levels can vary substantially due to their low circulating concentrations, the experimental techniques used to measure them, and inter-patient variability. When developing our model, we chose not to consider two enzymes that are active in RAS through the cancellation of their reaction rates: CHY and NEP (see Equations (17)-(18)). The CHY enzyme is expressed in mast cells present in interstitial lung connective tissues, and it cleaves AngI to form AngII. The addition of this enzymatic reaction in the model would not really influence the predictions since it would essentially be a reparametrization of ACE activity and of ACE-I action. It might, nevertheless, be interesting to add the CHY enzymatic reaction, which yields ACE-independent synthesis of AngII and has been suggested (although debated) to be upregulated in the case of long-term ACE-I administration [78]; this would enable an explanation of why ACE-I fails to inhibit AngII formation after some time [78,79].
The NEP enzyme is expressed in a wide range of tissues, being particularly abundant in kidney, and it cleaves AngI to form Ang1-7. It influences the counterregulatory RAS axis through its connection to Ang1-7 levels, thus affecting COVID-19 severity. However, NEP's role is far from clear, and the literature contains contradictory findings. Experimental data from rats with ARDS suggest that NEP is severely downregulated in both plasma and lung tissues [80]. Note that NEP also cleaves natriuretic peptides, which have both anti-inflammatory and anti-fibrotic effects [81]. Therefore, the combined administration of NEP-inhibiting and ARB drugs has been suggested to treat SARS-CoV-2 patients [82].
Our future work will include building more complexity into our model by explicitly considering the communication between local and systemic RASs [33,34] and by including the interaction between RAS and the immune system [83]. This model extension is necessary for an improved quantitative understanding of RAS dysregulation upon a variety of perturbations, including SARS-CoV-2 infection.
Our model and its predictions provide a valuable and robust framework for in silico testing of hypotheses regarding COVID-19's pathogenic mechanisms and the effect of drugs that are aimed at restoring RAS functionality. Our work also opens a broader discussion on the role of the full RAS in COVID-19, a topic that has received little attention to date, perhaps due to the current focus on the ACE2 enzyme, which, although very important as it is directly targeted by the virus, constitutes only one part of a much more complex system.