Quantitative prediction of erythrocyte sickling for anti-polymerization activities

Lu Lu

Zhen Li

He Li

Xuejin Li

Peter G. Vekilov

George Em Karniadakis

ickle cell disease (SCD), a genetic blood disorder, is originated from the mutation of a single nucleotide in the gene for hemoglobin, a group of proteins that bind oxygen molecules in red blood cells (RBCs, erythrocytes) . In the mutated form of hemoglobin (HbS), the native charged surface group, glutamic acid, at 𝛽6 site is replaced by a hydrophobic group, valine. This alteration induces polymerization of HbS molecules into long and stiff fibers under hypoxia condition. HbS fibers can distort RBCs into heterogeneous shapes, including elongated, granular, oval, holly-leaf, and crescent (classic sickle) shapes, depending on the intracellular mean corpuscular hemoglobin concentration (MCHC) and deoxygenation rates . In addition to the morphological changes, sickled RBCs are featured with increased stiffness and cell adhesion . Presence of intracellular HbS fibers not only compromises the deformability of the RBCs, but also damages the RBC membrane, resulting in adhesion of RBCs to vascular endothelium . Both the impaired RBC deformability and enhanced RBC-endothelium adhesion contribute to the initiation and propagation of vaso-occlusion (VOC), a hallmark of SCD . Recurrent and unpredictable episodes of VOC lead to stroke, frequent painful crisis events, and other serious complications, including acute chest syndrome, splenic sequestration and aplastic crises .

Sickling of RBCs is initiated with supersaturation of hemoglobin after deoxygenation, leading to possible formation of HbS nuclei. Once nuclei form, HbS molecules rapidly polymerize into fibers, which has been well described by the double nucleation mechanism . Subsequently, these HbS fibers interact with cell membrane and distort RBCs. These kinetic processes thus introduce a latency period, also known as delay time, between deoxygenation and RBC sickling. A prior study pointed out that the delay time is an inverse function of MCHC from the 30th to the 50th power, meaning that high-MCHC RBCs exhibit greater propensity to sickle . The relationship between the delay time and the RBC capillary transit time is considered to be one of the key determinants of pathophysiology of SCD . When the transit time is shorter than delay time, RBCs can pass through the microcirculation before sickling. On the contrary, when the transit time is longer than delay time, RBCs commence sickling when traveling inside the capillary and thus may get stuck in the capillary or adhere to the endothelium of postcapillary venules .

Different therapeutic treatments have been developed based on the increased understanding of pathophysiology of SCD in past decades . Transplantation of stem cells from a matched sibling donor has been used to cure SCD patients with high severity . In addition, other curative treatments, such as gene replacement and gene editing, are at the stage of clinical trials . These methods require operations at advanced medical facilities with high cost and thus are impractical for the majority of SCD patients. Drug therapy, therefore, is still the most feasible way to treat SCD patients. However, there is only one SCD drug, hydroxyurea, available in the past 60 years until last July when the Food and Drug Administration approved a new drug, Endari. Hydroxyurea inhibits RBC sickling through reducing the concentration of HbS by boosting the content of HbF, whereas Endari has a function of reducing the adhesion of sickled RBCs to endothelium. Other approaches such as increasing cell volume, blocking the binding sites on the HbS molecules or enhancing HbS oxygen affinity are still under laboratory studies or clinical trials.

Significant progress has been made in elucidating the kinetics of RBC sickling by analytic and numerical approaches . In particular, a universal curve that describes the correlation between the RBC sickling delay time and supersaturation has been discovered recently . This universal curve allows evaluation of RBC sickling delay-time based on various levels of supersaturation, via which the sickled fraction of RBCs with respect to time after deoxygenation can be estimated . Here, we present a detailed kinetic model to quantitatively predict RBC sickling and evaluate the efficacy of HbS-polymerization inhibitors. Distinguished from the prior model which is developed on the basis of the numerical fitting between the RBC sickling delay time and supersaturation , the current model explicitly describes the kinetics of homogeneous and heterogeneous nucleation of HbS monomers as well as dynamic growth of HbS fibers. This proposed model can evaluate the fraction of sickled RBCs according to the patient-specific MCHC distribution, oxygen tension and RBC capillary transit time, thereby examining the susceptibility of RBC sickling on multiple polymerization factors. In addition, the model can be used to predict the anti-polymerization effect of the existing and potential drugs for SCD. In this paper, we first introduce the kinetic model and the implicated numerical methods. Second, we validate the kinetic model against the available in vivo and in vitro experimental results. Third, we evaluate fraction of sickled RBCs under the inhibitory effects of SCD drugs and demonstrate the capability of the model in assessing therapeutic potential of future drugs. Forth, we perform sensitivity analysis on the parameters in the kinetic model, via which we capture the key quantities that can be used as the targets of the new anti-sickling drugs.

Schematics for the inputs and outputs of the kinetic model.

Results

Kinetic model for predicting RBC sickling in SCD

The proposed kinetic model integrates the kinetics of homogeneous and heterogeneous nucleation of HbS monomers with dynamic growth of HbS fibers. The driving force for HbS nucleation and polymerization is the supersaturation of deoxy-HbS, which is consistent with kinetic model applied in . However, the delay time in our current model is fundamentally different from the sickling delay time, defined in prior studies , which accounts for the time from the supersaturation of deoxy-HbS to the sickling of RBCs. The delay time in the present model represents the time period from supersaturation of deoxy-HbS to the emerging of first nucleus in HbS solution . This definition distinguish the time period before the formation of nuclei and the time for HbS fibers grow and distort the RBC membrane, thereby allowing more detailed analysis of initiation of RBC sickling. As shown in Fig. [fig:1], the inputs of the kinetic model include environmental variables such as temperature, deoxygenation time and oxygen tension, and variables of patients RBCs, such as cell volume, MCHC and concentration of other Hb variants. The outputs are numbers of nuclei from homogeneous and heterogeneous nucleation, the lengths of HbS fibers, the configuration of fiber domains and the sickling of RBCs. In the current work, guided by a prior analytic work , we assume that when an intracellular fiber bundle, consisting of 10 single HbS fibers or more, grow to or longer, the RBCs are considered sickled. A parametric study on the criteria that determines the RBC sickling can be found in SI.

(A) 𝑃𝑂2 decreases linearly from in , and the fraction of deoxy HbS increases from 4.3% to 49%

correspondingly. (B) Variation of supersaturation 𝛥𝜇/𝑘𝐵𝑇, delay time 𝜏𝑑 , homogeneous nucleus size 𝑛∗, homogeneous nucleation barrier 𝛥𝐺∗, and homogeneous nucleation rate 𝐽 with respect to the total hemoglobin concentration 𝐶𝑡 at the end of deoxygenation. (C) Dependence of kinetics of HbS polymerization on the HbS concentrations when 𝑃𝑂2 decreases

linearly from in . Area A: HbS does not become supersaturated in RBCs. Area B: the nucleation delay time is more than . Area C: Less than 1 nucleus is generated in due to the low homogeneous nucleation rates.

Sickling of RBCs in the hepatic vein

In this section, we model the kinetics of HbS polymerization in RBCs passing through the hepatic vein of SCD patients. The simulation results are compared to the experimental measurements for model validation. The distribution of MCHC is approximated by a Gaussian distribution with a mean of and standard deviation of (see Fig. [fig:2]C) . When blood flows from artery to vein through capillaries, we use the oxygen saturation of 51% measured from a SCD patient in , which corresponds to 𝑃𝑂2 = . Furthermore, we assume

𝑃𝑂2 decreases linearly from . The deoxygenation time is assumed to be , comparable to the

average capillary transit time of RBCs . Other model inputs can be found in Table S3 in SI. As shown in Fig. [fig:2]A, at the end of deoxygenation period, the concentration of deoxy HbS is 𝐶 ≈ 0.49𝐶𝑡, which is dependent on total HbS concentration 𝐶𝑡 and oxygen tension 𝑃𝑂2 .

Fig. [fig:2]B illustrates the dependences of supersaturation, nucleation delay time, nucleus size, nucleation barrier and nucleation rate, on the HbS concentration. Fig. [fig:2]B1 demonstrates that for RBCs with 𝐶𝑡 less than the critical concentration of (area A in Fig. [fig:2]C), supersaturation does not occur after deoxygenation (𝛥𝜇/𝑘𝐵𝑇 < 0). When 𝐶𝑡 of RBCs is larger than , Hb becomes supersaturated and nucleation will occur after a delay time of 𝜏𝑑 , which ranges from hundreds of seconds to 0 seconds depending on 𝐶𝑡 (see Fig. [fig:2]B2). It is expected that RBCs with 𝐶𝑡 less than (area B in Fig. [fig:2]C) are still able to pass through capillaries without sickling, thanks to the long delay time ($>\SI{10}{\second}$). When 𝐶𝑡 is larger than , the delay time is shortened to several seconds, comparable to the RBC capillary transit time . However, the probability of sickling is small for RBCs with 𝐶𝑡 less than , as the corresponding homogeneous nucleation rate 𝐽 is less than , meaning that less than 1 nucleus is generated in (area C in Fig. [fig:2]C). Hence, sickling of RBCs is more likely to occur on the RBCs with $C_t > \SI[per-mode=symbol]{34.3}{\gram\per\deci\litre}$ (area D in Fig. [fig:2]C), which accounts for ∼24.3% of RBCs. This finding is consistent with the fraction of sickled RBCs (22%) observed in the patient’s hepatic vein and thus demonstrates the capability of the present model to predict the fraction of RBC sickling in vivo.

Sickling of RBCs in microfluidic channel under the short-term and long-term deoxygenation processes

As a further validation of our model, we perform simulations following the in vitro experiment conduced by Du et al. where the fraction of sickling RBCs were evaluated during short-term and long-term deoxygenation processes. In this in vitro experiment, RBCs were categorized into four groups based on the cell density, which is linearly correlated with MCHC. No significant RBC sickling was observed from the first three groups of RBCs (MCHC < ), which verifies our findings in Fig. [fig:2]C. For the last group with MCHC > , we again assume that the MCHC of these RBCs follows a Gaussian distribution with a mean of and variance of . Guided by the experimental data reported in Ref. du2015kinetics, 𝑃𝑂2 is assumed to decrease linearly from to . Other model inputs can

be found in Table S3 in SI.

Profiles of fractions of sickled RBCs under (A) a short-term DeOxy process (∼25 s), and (B) a long-term DeOxy process (∼220 s) from 1000 RBC samples. Deoxygenation begins at 15 seconds for both cases.

We measure the evolution of the fraction of sickled RBCs during the short-term and long-term deoxygenation process, respectively. Fig. [fig:sickled_fraction] shows that our model can capture the change of sickling fractions for both the short-term and long-term deoxygenation process reported in . However, there are notable discrepancies between our simulation results and experiential data. These discrepancies are likely to result from the approximation of MCHC distribution by a Gaussian distribution. The actual MCHC distribution of the examined RBCs in Ref. du2015kinetics was not provided in the original work. In addition, the experimental data reported by Du et.al du2015kinetics were based on the observations of dozens of RBCs. This small number of samples could affect the statistics of the results.

Examination of sickling inhibitors that increase cell volume

Although the molecular basis of the SCD was discovered about seventy years ago , there is only one drug, hydroxyurea, available for the SCD patients until Endari was approved in 2017. Endari is designed to reduced the cell adhesion, a downstream consequence of RBC sickling, whereas hydroxyurea, which targets the root cause of SCD, inhibits HbS polymerization by reducing the MCHC through inducing the synthesis of fetal hemoglobin (HbF). Clinical evidence indicates that hydroxyurea exhibits heterogeneous outcomes and the causes are still under investigation . Therefore, alternative approaches are proposed to decrease the MCHC by increasing cell volume using compounds that interact with cell membrane . Two of these compounds, namely monensin A and gramicidin A, demonstrated strong therapeutic effect in recent screening assay . In this section, we implement our model to verify the efficacy of these two compounds.

First, we implement our kinetic model to predict the sickling fraction of sickle trait cells without influence of compound, following the prior photolysis experiments performed by Li et.al . The hemoglobin composition is selected to be 38% HbS and 62% HbA, comparable to the composition of the examined patient (38% HbS, 58% HbA, 3.7% HbA2, and 0.3% HbF) reported in . We employ the HbS concentration distribution for normal RBCs reported in . 𝑃𝑂2 is initial set to in our kinetic model. Other model inputs can be found in Table S3 in

SI. Our simulation results show a wide distribution of nucleation delay time primarily due to variation in MCHC. As illustrated in Fig. [fig:4]A (blue curve), this wide distribution of nucleation delay time leads to a gradual increased fraction of sickled RBCs with respect to the time. It is noted that the fraction of sickled RBCs predicted from our kinetic model is consistent with experimental data reported in (black curve in Fig. [fig:4]A). Moreover, when the deoxygenation time is less than 20 s, our model (blue curve) provides a more accurate prediction of the sickling fraction than a prior model (red curve) based on the universal curve of delay time vs. supersaturation . This discrepancy is likely attributed to the fact that the present model considers the number of HbS molecules in the nucleus 𝑛∗ as discrete integer numbers whereas the model applied in represents 𝑛∗ with continuous real numbers. Representation of 𝑛∗ with continuous real numbers leads to infinitely high HbS polymerization rates when the delay time is zero and thus results in overestimation of the RBC sickling fraction at early deoxygenation time.

(A) Sickling fraction of sickle trait cells with hemoglobin composition of 38% HbS and 62% HbA, increases with respect to time after the 𝑃𝑂2 is decreased to . The black curve highlights

the experimental results reported in . The green dashed curve signifies the prediction based on the universal curve of delay time vs. supersaturation in . (B) Fraction of sickled RBCs decreases with increased MCV at 60 s after photolysis. The swelling of RBCs is induced by (B1) monensin A and (B2) gramicidin A. Blue curves represent our simulation results and black curves denote the experimental results reported in . Solid symbols are for 4 h incubations and open symbols are for 24h incubations.

Next, we examine the efficacy of two sickling inhibitors, namely monensin A and gramicidin A, both of which function through reducing the concentration of HbS by increasing cell volume, thereby delaying the sickling of RBCs. For the increased RBC volume by these two compounds, we follow the mean corpuscular volume (MCV) measured with Coulter Counter in . As shown in Fig. [fig:4]B1, our kinetic model predicts that as the concentration of the monensin A increases, the sickling fraction decreases in the same pattern as the experimental data reported in screening assay for both 4 h and 24 h incubations,

respectively. This finding demonstrates the applicability of the present model on testing the efficacy of the SCD drug candidates. Our predictions for gramicidin A after 4 h incubations (Fig. [fig:4]B2) are also consistent with the screening assay. But in the case of 24 h incubations, our simulation results predict much stronger efficacy at lower end of the examined concentration range, which is not evidenced from the experimental results (Fig. [fig:4]B2) . One of the possible causes of this discrepancy is that RBCs under the influence of gramicidin A exhibit a non-monotonic correlation between the increased cell volume and decreased sickling fraction . This finding implies that besides increasing RBC volume, gramicidin A has additional effect on promoting the HbS polymerization, but it is overwhelmed by the effect of increased cell volume as the concentration of gramicidin A increases.

Discovery of targets for sickling inhibitors

Search for SCD drugs that inhibit intracellular polymerization of HbS, other than by swelling the RBCs, has been hampered by the lack of understanding of key quantities that control RBC sickling . Kinetics of HbS polymerization involves unbinding oxygen from the HbS molecules, hemoglobin supersaturation, formation of HbS nucleus after the delay time and the dynamic growth of HbS fibers. To accurately describe each step of this process, multiple parameters are introduced into the kinetic model. A comprehensive understanding of the dependence of RBC sickling on each of these parameters can facilitate the determination of the potentially druggable targets and thus provide guidance on the development of new drugs. The default values of the parameters applied in the kinetic model are selected according to the prior experimental and clinical studies. Detailed information can be found in SI. However, many of these parameters are patient-specific and thus may vary from patient to patient. Therefore, a quantitative evaluation of effects of varying these parameters within their physiological ranges on the eventual results of RBC sickling is necessary.

In this section, we perform sensitivity analysis on the kinetic parameters employed in the proposed model, via which we identify the key quantities that modulate the sickling of RBCs. The parameter sensitivity analysis is performed by using the analysis of variance (ANOVA) functional decomposition , where we vary thirteen parameters simultaneously. The examined parameters include oxygen affinity parameters (𝑐, 𝐾𝑅 , 𝐿), solubility (𝐶𝑒0), activity coefficient factor (𝐵2), delay time factors (𝜏0, 𝑘𝜏), the number of HbS molecules in the nucleus 𝑛∗, homogeneous nucleation factor (𝐽0), ratio of heterogeneous nucleation factor over homogeneous nucleation factor (𝐽0𝑠/𝐽0), ratio of heterogeneous nucleation barrier over the homogeneous nucleation barrier (𝛥𝐺𝑠

∗/𝛥𝐺∗), factors for fiber growth (𝛽0, 𝐸𝛼). The definitions of these parameters and their varying range can be found in SI.

To illustrate the result of the sensitivity analysis, we place the thirteen parameters in a large ring as shown in Fig. [fig:5]A and B. The sensitivity of a single parameter is plotted as

a circle, with the diameter equal to the sensitivity of that parameter. The green lines indicate the interaction of two parameters, which describes how the fraction of sickled RBCs changes when two parameters are varied simultaneously. Thickness of green lines indicates sensitivity of two-parameter interaction pair. First, we examine the sensitivity of the parameters with model inputs following the screening assay in . Fig. [fig:5]A suggests that the fraction of sickled RBCs is most sensitive to 𝐶𝑒0, followed by 𝐽0 and 𝐵2. The sensitivity of the interaction pairs between these thirteen parameters are similar and much weaker than the sensitivity of single parameters of 𝐶𝑒0,𝐽0 and 𝐵2. This finding provides an analytic basis for inhibiting the HbS polymerization through increasing the solubility of HbS by 2,3-Diphosphoglycerate . Our results also suggest suppressing the activity of HbS and nucleation factor can be two potential druggable targets to efficiently inhibit the polymerization of HbS. Notably, the screening assay in was performed in absence of oxygen and thus the effect of oxygen affinity parameters, such as 𝑐, 𝐾𝑅 , 𝐿, can not be considered. Therefore, we perform another sensitivity analysis with the model inputs following the conditions of RBC sickling in hepatic vein . As shown in Fig. [fig:5]B, 𝐽0 is found to be most sensitive to the sickling fraction of RBCs, followed by 𝐶𝑒0 and 𝐾𝑅 . In particular, the effect of 𝐾𝑅 overwhelms that of 𝐵2 when oxygen binding is considered, implying that increasing the binding affinity of hemoglobin could be a more efficient approach than suppressing the activity of hemoglobin. Furthermore, we quantitatively illustrate the effects of changing 𝐶𝑒0, 𝐽0, 𝐵2 and 𝐾𝑅 on reducing the fraction of sickled RBCs. Fig. [fig:5](C-F) show that the fractions of sickled RBCs decrease with MCV at three values of 𝐶𝑒0, 𝐽0, 𝐵2 and 𝐾𝑅 , respectively. These results demonstrate that therapeutic efficacy can be achieved by tuning these four kinetic parameters.

Discussion

RBC sickling is an integrated process of unbinding oxygen from the HbS molecules, hemoglobin supersaturation, formation of HbS nucleus after the delay time and the dynamic growth of HbS fibers. In this work, we develop a kinetic model to describe the kinetics of HbS polymerization and predict the fraction of sickled RBCs based on patient-specific inputs. We implement our model to predict the fraction of sickling RBCs, following prior in vivo Serjeant1973 and in vitro experiments, respectively, for model validation. The prediction of fraction of sickled RBCs from our kinetic model is consistent with the measurements of blood samples from the hepatic vein of 4 SCD patients Serjeant1973. We also demonstrate that our model is capable of capturing the fraction of sickled RBCs during both the short-term and long-term deoxygenation processes in microfluidic experiments du2015kinetics and screening assay . It is noted that our kinetic model can reproduce various experimental results without tuning the model parameters, indicating the applicability of our model to confirm and predict the laboratory and clinical findings.

The proposed kinetic model also can be implemented to examine the efficacy of potential drug candidates. Our simulation results confirm the efficacy of the monensin A, a compound that inhibits the RBC sickling by increasing the cell volume . Our kinetic model, however, overestimates the efficacy of gramicidin A, which functions in a similar manner as monensin A. The deviation of our simulation results from the screening assay indicates that

in addition to increasing cell volume, gramicidin A has additional effects that compromises its anti-sickling activities. This finding is confirmed by the non-monotonic correlation between the increased cell volume and decreased sickling fraction reported in . These results demonstrate the capability of the present model to predict the efficacy of the sickling inhibitors and identify potential side effects of existing and future drugs.

Both the prior screening assay and our simulations suggest that increasing the cell volume can be an efficient approach to inhibit RBC sickling. However, increased cell volume compromises the deformability of the RBCs due to reduced surface area to volume ratio of RBCs, thereby increasing the possibility of hemolysis and splenic retention. Therefore, new drugs that delay RBC sickling through altering the kinetics of HbS polymerization should be considered. Our sensitivity analysis identifies that sickling of RBCs is most sensitive to the solubility (𝐶𝑒0) of HbS, thereby providing an analytical basis for prior attempts on inhibiting RBC sickling through increasing the solubility of HbS by 2,3-Diphosphoglycerate . Our simulation results also indicate that increasing hemoglobin oxygen binding affinity, reducing nucleation rate factor and hemoglobin activity are three effective drug targets. Our simulation results further demonstrate that modulation of each of these four kinetic parameters can achieve anti-sickling efficacy that is comparable to increasing the cell volume. It is noted that when we predict the efficacy of potential drug candidates that target the kinetics of the HbS polymerization, we did not consider the effect of drug molecule diffusion and the RBC membrane permissibility, which could significantly compromise the efficacy of the drugs. Therefore, future anti-sickling drugs that target both sickling kinetics and RBC volume should be considered to optimize their therapeutic efficacy.

Hydroxyurea is a toxic anti-cancer chemotherapy drug. Endari is not toxic, but has potential side effects, such as constipation, nausea, headache, abdominal pain, cough, pain in the extremities, back pain and chest pain. Therefore, reducing the side effects of the drugs is a topic of significance in SCD. A novel feature of the proposed kinetic model is that the inputs of the model are patient-specific quantities such as MCHC, oxygen pressure, cell volume and so on. Therefore, this model can be used to guide the development of precision medicine for SCD patients and thus eliminate the side effects of existing and future drugs.

Methods

Given the five inputs shown in Fig. [fig:1], the computations in our kinetic model can be decomposed into the following six steps: 1. get the fraction of deoxy hemoglobin under 𝑂2 pressure; 2. compute the hemoglobin solubility and then supersaturation; 3. nucleation delay time; 4. homogeneous and heterogeneous nucleation rate; 5. fiber growth rate; 6. fiber fiber growth and branching, and the final RBC sickling time. See SI Appendix for details of methods.