Role of channels in the O2permeability of murine red blood cells III. Mathematical modeling

Role of channels in the O2 permeability of murine red blood cells III. 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Mathematical modeling View ORCID Profile Rossana Occhipinti , View ORCID Profile Pan Zhao , View ORCID Profile Fraser J. Moss , View ORCID Profile Walter F. Boron doi: https://doi.org/10.1101/2025.03.05.639964 Rossana Occhipinti 1 Department of Physiology & Biophysics Case Western Reserve University School of Medicine Cleveland , OH 44106 Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Rossana Occhipinti For correspondence: rossana.occhipinti{at}case.edu Walter.Boron{at}case.edu Pan Zhao 1 Department of Physiology & Biophysics Case Western Reserve University School of Medicine Cleveland , OH 44106 Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Pan Zhao Fraser J. Moss 1 Department of Physiology & Biophysics Case Western Reserve University School of Medicine Cleveland , OH 44106 Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Fraser J. Moss Walter F. Boron 1 Department of Physiology & Biophysics Case Western Reserve University School of Medicine Cleveland , OH 44106 2 Department of Medicine, and Department of Biochemistry Case Western Reserve University School of Medicine Cleveland , OH 44106 Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Walter F. Boron For correspondence: rossana.occhipinti{at}case.edu Walter.Boron{at}case.edu Abstract Full Text Info/History Metrics Preview PDF Abstract In this third of three papers, we develop a reaction-diffusion model for O 2 offloading from a red blood cell (RBC), treated as a sphere with diameter approximating RBC thickness. Stopped-flow (SF) analysis (paper #1) of hemoglobin/oxyhemoglobin (Hb/HbO 2 ) absorbance spectra during O 2 efflux from intact murine RBCs show that membrane-impermeant inhibitor p-chloromercuribenzenesulfonate (pCMBS) reduces the HbO 2 -deoxygenation rate constant ( k HbO2 ) by ∼61%. SF experiments show that k HbO2 falls by (1) 9% for aquaporin-1 knockouts (AQP1-KOs), (2) 17% for Rhesus A-glycoprotein knockouts (RhAG-KOs), (3) 30% for double knockouts (dKOs), and (4) ∼78% in dKOs/pCMBS. Here, we simulate HbO 2 dissociation in the intact RBC (i.e., k HbO2 ); HbO 2 , Hb, and O 2 diffusion through RBC cytosol; transmembrane O 2 diffusion; and O 2 diffusion through extracellular unconvected fluid (EUF) to bulk extracellular fluid. Informed by automated-hematology data (paper #1) and imaging–flow-cytometry data (paper #2), simulations predict that observed k HbO2 decreases cannot reflect changes only in RBC size/shape or [Hb/HbO 2 ]. Instead, membrane O 2 permeability ( P M,O2 ) must fall by (1) 22% to account for AQP1-KO data, (2) 36% for RhAG-KOs, (3) 55% for dKOs, and (4) 91% for dKOs/pCMBS. Exploring predicted k HbO2 sensitivities to eight key parameters (e.g., [Hb/HbO 2 ], diffusion constants, k HbO2→Hb+O2 , thickness EUF , diameter Sphere ) shows that no reasonable changes explain the k HbO2 data. We introduce a linear-combination approach to accommodate for the presence of poikilocytes. Finally, contrary to common beliefs, the model predicts that, in the absence of inhibitors, the RBC membrane represents >30% of total diffusive “resistance” to O 2 offloading, even for a WT mouse. Key Points In this third of three papers, we develop a novel reaction-diffusion model for O 2 offloading from a red blood cell (RBC), treated as a sphere with diameter approximating RBC thickness. Using physical constants and parameter values from the literature and papers #1 and #2, we generate simulations that reproduce observed hemoglobin desaturation (rate constant, k HbO2 ) of intact RBCs from wild-type mice. To simulate k HbO2 decreases (9%, 17%, 30%) observed in RBCs from knockouts (KOs: aquaporin-1, Rhesus A-glycoprotein, both) we must decrease membrane O 2 permeability ( P M,O2 ) by far greater percentages (22%, 36%, 55%). Analyses of simulated- k HbO2 sensitivity to kinetic and geometric parameters suggest that reasonable parameter-value changes cannot explain experimentally observed k HbO2 decreases in RBCs from KOs. Thus, experimentally observed k HbO2 decreases must reflect decreases in P M,O2 . To accommodate for poikilocytes, we develop a linear-combination approach (poikilocytes + biconcave disks, BCDs) to extract the k HbO2 and P M,O2 of just BCDs. Introduction Red blood cells (RBCs) are uniquely suited to perform the vital task of taking up oxygen (O 2 ) in the pulmonary capillaries, carrying this O 2 to the systemic capillaries, and there offloading the O 2 for consumption in the peripheral tissues. This journey of O 2 within the vertebrate body— and the oppositely directed journey of carbon dioxide (CO 2 )—involves a series of convective and diffusive steps within and across the various components of the respiratory system. Integral to the above sequence of events is an important series of membrane-permeation steps, both for O 2 delivery and CO 2 removal. Among the membranes that O 2 and CO 2 must traverse is the plasma membrane of RBCs. The mechanisms by which these gases move across cell membranes, including the RBC membranes, have been the subject of numerous investigations ( Graham, 1829 , 1833 ; Mitchell, 1830 , 1833 ; Fick, 1855 ; Overton, 1895 ; Wroblewski, 1879 ; Hartridge & Roughton, 1927 ; Forster, 1964 ; Waisbren et al ., 1994 ; Nakhoul et al ., 1998 ; Cooper & Boron, 1998 ) and reviews ( Stannett, 1978 ; Geers & Gros, 2000 ; Boron, 2010 ; Boron et al ., 2011 ; Endeward et al ., 2014 ; Cooper et al ., 2015 ; Michenkova et al ., 2021 ). Prior to the 1990s, two major schools of thought emerged. One, influenced by Krogh’s cylinder model of O 2 diffusion from capillary to tissue, implicitly assumes that the RBC membrane offers no resistance to O 2 diffusion ( Krogh, 1919 ; Kreuzer, 1982 )—that is, the membrane behaves as a layer of water of equivalent thickness. The other school of thought, influenced by Overton’s work on the lipoid nature of plasma membranes, is that the permeability of a membrane to a solute like O 2 depends only on the solubility of the solute in the lipid phase of the membrane ( Overton, 1895 , 1899 ). The present contribution is the third in a series of three interconnected papers 1 that investigate the hypothesis that two membrane proteins that are abundant in the murine RBC membrane, aquaporin-1 (AQP1) and the Rh complex (Rh Cx )—comprising Rhesus blood group-associated A-glycoprotein (RhAG) and + Rhesus blood group D antigen (RhD)—represent a large fraction of the membrane permeability to O 2 ( P M,O2 ). In the first paper ( paper #1 ; Zhao et al ., 2025 ), we employ stopped-flow (SF) analysis of hemoglobin/oxyhemoglobin (Hb/HbO 2 ) absorbance spectra to measure the rate constant of Hb deoxygenation ( k HbO2 ) in RBCs from mice genetically deficient in AQP1 ( Aqp1−/− ), RhAG ( Rhag−/− ), or both ( Aqp1−/−Rhag−/− ). In both wild-type (WT) mice and double knockouts (dKO), we also examine the effects of the inhibitors p-chloromercuribenzenesulfonate (pCMBS) and 4,4’-diisothiocyanatostilbene-2,2’-disulfonate (DIDS). Because k HbO2 depends on factors other than P M,O2 , we also assess in paper #1 the rate constant of the dissociation HbO 2 → Hb + O 2 in RBC hemolysates ( k HbO2→Hb+O2 ), and automated-hematological parameters, including mean corpuscular volume (MCV), mean corpuscular hemoglobin (MCH), and mean corpuscular hemoglobin concentration (MCHC). We find that—compared to WT mice— k HbO2 decreases by 9% in RBCs from AQP1-KOs, by 17% in RhAG-KOs, and by 30% in dKO. This work thus points towards roles for AQP1 and RhAG in enhancing O 2 permeation through RBC membranes. In the second paper (i.e. paper #2 ; Moss et al ., 2025 ),, we examine RBC morphology at a microscopic scale, and confirm that the RBCs from the KO and dKO mice are predominantly biconcave disks (BCDs). We also use imaging flow cytometry (IFC) to assess RBC major diameter (Ø Major ). Together with MCV data from paper #1, Ø Major allows us to estimate RBC thickness—a critical piece of information for the simulations in this third (the present) paper in the series. The IFC data also shows that a small fraction of RBCs, more so in drug-treated cells, are poikilocytes, that is, non-biconcave disks (nBCDs). This is also critical information for the present paper because it sets the stage for a novel correction methodology that we apply here. Finally, paper #2 includes a proteomic study that shows that the KO of one or both channels does not affect the apparent abundance of any of the 100 proteins with the greatest inferred abundance. The purpose of this third (the present) paper in the series is to develop a reaction-diffusion model of O 2 offloading from RBCs, and to use this model—based only on first principles and informed by MCV, MCH, MCHC, k HbO2→Hb+O2 , and Ø Major data from the first two papers—to simulate the SF data from the paper #1. We find that the simulations agree reasonably well with the time course of deoxygenation for RBCs from WT mice (measured as k HbO2 values) in paper #1. Note that our approach is not merely to adjust a variety of fitting parameters in an effort to reproduce the observed k HbO2 . Except for sensitivity analyses—when we examine how sensitive the simulations are to variations in specific parameters—the only parameter that we vary is P M,O2 , which we do to predict the P M,O2 values that reproduce the experimentally determined time courses. Our graphical user interface ( Huffman et al ., 2025 ) expands the power of the model, especially for non-experts. Taken together, the three papers argue strongly against each of two previously dominant— and quite opposite—schools of thought (addressed in the Discussion). One holds that RBC membranes offer no resistance whatsoever to O 2 diffusion. The other, posits that they do offer resistance, but that this resistance depends solely on the solubility of O 2 in RBC membrane lipids. Instead, our data are consistent with the notion that RBC membranes from WT mice offer ∼150-fold more resistance to O 2 diffusion than an equivalently thick layer of water, and that channels—AQP1, RhCx, and an unidentified channel(s)—account for >90% of P M,O2 under the conditions of our experiments 2 . Methods Figure 1 shows the workflow for the three-paper project. The present paper focuses on steps #10, #12 and #12′, #13, #14 and #14′, #16, #17 and #17′, #18 through #22. Download figure Open in new tab Figure 1. Workflow The numerals 1 through 22 indicate the steps summarized in paper #1 —and presented in detail, as appropriate, in the present paper ( paper #3 ), paper #1 , or paper #2 —to arrive, first, at a hemolysis-corrected (HC) k HbO2 and, ultimately, at a shape-corrected (SC) k HbO2 . At each step, we provide example values, if possible, and example figure/table numbers referenced to paper #1 , paper #2 , or the present paper. We repeat workflow steps 1–5 for each mouse, ultimately arriving at an HC- k HbO2 value for each. Step 6 applies to biconcave disks (BCDs), whereas step 6′ applies to non-biconcave disks (nBCDs). The same is true for steps 12 vs. 12′, 14 vs. 14′, 15 vs. 15′, and for 17 vs. 17′. F SC , shape-correction factor; Hb, hemoglobin; IFC, imaging flow cytometry; MCH, mean corpuscular hemoglobin; k HbO2 , rate constant for deoxygenation of HbO 2 within intact RBCs; k HbO2→Hb+O2 , rate constant for deoxygenation of HbO 2 in free solution; LC, linear combination; MCH, mean corpuscular hemoglobin; MCHC, mean corpuscular hemoglobin concentration; MCV, mean corpuscular volume; MMM-k HbO2 , macroscopic mathematically modeled rate constant for deoxygenation of intracellular HbO 2 ; P M,O2 , membrane permeability to O 2 ; Prov., provisional; Ø Major , major diameter (of BCD or nBCD); SC-k HbO2 , shape-corrected rate constant for deoxygenation of intracellular HbO 2 . Mathematical modeling and simulations Model formulation To examine the possibility that changes in RBC parameters—decreases in k HbO2→Hb+O2 , increases in MCHC (i.e., [Hb] i + [HbO 2 ] i = [Hb Total ] i ) 3 , decreases in diffusion constants (i.e., intra-/extracellular D for O 2 ; intracellular D for HbO 2 , Hb), increases in the thickness of the unconvected-fluid layer that surrounds the cell, or increases in intracellular diffusion distances—can explain our observed decreases in k HbO2 , we developed a distributed (i.e., space-dependent) and dynamic (i.e., time-dependent) reaction-diffusion mathematical model of an RBC—simplified as a perfectly symmetric sphere with diameter equal (Ø Sphere ) to the thickness of the RBC. Previous modelers have used a sphere in their simulations of an RBC ( Vandegriff & Olson, 1984 a ; Williams & Kutchai, 1986 ; Deonikar & Kavdia, 2013 ; Deonikar et al ., 2014 ), matching the diameter of the sphere to Ø Major . We avoid equating our Ø Sphere with the major diameter of the BCD because this would generate unrealistically large diffusion distances and thus low values of k HbO2 and P M,O2 . As described below (see below 4 ), our geometrical approach is also unique in that we chose Ø Sphere to match the minor diameter of a torus that has the same volume as the RBC. In our model (see Figure 2A ), the spherical RBC is enveloped by a layer of extracellular unconvected fluid (EUF) that is in turn enveloped by the bulk extracellular fluid (bECF). The model accounts for diffusion of O 2 , Hb, and HbO 2 within the intracellular fluid (ICF). Only O 2 can move—via diffusion—between the ICF and EUF (i.e., across the plasma membrane, M), within the EUF (which lacks Hb and HbO 2 ), and between the EUF and bECF (which also lacks Hb and HbO 2 ). Reactions among O 2 , Hb, and HbO 2 occur only in the ICF. Download figure Open in new tab Figure 2. Mathematical model of O 2 efflux from an RBC A, Major components of the mathematical model of spherical RBC cell, the diameter of which matches the computed thickness of an average RBC from a WT or KO mouse. See text for details on the model. B, The computational domain. For panel ‘A’, the abbreviations are: bECF, bulk extracellular fluid; EUF, extracellular unconvected fluid; ICF, intracellular fluid; M, plasma membrane; Ø Sphere , diameter of sphere (matches thickness of RBC); ℓ EUF , thickness of EUF; 3 values of are average distance from M to the following 3 molecules in ICF: (a) O 2 (black dot and lettering), (b) Hb (violet), or HbO 2 (red) in ICF; [1] R EUF , effective resistance offered by EUF to O 2 diffusion from M to bECF; [2] R M , effective resistance offered by M to O 2 diffusion; [3] R ICF , effective resistance offered by ICF to combined diffusion of O 2 and HbO 2 . 1/ R ICF = 1/ R ICFa + 1/ R ICFb , where R ICFa is effective resistance offered by ICF to O 2 diffusion over the distance − ICF , from the position of the average O 2 to M; R ICFb is effective resistance offered by ICF to HbO 2 diffusion over the distance − ICF , from the position of the average HbO 2 to M. For panel ‘B’, the abbreviations are: R ∞ , maximal radial distance from the center of the sphere ( R ∞ = R sphere + ℓ EUF ); r , radial distance from the center of the cell. Assuming spherical-radial symmetry, then for each solute S (i.e., O 2 , Hb, and HbO 2 ), the concentration ( C S ) changes in time ( t ) and space according to the reaction-diffusion equation where r (without a subscript 5 ) is the radial distance from the center of the RBC, and D S is the diffusion coefficient of solute S . Figure 2B shows the computational domain. Kinetics of the oxygen-hemoglobin reaction We model the binding of O 2 to Hb (and dissociation of O 2 from HbO 2 )—that is, the reaction term in Equation (1) —as a simple, one-step reaction in which we use a version of the variable-rate-coefficient (VRC) approach proposed by Moll ( Moll, 1968 ). Thus, we assume that each hemoglobin tetramer (Hb T ) is replaced by four independent hemoglobin monomers (4 × Hb M ), and that each hemoglobin monomer can combine with an O 2 via the one-step reaction Here, the dissociation rate constant k has the same meaning as k HbO2→Hb+O2 in paper #1 6 , namely, the rate constant for deoxygenation of HbO 2 in an RBC hemolysate; we will use the two notations interchangeably. The rate constant k ′ for the association reaction is equivalent to k Hb+O2→HbO2 . Rather than assuming that both k and k′ are constant—an assumption that would lead to a non-realistic hyperbolic hemoglobin-oxygen saturation curve—we follow a version of Moll’s VRC approach, in which we assume that only one of the two reaction rates is constant, while the other one becomes a function of the partial pressure of O 2 (P O2 ) so that the mathematical representation of the hemoglobin-oxygen saturation curve corresponds to the measured sigmoidal curve. Yap and Hellums (1987) showed that the VRC model is appropriate in the physiological range, and avoids the use of the more complex Adair model. In his VRC approach, Moll assumed that k′ is constant and k is a function of P O2 . However, Clark et al . (1985) pointed out that the function used by Moll to describe k as a function of P O2 approaches infinity as P O2 approaches zero. These last authors overcame this problem by assuming that k is constant and k′ is a function of P O2 , as described by where α is the solubility coefficient for O 2 in water, P 50 is P O2 when [Hb] = [HbO 2 ], and n is the Hill coefficient. With this choice and n > 1, k ′ approaches zero as P O2 approaches zero. Thus, in our model, we follow Clark’s variation of the VRC approach, which we implement with the slight modification of substituting [O 2 ] for P O2 : Calculation of RBC thickness based on the geometry of a torus For each mouse strain, we calculate the RBC thickness as follows ( Figure 3 ). We start from the mean major diameter of all RBC-related cell types 7 (e.g., Ø Major = 6.80 μm for WT cells; see paper #2 8 )—a measured value— and assume that the RBC is a torus, the outer diameter ( OD Torus ) of which equals the aforementioned Ø Major (i.e., 6.80 μm in our example). Next, we use the mean corpuscular volume (e.g., MCV = 47.9 pl for WT cells; see paper #2 9 )—also a measured value, which we assume to be equal to the volume of the torus ( V Torus )—to calculate the minor radius ( r Torus ) of the torus (equivalent to half the thickness of the RBC). Thus, r Torus is a root of the polynomial: Download figure Open in new tab Figure 3. Conversion of biconcave disk geometry into torus geometry A, Biconcave disk, tilted forward. B, Torus, tilted forward. C, Torus, top view. D, Torus, side view. E, Equivalent sphere. Ø Major , major diameter of biconcave disk; Ø Sphere , diameter of equivalent sphere; MCV, mean corpuscular volume; OD Torus , outer diameter of torus; r Torus , minor radius of torus; R Torus , major radius of torus; V Torus , volume of torus (made equal to MCV). The calculated value for the WT cells is r Torus = 1.01 μm. In Table 2 , we provide—for each of the 4 mouse strains—the computed values for r Torus , the major radius of the torus ( R Torus ), and the diameter of the equivalent sphere, where Ø Sphere = 2 r Torus (i.e., RBC thickness). The equivalent sphere diameters serve as inputs to our mathematical model. Calculation of Hb content For each mouse strain, we calculate the total concentration (both O 2 -bound and free, in mM) of Hb tetramers (T)—that is, ([Hb T,Total ]) 10 —as follows. Based on the hematology data 11 reported in paper #2 9 —specifically, the MCH (in pg) and MCV (in fl)—and using a value of 64316 g/mol for the molecular weight of mouse Hb, we calculate the MCHC (in mM). For WT mice, the concentration of total Hb tetramers was ∼4.7 mM. However, because our mathematical model deals with monomers, we calculate the concentration of total of Hb monomers ([Hb M,Total ]) 12 using 4 × MCHC. In the case of RBCs from WT mice, this figure is 18.73 mM . Table 2 lists this value, as well as the comparable values for the other three genotypes. These values inform our mathematical model. Computational model In order to solve the system of the three reaction-diffusion equations (1)—one for O 2 , one for Hb monomers, and one for HbO 2 monomers—we set the following boundary conditions. At the center of the RBC, we posit the no-flux Neumann boundary condition: At the interface between the ICF and the EUF (i.e., at the M), we establish continuity of Fick’s diffusive flux: Here, R Sphere is the radius of the sphere, C S ( t , R Sphere –) is the time-dependent concentration of solute S in the aqueous phase adjacent to the intracellular (−) side of the M; C S ( t , R Sphere +), the comparable value on the extracellular (+) side; and P M ,S is the true permeability 13 of the membrane to solute S . Because O 2 is the only solute that can diffuse across the M, P M ,S ≠ 0 only for S = O 2 . 14 In the bECF, we assign Dirichlet boundary conditions and assume that the concentration of O 2 is constant and equal to zero. This assumption is the mathematical equivalent of the annihilation of O 2 by sodium dithionite in our SF reaction cell (see paper #1 15 ). Finally, we set the initial conditions by assuming that, at the beginning of the experiment (i.e., our simulation), no O 2 is present in the EUF, and the free [O 2 ] everywhere inside the RBC is that dictated by Henry’s law for a gas mixture containing 21% O 2 . The initial concentrations of O 2 , Hb, and HbO 2 in the ICF are those reported in Table 1 for RBCs from WT mice, and in Table 2 , for RBCs from mice of each of four genotypes. View this table: View inline View popup Table 1. Parameter values used in mathematical simulations of control RBCs from WT mice * View this table: View inline View popup Table 2. Genotype-specific parameter values used in mathematical simulations of control RBCs in Figure 5 We solve the system of the three reaction-diffusion equations (i.e., Eqn (1) for the solutes O 2 , Hb, and HbO 2 ) using the method of lines and the approach described by Somersalo et al for the numerical implementation of the diffusion component ( Somersalo et al ., 2012 ). We solve the resulting system of time-dependent ordinary differential equations in Matlab R2015a using the stiff solver ode15s with AbsTol=RelTol=1e–12. The Matlab code is based on the original implementation of Somersalo et al. Parameter values Table 1 lists the parameter values used in the simulation for RBCs from WT mice. Table 2 lists the parameters that we measured in the present study, specific for each of the 4 genotypes; we use these values for computing values for Figure 5 . All parameter values correspond to a temperature of 10°C, the temperature at which we perform the stopped-flow experiments (see paper #1 16 ). For parameter values not available at the temperature of 10°C, we derive them using Arrhenius equation and values at 25°C and 37°C from references indicated in Table 1 . We use this approach for calculating the diffusion coefficient of HbO 2 and HbO 2 in the bulk ICF. For the diffusion coefficient of O 2 in the bulk ICF, the commonly used value (adjusted for 10°C using the Arrhenius equation, as described above) is 5.09 × 10 −6 cm 2 s −1 ( Clark et al ., 1985 ). More recently, Richardson et al ( Richardson et al ., 2020 ) obtained the much lower value of 0.45907 × 10 −6 cm 2 s −1 . However, we are concerned that this latter value is an underestimate (see Discussion ) 17 . We chose to use the arithmetic mean of the commonly used value of 5.09 × 10 −6 cm 2 s −1 and the more recent value of 0.45907 × 10 −6 cm 2 s −1 , namely 2.7745 × 10 −6 cm 2 s −1 (i.e., somewhat more than half the value reported by Clark et al ., 1985 ; see Table 1 ). Because the molecular weight of O 2 is much less than the molecular weight of Hb, and because the diffusion coefficient mainly depends on molecular weight, we assume that the diffusion coefficients of monomeric HbO 2 and Hb are the same ( Popel, 1989 ). We assume that the permeability of the RBC plasma membrane to O 2 is 0.15 cm s −1 , the value estimated by Endeward et al ( Endeward et al ., 2006 ) for the permeability of the RBC plasma membrane to CO 2 . Simulation of time course of HbO2 deoxygenation, and estimation of k HbO2 Using the reaction-diffusion model, we simulate the time course of the decline of HbO 2 saturation (i.e., HbSat = [HbO 2 ]/([Hb]+[HbO 2 ]) vs. time, employing hematological and morphological data that we gathered for RBCs from WT mice (see Table 1 ). Because neither the actual nor the simulated time courses of HbO 2 desaturation are precisely exponential, we compute k HbO2 from simulations in a manner analogous to that for our physiological data (see paper #1 18 ): we determine the time for the simulated HbSat to fall to 1/e ≅ 37% of its initial value (i.e., τ Quasi ≅ t 37 ), and then compute k HbO2 as 1/ t 37 . Data availability The data supporting the findings of this study are available within the paper. Any further relevant data are available from the corresponding author upon reasonable request. Results The overarching hypothesis of papers #1 and #2 is that AQP1, RhAG, and at least one other RBC membrane protein contribute in a major way to the O 2 permeability of murine RBC membranes. From a technical perspective, the purpose of this third paper is to develop a reaction-diffusion model—our shorthand for this macroscopic mathematical model is MMM—that is applicable to RBCs in the stopped-flow studies of paper #1. Because our MMM incorporates Fick’s first law of diffusion to describe transmembrane O 2 efflux (see Methods ) 19 we can relate the input parameter P M,O2 —incorporated into Eqns (6) and (7)—to the output parameter MMM- k HbO2 . As noted in the Introduction, in our simulations, we do not adjust myriad parameter values to match physiological data, but rather use the model to predict—from first principles—the time course of HbO 2 desaturation, which yields MMM- k HbO2 . The input parameters for the simulations are physical constants and values obtained from the literature (see Table 1 ) as well as values obtained in automated-hematology studies and imaging flow cytometry in papers #1 & #2 (see Table 2 ). The latter—which include MCV, MCH, and Ø Major —allow us to determine values for RBC thickness and total Hb concentration and thereby perform simulations that mimic as closely as possible the conditions of our physiological experiments. From a philosophical perspective, our purpose in the present paper is to use simulations based on the MMM to make a series of estimates of P M,O2 that generate a series of MMM- k HbO2 values that allow us to hone in on our best estimate of P M,O2 in a particular set of stopped-flow experiments in paper #1. If our hypothesis is correct, RBCs from mice genetically deficient in AQP1, RhAG, or both ought to have lower P M,O2 values when compared to RBCs from WT mice. In order to test our hypothesis, we must assess P M,O2 from experimental data. In paper #1 20 , we use SF absorbance spectroscopy to measure k HbO2 of RBCs from mice genetically deficient in AQP1, RhAG, or both. We find that, when compared to WTs, k HbO2 decreases by 9% in AQP1-KOs, 17% in RhAG-KOs, and 30% in dKOs. Moreover, k HbO2 decreases by 53% in dKO RBCs treated with DIDS, and by 78% in dKO RBCs treated with PCMBS. One might immediately ask why the sum 9% + 17% is not precisely 30%? The answer, aside from experimental and rounding errors, is that P M,O2 —the presumed target of our genetic and pharmacological manipulations—is only one element that contributes to MMM- k HbO2 (see Figure 2A ), resulting (as we shall see later) in a curvilinear dependence of P M,O2 on log( k HbO2 ). In the following sections we show that, by informing the model with reasonable parameter values and internally consistent (i.e., gathered in our laboratory) hematology and flow cytometry data, our MMM model provides reasonable estimates of the time course of HbO 2 desaturation and corresponding MMM- k HbO2 . Moreover, the model confirms that decreases in the value of k HbO2 correspond to decreases in P M,O2 , thereby supporting the hypothesis that membrane proteins like AQP1 and the mouse Rh complex (Rh Cx = RhAG + RhD) contribute to P M,O2 . Simulations of O2 efflux from a control RBC of WT mouse Our first goal was to establish an in-silico experiment that would mimic O 2 efflux from a control (Ctrl; i.e., not pretreated with inhibitors) RBC of a WT mouse (WT/Ctrl). For this simulation, our input parameters are those reported in Table 1 , which includes physical constants and values obtained from the literature as well as values for RBC thickness and total Hb concentration obtained from our hematological and morphological data (i.e., MCV, MCH, Ø Major ) of WT mice (see Table 2 ), first column). Figure 4 shows results of this simulation. At the top of the figure, panels A , B , and C show the time courses of the intracellular (i) concentrations of free O 2 , HbO 2 , and the sum of free O 2 + HbO 2 —integrated over the entire volume of the cytoplasm—as O 2 diffuses out of the cell in a simulated O 2 -offloading experiment, as in paper #1 21 . Download figure Open in new tab Figure 4. Simulated time courses of O 2 /Hb-related parameters and transmembrane O 2 flux for cells mimicking WT/Ctrl (control/no inhibitors) For this simulation, the input parameters are those reported in Table 1 . See text for details on the model and simulations. All concentrations are in mM. A, Concentration of free O 2 , integrated over the entire volume of cytoplasm. B, Concentration of oxyhemoglobin (HbO 2 ) monomers, integrated over the entire volume of cytoplasm. C, Concentration of the sum of free O 2 and HbO 2 monomers, integrated over the entire volume of cytoplasm. D, Hemoglobin saturation (HbSat), integrated over the entire volume of cytoplasm. E, Concentration of free hemoglobin (Hb) monomers, integrated over the entire volume of cytoplasm. F, Transmembrane O 2 flux ( J M,O2 ). Note that the units are nmol cm −2 s −1 (rather than μmol cm −2 s −1 , as in the text of the Discussion). The inset shows a magnification of the first 1 ms of the simulated J M,O2 Download figure Open in new tab Figure 5. MMM- k HbO2 values predicted from simulations of RBCs from WT and KO mice Informed by hematological, morphological, and kinetic data, which provide genotype-specific information on MCV, MCH, Ø Major , and k HbO2→Hb+O2 (see Table 2 ) we perform four simulations, one per genotype, to test whether the collection of parameter changes can explain the physiologically observed decreases in shape-corrected (SC) k HbO2 values reported in figure 5 b of paper #1 : about –9% (AQP1-KO), –17% (RhAG-KO), and –30% (double KO). The values in red above each knockout bar indicate the % change predicted by MMM- k HbO2 vs. the observed change. Thus, the hematological/morphological changes cannot explain our observed k HbO2 data. A, k HbO2 → Hb+O2 assumed to be 11.60 s −1 for all genotypes. B, k HbO2 → Hb+O2 assumed to be specific for each genotype. Except as noted otherwise, in these simulations we used the values summarized in Table 1 , including P M,O2 = 0.15 cm s −1 . We assume that all cells are intact (analogous to a hemolysis correction), and that all cells are biconcave disks (analogous to a shape correction). Thus, the two green bars have MMM- k HbO2 values of ∼3.99 s −1 . MMM- k HbO2 , macroscopic mathematically modeled rate constant for deoxygenation of intracellular HbO 2 ; Aqp1 , the gene encoding murine aquaporin 1; Rhag , the gene encoding murine RhAG: +, gene present; –, gene deleted. MCV, mean corpuscular volume; MCH, mean corpuscular hemoglobin; Ø Major , major diameter of biconcave disk; k HbO2→Hb+O2 , rate constant for deoxygenation of HbO 2 in free solution; P M,O2 , membrane permeability to O 2 . [O2]i As expected, because O 2 is leaving the cell, [O 2 ] i decreases ( Figure 4A ). [HbO2]i and [O2]i + [HbO2]i At the same time as [O 2 ] i decreases, [HbO 2 ] i decreases ( Figure 4B ) in parallel with intracellular [O 2 ] i + [HbO 2 ] i ( Figure 4C ). These three time courses in Figure 4A-C reflect (1) the efflux of free O 2 ; (2) the diffusion of free O 2 toward the membrane; (3) the net reaction HbO 2 → Hb + O 2 , which consumes HbO 2 to produce free Hb and free O 2 and continues until HbO 2 is fully depleted; (4) the diffusion of HbO 2 toward the membrane, and (5) the diffusion of free Hb away from the membrane. The time courses of [HbO 2 ] i and of [O 2 ] i + [HbO 2 ] i are nearly identical because [O 2 ] i << [HbO 2 ] i . At the bottom of the figure, panels D , E , and F show the time courses of hemoglobin saturation, [Hb] i , and the O 2 efflux across the plasma membrane ( J M,O2 ). HbSat As expected, HbSat falls ( Figure 4D ) with a time course that is similar to that of [HbO 2 ] i (see Figure 4B ). It is from this time course of HbSat that we calculate the MMM- k HbO2 of WT/Ctrl RBCs (see Methods 22 ). The model predicts a MMM- k HbO2 of ∼3.99 s −1 , which will become our “Provisional MMM- k HbO2 ” in our accommodation for nBCDs (see below 23 ). [Hb]i As anticipated, [Hb] i increases ( Figure 4 E ) with a time course that is the inverse of that of [HbO 2 ] i . J M,O 2 Finally, J M,O2 initially zero, spikes rapidly to a maximal value, decays rapidly to a value somewhat above zero, and finally drifts toward zero ( Figure 4 F ). The J M,O2 upward spike reflects the large initial transmembrane O 2 gradient. The rapid decay reflects O 2 depletion near the inner surface of the membrane. The O 2 efflux during the slower decay is supported by O 2 that diffuses to the inner surface of the membrane from the depths of the RBC and, to a much lesser extent, the parallel diffusion and dissociation of HbO 2 . Predicted k HbO 2 values from simulations of control RBCs from WT and KO mice, as informed by genotype-specific hematological, morphological, and kinetic data Besides P M,O2 , the MMM- k HbO2 that describes O 2 -offloading from RBCs depends on several genotype-specific parameters: Ø Sphere (which in turn depends on MCV, Ø Major , and RBC shape), [Hb Total ] i (computed from MCHC), and k HbO2→Hb+O2 (i.e., the same as k in Eqn (2) ; determined on SF experiments on RBC hemolysates). To investigate whether the decreases in k HbO2 observed in RBCs from KOs (see paper #1 24 ) could be explained by changes in parameters other than P M,O2 , we as a first step gathered the requisite genotype-specific data in paper #1 and paper #2. We provide these parameter values in Table 2 . Note the trend (from left to right) for MCV to increase in the KOs, but for MCHC to decrease. In addition, using a battery of methodologies to assess RBCs shape (blood smears, still/video images of living RBCs, and imaging flow cytometry), we find that ∼98% of control RBCs, both from WTs and dKOs, are biconcave disks (see paper #2 25 ). To assess the effects of the genotype-specific differences summarized in Table 2 , we perform simulations in which we mimic control RBCs (i.e., not pretreated with inhibitors) from WT, AQP1-KO, RhAG-KO and dKO mice. Note that the simulation for a control WT RBC is the same one that yielded the MMM- k HbO2 of 3.99 s −1 in the previous section. We simulate O 2 efflux from RBCs of AQP1-KO/Ctrl, RhAG-KO/Ctrl, and dKO/Ctrl mice using the same parameter values (including P M,O2 ) that we used for WT/Ctrl RBC (see Table 1 ) with the exception of the genotype-specific values in Table 2 for Ø Sphere (mimicking RBC thickness) and [Hb M,Total ] i (a surrogate for MCHC). Figure 5 shows a pair of MMM- k HbO2 values for each of the four genotypes. In these simulations we assume that all RBCs are intact, as they are for the hemolysis-corrected (HC) k HbO2 values in paper #1, and that all cells are BCDs, as they are for the shape-corrected (SC) k HbO2 values for SF experiments in paper #1 26 . In Figure 5 A , we compute MMM- k HbO2 values using our standard k HbO2→Hb+O2 value of 11.6 s −1 (see Table 1 ), which is an average for all four genotypes. We use this standard value for all simulations here in paper #3 (including Figure 5 A ) except for those in Figure 5 B , where we use four separate, genotype-specific k HbO2→Hb+O2 values (see Table 2 ). The solid green bars in both Figure 5 A and B correspond to an MMM- k HbO2 of ∼3.99 s −1 for a WT/Ctrl RBC (i.e., no drugs). Both panels have the same MMM- k HbO2 value because, by serendipity, the standard k HbO2→Hb+O2 is virtually the same as the genotype-specific value for WT/Ctrl RBCs. The pink bar in Figure 5 A simulates an AQP1-KO/Ctrl RBC; MMM- k HbO2 ≅ 4.04 s −1 , which is a predicted 1.1% increase vs. 3.99 s −1 , rather than the 9% decrease observed in paper #1. In Figure 5 B , MMM- k HbO2 is ∼4.10 s −1 , which represents an even larger increase, 2.6%. The blue bar in Figure 5 A simulates a RhAG-KO/Ctrl RBC; its MMM- k HbO2 of 3.82 s −1 represents a predicted decrease of 4.4%, rather than the 17% decrease observed in paper #1. The corresponding MMM- k HbO2 in Figure 5 B , is ∼3.85 s −1 , which predicts a somewhat smaller decrease of 3.5%. Finally, the purple bar in Figure 5 A simulates a dKO/Ctrl RBC; its MMM- k HbO2 of 3.85 s −1 is a decrease of only 3.7% vs. WT/Ctrl, rather than the ∼30% decrease in paper #1. In Figure 5 B , MMM- k HbO2 is ∼3.74 s −1 , which predicts a decrease of 6.3%, which is still far smaller than what we observed in paper #1. Thus, we conclude that the genotype-specific changes reported in paper #1 and paper #2— and summarized here in Table 2 —cannot explain the observed KO-induced decreases in SC- k HbO2 in paper #1. We intuit that the effects of anti-parallel increases in MCV and decreases in MCHC tend to cancel, and thereby tend to minimize the impact on MMM- k HbO2 values. We examine this possibility in the sensitivity analyses, below 27 . We conclude that KO-induced variations in RBC dimensions, [Hb Total ] i , and k HbO2→ Hb+O2 cannot account for the KO-induced decreases in SC- k HbO2 observed in paper #1. Predicted dependence of O2 offloading ( k HbO2) on the O2 permeability of RBC membrane ( P M,O 2 ) In the previous sections, we show that our MMM—with a fixed P M,O2 and genotype-specific values for other parameters—can mimic the time course of O 2 offloading (i.e., MMM- k HbO2 ≅ SC- k HbO2 ) for a control RBC from a WT mouse, but not for control RBCs from any of our three KO groups. Clearly, we must invoke some other parameter to account for the control and KO data in paper #1 and, by extension, the paper-#1 data on inhibitors and KO+inhibitors. We hypothesize that this other parameter is P M,O2 . (In the case of drug-treated RBCs, we will apply a shape correction, described below, to accommodate the appearance of nBCDs, i.e., poikilocytes). Having established an in-silico approach for WT/Ctrl experiments, we can now employ the model to answer the two key questions that are at the core of our investigation: What is the dependence of MMM- k HbO2 on P M,O2 ? This will help us understand the extent to which observed decreases in SC- k HbO2 in paper #1 correspond to decreases in P M,O2 ? Can we use physiological k HbO2 measurements to make predictions about P M,O2 ? To answer these questions, we employ the model that mimics a control WT RBC (i.e., input parameter values listed in Table 1 ) and perform a series of more than 70 simulations in which we systematically vary P M,O2 while holding all other parameters constant. Throughout, we assume that the RBCs are biconcave disks so that we need not accommodate for shape changes. Thus, by definition, the MMM- k HbO2 is analogous to the SC- k HbO2 . The continuous curve in Figure 6 is a semi-logarithmic plot showing the dependence of MMM- k HbO2 (y-axis) on P M,O2 (x-axis, scaled logarithmically). The shape of this curve depends on the parameters in Table 1 , and thus could vary with each strain of mouse, and each species (see Discussion 28 ). Download figure Open in new tab Figure 6. Predicted dependence of MMM- k HbO2 on P M,O2 Using our macroscopic mathematical model (MMM), we systematically varied membrane O 2 permeability ( P M,O2 ; logarithmic x-axis) to compute the corresponding rate constant for deoxygenation of HbO 2 ( k HbO2 ; linear y-axis). Except for P M,O2 , we used the values summarized in Table 1 for these 70 + simulations. The resulting sigmoidal curve represents all possible combinations of P M,O2 and k HbO2 for P M,O2 > 0.001 cm s −1 . The point labeled “H 2 O membrane” is the result of a simulation in which we assumed that the diffusion constant of O 2 in the plasma membrane is the same as in water; it has the coordinates ( P M,O2 = 22.63 cm s− 1 , k HbO2 = 5.9953 s− 1 ). The point labeled WT/Ctrl (i.e., RBCs from wild-type mice in the absence of drugs) has the coordinates ( P M,O2 = 0.1546 cm s −1 , k HbO2 = 4.0358 s −1 ), and represents the shape-corrected value obtained by the linear-combination procedure (steps #11 through #22 in Figure 1 ). The other labeled points also represent shape-corrected values, as summarized in Table 3 . Note that the dKO/Ctrl and the WT/DIDS points nearly overlie each other. View this table: View inline View popup Download powerpoint Table 3. Parameter values predicted by macroscopic mathematical model The point at the extreme upper-right—labeled “H 2 O membrane”—represents the hypothetical situation in which we assume that the plasma membrane—as is implicit in Krogh’s calculations—is a thin film of H 2 O of equivalent thickness. That is, for the plasma membrane, we assume that the diffusion constant D O2 has the same value as in free water, namely 1.3313 × 10 −5 cm 2 s −1 , which translates to a P M,O2 of ∼22.63 cm s −1 . In our simulations, this value generates a MMM- k HbO2 of ∼6.00 s −1 . For the given geometric, hematologic, and kinetic parameters, this point represents the maximum-possible values for P M,O2 and thus the maximum-possible rate of O 2 offloading. What is the position of a WT/Ctrl RBC on this curve? Note that (1) WT/Ctrl RBCs have an nBCD prevalence of 1.41% (see paper #2 29 ) and (2) nBCDs (because of their larger spherical diameter; see paper #2 29 ) are expected to have a lower O 2 -offloading rate that BCDs. The MMM- k HbO2 of 3.99 s −1 —which we have already generated using P M,O2 = 0.15 cm s −1 —in fact pertains not to a pure population of BCDs, but rather to a mixture of 98.59% BCDs and 1.41% nBCDs. Our accommodation for nBCDs presented below 30 ) leads to the conclusions that (1) the actual k HbO2 for the BCD component is ∼4.04 s −1 vs. 3.99 s −1 for the mixture, and (2) the actual P M,O2 for both BCDs and nBCDs is 0.1546 cm s −1 , which is ∼3.1% higher than our provisional value of 0.15 cm s −1 . Thus, the point labeled “WT/Ctrl” in Figure 6 has the shape-corrected coordinates ( P M,O2 = 0.1546 cm s −1 , k HbO2 = 4.0358 s −1 ) and is our best estimate of a pure population of BCDs. Note that the P M,O2 of 0.1546 cm s −1 for WT/Ctrl BCDs is only ∼0.7% as large (or ∼1/150) as the hypothetical P M,O2 of Krogh’s pure-water membrane. In other words, the RBC membrane offers 150-fold more resistance to O 2 diffusion than would a film of H 2 O. Besides WT/Ctrl, each of the seven other labeled points in Figure 6 corresponds to one of the other experimental conditions. For each point, we determine the y-axis value (MMM- k HbO2 ) by performing the shape correction described below 30 (steps #13 – #22 in Figure 1 ), based on experimentally determined parameter values. We determine the x-axis value ( P M,O2 ) by interpolation (step #19 in Figure 1 ), thereby placing the point on the sigmoidal curve. Note that the dKO/Ctrl and the WT/DIDS points nearly overlap. Table 3 summarizes the key numerical values corresponding to the eight experimental points in Figure 6 . Our standard condition is WT/Ctrl (upper-left corner of table), for which the shape-corrected values are SC- P M,O2 = 4.0358 s −1 and SC- P M,O2 = 0.1546 cm s −1 . Compared to these standard SC values, the AQP1-KO/Ctrl has an SC- P M,O2 that is 21.56% lower, and an SC- k HbO2 that is 8.85% lower—and so on for the other conditions. Note that the sum of the computed % decreases in P M,O2 for AQP1-KO/Ctrl (∼22%) and RhAG-KO/Ctrl (∼36%) is nearly equal to the computed P M,O2 % decrease for the dKO/Ctrl (∼55%). The largest % decrease in P M,O2 occurs with dKO/pCMBS (∼91%). The effects of the membrane-protein knockouts and inhibitors on P M,O2 —along with the proteomics data in paper #2 31 —supports the hypothesis that the vast majority of O 2 moves through channels in RBC plasma membrane: AQP1, Rh Cx , and at least one additional channel that is blocked by pCMBS. In other words, under our experimental conditions, “Overton’s rule” applies, at most, to ∼9% of the O 2 traffic across the murine RBC membrane. Predicted effects of dKO/pCMBS on the kinetics of parameters related to intracellular O2 and Hb In Figure 4 A-F , we examined the simulated time courses of six parameters for control (drug-free) WT RBCs: [O 2 ] i , [HbO 2 ] i , [O 2 ] i + [HbO 2 ] i , HbSat, [Hb] i , and J M,O2 . In Figure 7 A-F , we extend this WT/Ctrl analysis (green curves) by simulating the slowest O 2 offloading protocol observed in paper #1 32 : a dKO RBC pre-treated with pCMBS (purple curves). Download figure Open in new tab Figure 7. Simulated time courses of O 2 /Hb-related parameters and transmembrane O 2 flux for cells mimicking WT/Ctrl (control/no inhibitors) vs. dKO/pCMBS We extend the analysis of the WT/Ctrl condition (untreated RBCs from wild-type mice; green curves) shown in Figure 4 by adding the simulations of the slowest O 2 -offloading protocol observed in paper #1 (i.e., figure 5 b , light purple bar) : the dKO/pCMBS condition (i.e., RBCs pre-treated with pCMBS; purple curves). For the WT/Ctrl simulation, the input parameters are those reported in Table 1 . For the dKO/pCMBS simulation, the input parameters are the same, with the exception of P M,O2 = 0.01365 cm s −1 . See text for details on the model and simulations. All concentrations are in mM. A, Concentration of free O 2 , integrated over the entire volume of cytoplasm. The inset shows a magnification of the first 1 ms of the simulated [O 2 ] i . B, Concentration of oxyhemoglobin (HbO 2 ) monomers, integrated over the entire volume of cytoplasm. C, Concentration of the sum of free O 2 and HbO 2 monomer, integrated over the entire volume of cytoplasm. D, Hemoglobin saturation (HbSat) integrated over the entire volume of cytoplasm. E, Concentration of free hemoglobin (Hb), integrated over the entire volume of cytoplasm. F, Transmembrane O 2 flux ( J M,O2 ). Note that the units are nmol cm −2 s −1 (rather than μmol cm −2 s −1 , as in the text of the Discussion). The inset shows a magnification of the first 1 ms of the simulated J M,O2 On the time scale of the main part of Figure 7 A , the initial rates of decline for [O 2 ] i appear to be nearly the same for dKO/pCMBS and WT/Ctrl. The inset—a blow-up of the initial 1 ms of the simulation—shows that, in fact, the time course for dKO/pCMBS is much slower than for WT/Ctrl, reflecting a P M,O2 that is reduced by ∼91%. For [HbO 2 ] in Figure 7 B , the time course of the decline for dKO/pCMBS is markedly slower than for WT/Ctrl. The time courses of [O 2 ] i + [HbO 2 ] i in Figure 7 C are nearly the same as for [HbO 2 ] alone in Figure 7 B because, of the total O 2 in the RBC, ∼98% is bound to Hb as HbO 2 . The time courses of HbSat in Figure 7 D , when appropriately scaled, are nearly the same as for [HbO 2 ] i in Figure 7 B because [HbO 2 ] i is the numerator in the calculation of HbSat. Because the sum [Hb] i + [HbO 2 ] i is constant, the time courses of [Hb] i in Figure 7 E is the inverse of [HbO 2 ] i in Figure 7 B . Finally, in Figure 7 F , the transmembrane flux J M,O2 is initially ∼10-fold higher for WT/Ctrl than for dKO/pCMBS. For dKO/pCMBS, the J M,O2 time course has a similar shape, but is scaled down by 91%, reflecting the reduced P M,O2 . Mathematical simulations exploring the predicted sensitivity of k HbO2 to eight key kinetic and geometric parameters To understand better the dependence of O 2 offloading on a range of parameters in Table 1 , we performed a sensitivity analysis, studying eight key parameters. See Figure 2 for an overview of the model. Our approach was to start with the parameter values in Table 1 ; that is, with the provisional value P M,O2 = 0.15 cm s −1 , our simulation yields the provisional MMM- k HbO2 = 3.99 s −1 . Next, as described in the following five sub-sections, we computed k HbO2 as we systematically varied, one at a time, each of the following: k HbO2→Hb+O2 in Figure 8 , [Hb Total ] i in Figure 9 , each of the four diffusion constants in Figure 10 A–D , ℓ EUF in Figure 11 , and Ø Sphere in Figure 12 . Download figure Open in new tab Figure 8. Mathematical simulations exploring the predicted sensitivity of k HbO2 to changes in the rate constant of the reaction HbO 2 → Hb + O 2 . Using our macroscopic mathematical model (MMM), we systematically varied the rate constant for deoxygenation of HbO 2 in free solutions: HbO 2 → Hb + O 2 ( k HbO2→Hb+O2 ) to compute the corresponding the rate constant for deoxygenation of HbO 2 in intact RBCs ( k HbO2 ). We computed the MMM- k HbO2 corresponding to the red dot—which corresponds to 100% of our standard k HbO2 → Hb+O2 —by using the parameter values that come from Table 1 . This is our standard condition for RBCs from wild-type mice in the absence of drugs (WT/Ctrl), and also is represented by the dark green circle and associated horizontal dashed green line. We performed nine additional simulations (blue dots) by decreasing k HbO2 → Hb+O2 by the indicated percentages, keeping all other parameter values constant. The seven other colored circles and associated horizontal dashed lines indicate the MMM- k HbO2 values, relative to WT/Ctrl, for each of the other 7 experimental conditions. As summarized in the legend on the left, for each condition, we multiplied the standard MMM- k HbO2 (i.e., 3.99 s −1 ) by the shape-corrected decrease in the experimentally observed k HbO2 (e.g., ∼9% for AQP1-KO/Ctrl cells) to obtain the y-axis coordinate for that condition (e.g., 3.64 s −1 for AQP1-KO/Ctrl). These % decreases are also listed in the last row of Table 3 . The left downward arrow (tail on dKO/pCMBS) indicates the % decrease necessary to account for the effect of this condition on MMM- k HbO2 . The right downward arrow is the same for AQP1-KO/Ctrl. Download figure Open in new tab Figure 9. Mathematical simulations exploring the predicted sensitivity of k HbO2 to changes in the total intracellular Hb concentration Using our macroscopic mathematical model (MMM), we systematically varied the total intracellular Hb concentration, [Hb Total ] i , to compute the corresponding rate constant for deoxygenation of HbO 2 in intact RBCs ( k HbO2 ). We computed the MMM- k HbO2 corresponding to the red dot—which corresponds to 100% of our standard [Hb Total ] i —by using the parameter values that come from Table 1 . This is our standard condition for RBCs from wild-type mice in the absence of drugs (WT/Ctrl), and also is represented by the dark green circle and associated horizontal dashed green line. We performed ten additional simulations (blue dots) by decreasing [Hb Total ] i by the indicated percentages, keeping all other parameter values constant. The seven other colored circles and associated horizontal dashed lines indicate the MMM- k HbO2 values, relative to WT/Ctrl, for each of the other 7 experimental conditions. As summarized in the legend on the left, for each condition, we multiplied the standard MMM- k HbO2 (i.e., 3.99 s −1 ) by the shape-corrected decrease in the experimentally observed k HbO2 (e.g., ∼9% for AQP1-KO/Ctrl cells) to obtain the y-axis coordinate for that condition (e.g., 3.64 s −1 for AQP1-KO/Ctrl). These % decreases are also listed in the last row of Table 3 . The downward arrow (tail on AQP1-KO/Ctrl) indicates the % decrease necessary to account for the effect of this condition on MMM- k HbO2 . Download figure Open in new tab Figure 10. Mathematical simulations exploring the predicted sensitivity of k HbO2 to changes in diffusion constants of solutes Using our macroscopic mathematical model (MMM), we systematically varied the four relevant diffusion constants ( D ), one at a time, to compute the corresponding rate constant for deoxygenation of HbO 2 in intact RBCs ( k HbO2 ). These four diffusion constants are: D for extracellular O 2 —( D O2 ) o in panel A ; and the three intracellular diffusion constants for the components of the equilibrium HbO 2 ⇒ Hb + O 2 , namely, ( D O2 ) i in panel B , ( D HbO2 ) i in panel C, and ( D Hb ) i in panel D . We computed the MMM- k HbO2 corresponding to the red dot—which corresponds to 100% of the value of the varied parameter—by using the parameter values that come from Table 1 . This is our standard condition for RBCs from wild-type mice in the absence of drugs (WT/Ctrl), and also is represented by the dark green circle and associated horizontal dashed green line. We performed additional simulations (blue dots) by decreasing the varied parameter by the indicated percentages, keeping all other parameter values constant. The seven other colored circles and associated horizontal dashed lines indicate the MMM- k HbO2 values, relative to WT/Ctrl, for each of the other 7 experimental conditions. As summarized in the legend on the left, for each condition, we multiplied the standard MMM- k HbO2 (i.e., 3.99 s −1 ) by the shape-corrected decrease in the experimentally observed k HbO2 (e.g., ∼9% for AQP1-KO/Ctrl cells) to obtain the y-axis coordinate for that condition (e.g., 3.64 s −1 for AQP1-KO/Ctrl). These % decreases are also listed in the last row of Table 3 . The downward arrows (tails on AQP1-KO/Ctrl) indicates the % decrease necessary to account for the effect of the respective condition on MMM- k HbO2 . Download figure Open in new tab Figure 11. Mathematical simulations exploring the predicted sensitivity of k HbO2 to changes in thickness of the EUF Using our macroscopic mathematical model (MMM), we systematically varied the thickness of the EUF, ℓ EUF , to compute the corresponding rate constant for deoxygenation of HbO 2 in intact RBCs ( k HbO2 ). We computed the MMM- k HbO2 corresponding to the red dot—which corresponds to 100% of our standard ℓ EUF —by using the parameter values that come from Table 1 . This is our standard condition for RBCs from wild-type mice in the absence of drugs (WT/Ctrl), and also is represented by the dark green circle and associated horizontal dashed green line. We performed eleven additional simulations (blue dots) by decreasing ℓ EUF by the indicated percentages, keeping all other parameter values constant. The seven other colored circles and associated horizontal dashed lines indicate the MMM- k HbO2 values, relative to WT/Ctrl, for each of the other 7 experimental conditions. As summarized in the legend on the left, for each condition, we multiplied the standard MMM- k HbO2 (i.e., 3.99 s −1 ) by the shape-corrected decrease in the experimentally observed k HbO2 (e.g., ∼9% for AQP1-KO/Ctrl cells) to obtain the y-axis coordinate for that condition (e.g., 3.64 s −1 for AQP1-KO/Ctrl). These % decreases are also listed in the last row of Table 3 . The downward arrow (tail on AQP1-KO/Ctrl) indicates the % decrease necessary to account for the effect of this condition on MMM- k HbO2 . Download figure Open in new tab Figure 12. Mathematical simulations exploring the predicted sensitivity of k HbO2 to changes in sphere diameter Using our macroscopic mathematical model (MMM), we systematically varied the diameter of the sphere, Ø Sphere , to compute the corresponding rate constant for deoxygenation of HbO 2 in intact RBCs ( k HbO2 ). We computed the MMM- k HbO2 corresponding to the red dot—which corresponds to 100% of our standard Ø Sphere —by using the parameter values that come from Table 1 . This is our standard condition for RBCs from wild-type mice in the absence of drugs (WT/Ctrl), and also is represented by the dark green circle and associated horizontal dashed green line. We performed additional simulations (blue dots) by decreasing Ø Sphere by the indicated percentages, keeping all other parameter values constant. The seven other colored circles and associated horizontal dashed lines indicate the MMM- k HbO2 values, relative to WT/Ctrl, for each of the other 7 experimental conditions. As summarized in the legend on the left, for each condition, we multiplied the standard MMM- k HbO2 (i.e., 3.99 s −1 ) by the shape-corrected decrease in the experimentally observed k HbO2 (e.g., ∼9% for AQP1-KO/Ctrl cells) to obtain the y-axis coordinate for that condition (e.g., 3.64 s −1 for AQP1-KO/Ctrl). These % decreases are also listed in the last row of Table 3 . Note that main panel shows the y-axis from 0 to 5 s −1 , whereas the inset expands the y-axis to extend from 0 to 12 s −1 . The downward arrows (tails on AQP1-KO/Ctrl, etc.) indicate the % decreases necessary to account for the effect of the respective condition on MMM- k HbO2 . In each of these figures or panels, we plot MMM- k HbO2 on the y-axis and the varied parameter (as a % of the standard parameter value) on the x-axis. Thus, when the standard value of the parameter = 100%, MMM- k HbO2 = 3.99 s −1 . As noted in our introduction of Figure 5 , here we assume that all RBCs are intact BCDs, as for SC- k HbO2 values in paper #1. In each figure/panel, we present eight dashed horizontal lines, one for each of the experimental conditions, each coded with a colored circle with same meaning as in Figure 6 . For the y-axis value of a horizontal dashed line, we scale the hemolysis-corrected MMM- k HbO2 of 3.99 s −1 for the WT/Ctrl by the experimental condition’s % decrease in SC- k HbO2 vs. WT/Ctrl ( Table 3 , bottom line). Thus, MMM- k HbO2 = 3.99 s −1 for WT/Ctrl (0% decrease) and falls to ∼0.9 s −1 for dKO/pCMBS (78% decrease in MMM- k HbO2 ). Figure 8 shows the dependence of MMM- k HbO2 on k HbO2→Hb+O2 . As k HbO2→Hb+O2 rises, MMM- k HbO2 rises at first rapidly and then more slowly. The red dot represents the provisional control condition (i.e., WT/Ctrl) in which k HbO2→Hb+O2 is 100% of the standard parameter value (labeled in green on the x-axis) and MMM- k HbO2 is the familiar hemolysis-corrected value of 3.99 s −1 . The condition with our smallest-observed % decrease in SC- k HbO2 is AQP1-KO/Ctrl (i.e., ∼9% lower than WT/Ctrl; Table 3 ). In order for a decrease in k HbO2→Hb+O2 , by itself, to account for the observed ∼9% decrease in k HbO2 , k HbO2→Hb+O2 would have to fall by ∼27% (i.e., x ≅ 73%; right downward dashed arrow). Our largest-observed % decrease in SC- k HbO2 was ∼78%, for dKO/pCMBS (see Table 3 ). In order for a decrease in k HbO2→Hb+O2 , by itself, to lower the observed SC- k HbO2 by ∼78% (violet point and its horizontal dashed line), k HbO2→Hb+O2 would have to have fallen by ∼92% (i.e., x ≅ 8%; left downward dashed arrow). In fact, our largest-observed decrease in k HbO2→Hb+O2 was ∼10% (see Table 2 ), for dKO/Ctrl. We conclude that the sensitivity of MMM- k HbO2 to decreases in k HbO2→Hb+O2 is far too small have an appreciable impact on our O 2 -offloading data. [Hb Total ] i Figure 9 shows the dependence of MMM- k HbO2 on [Hb Total ] i . Starting from our lowest plotted value of x = 50% of the standard parameter value, increases in [Hb Total ] i cause MMM- k HbO2 to fall at first steeply and then more shallowly. The reasons for this slowing of O 2 offloading are two-fold: (1) oxygen diffuses through the RBC far more slowly when bound to the large HbO 2 tetramer than when free as O 2 and (2) as [Hb Total ] i increases, total O 2 increases—although [O 2 ] i does not—meaning that more time is required to rid the cell of O 2 . In order to account for even our smallest-observed decrease in SC- k HbO2 —that is, 9% ( Table 3 ) in AQP1-KO/Ctrl—[Hb Total ] i would have to increase by 17% (i.e., x ≅ 117%; downward dashed arrow). In fact, in our genotype-specific comparisons (see Table 2 ), MCHC (analogous to [Hb Total ] i ) tended to fall in the KOs, with the greatest decrease being ∼5% in the AQP1-KOs/Ctrl. Thus, by themselves these decreases in MCHC would have caused an increase, rather than the observed decrease in SC- k HbO2 . However, as noted in paper #2 33 , the decreases in MCHC were accompanied by increases in MCV, which have the opposite effect on SC- k HbO2 (as we will see below 27 ). Thus, the MCHC and MCV effects tend to cancel. In our drug studies (see paper #2 34 ), pCMBS in dKO cells produced a 2% decrease—again in the wrong direction to explain a decrease in SC- k HbO2 . On the other hand, pCMBS in WT cells produced a 4.4% increase in MCHC. However, examination of Figure 9 shows that increasing— [Hb Total ] i by 4.4% (i.e., x = 104.4%) would have only a trivial effect on MMM- k HbO2 . Diffusion constants Figure 10 shows the dependence of MMM- k HbO2 on the four relevant diffusion constants, namely, D for extracellular O 2 (the only relevant extracellular solute), and the three intracellular diffusion constants for the components of the equilibrium HbO 2 ⇒ Hb + O 2 . Extracellular D O 2 Figure 10A shows that, starting from our standard parameter value for extracellular D O2 (i.e., 1.3313 × 10 −5 cm 2 s −1 ; see Table 1 )—indicated by the red dot—decreasing ( D O2 ) o at first has little effect, and then causes MMM- k HbO2 to fall steeply. However, ( D O2 ) o — which almost certainly does not vary substantially with genotype—would have to fall by ∼34% (i.e., x ≅ 66%; right downward dashed arrow) to account for the fall in SC- k HbO2 in our AQP1-KO/Ctrl experiments, and by ∼94% (i.e., x ≅ 6%; left downward dashed arrow) to account for the dKO/pCMBS data. Because ( D O2 ) o governs the ability of O 2 to diffuse through the EUF, away from the membrane surface to the bECF—where [O 2 ] = 0 to mimic the annihilation of O 2 by sodium dithionite or (Na + ) 2 S 2 O = (NDT)—we are not surprised that MMM- k HbO4 approaches zero as ( D O2 ) o2 approaches zero. Thus, if ( D O2 ) o were zero, O 2 offloading would stop as soon as [O 2 ] at the outer surface of the membrane rose to match [O 2 ] i . Note that our model does not explicitly take into account the diffusion of NDT from the bECF into the EUF. Although the diffusion constant for S 2 O = is doubtlessly lower than that for O 2 , is ∼71-fold greater than the initial [O 2 ] i . Thus, we expect the EUF to be flooded with S 2 O = —to annihilate O 2 as it exits the RBC) very early in the physiological experiment. It is for this reason that others believe that ℓ EUF is probably <1 μm ( Huxley & Kutchai, 1981 ; Vandegriff & Olson, 1984 b ; Holland et al ., 1985 ), which, if true, means that ( D O2 ) o is even less impactful than indicated in Figure 10 A . Intracellular D O 2 Figure 10B shows that the dependence of MMM- k HbO2 on ( D O2 ) i is similar to that of ( D O2 ) o in Figure 10 A . In fact, starting from the red dot in Figure 10 B , which represents our standard parameter value for intracellular D O2 (i.e., 2.7745 × 10 −6 cm 2 s −1 ; see Table 1 ), decreasing ( D O2 ) i causes MMM- k HbO2 to fall somewhat less steeply than does decreasing ( D O2 ) o in panel A . Thus, ( D O2 ) i —which again we regard as unlikely to change substantially with genotype— would need to fall by ∼40% (i.e., x ≅ 60%; right downward dashed arrow) to account for the fall in SC- k HbO2 in our AQP1-KO/Ctrl experiments, and by ∼98% (i.e., x ≅ 2%; left downward dashed arrow) to account for the dKO/pCMBS data. Note that, as noted in Methods 35 , our choice of ( D O2 ) i is already, historically, a rather low value. If ( D O2 ) i is in fact greater than we estimated, then the red dot would slide further to the right (i.e., on an even flatter part of the curve, then the MMM- k HbO2 would be even less sensitive to changes in ( D O2 ) i . Intracellular D HbO 2 Figure 10C shows that MMM- k HbO2 is nearly insensitive to changes in D HbO2 until, starting at the red dot, this parameter falls to a tiny fraction of its standard parameter value (i.e., 6.07 × 10 −8 cm 2 s −1 cm s −1 ; see Table 1 ). The reason is that the diffusion constant of tetrameric HbO 2 is so low that—even though the concentration of HbO 2 is high—the movement of HbO 2 per se toward the membrane during O 2 offloading has little effect on O 2 delivery, and thus to MMM- k HbO2 . Of course, in Figure 9 we saw that decreases in [Hb Total ] i can greatly increase MMM- k HbO2 . The reason is that Hb acts as a near-stationary buffer that binds O 2 and nearly immobilizes it until O 2 release from HbO 2 permits the now-free O 2 to resume its net diffusion to the membrane or to the next unoccupied Hb. Thus, we expect genotype-dependent changes in D HbO2 to have only miniscule effects on SC- k HbO2 until D HbO2 becomes so small that HbO 2 approaches absolute immobility. Intracellular D Hb Figure 10D shows that MMM- k HbO2 is even less sensitive to changes in D Hb than to D HbO2 . The reasons are: (1) The diffusion constant for the deoxygenated Hb tetramer (i.e., standard parameter value = 6.07 × 10 −8 cm 2 s −1 ; see Table 1 ) is nearly as low as it is for the fully oxygenated tetramer. And (2) Hb is moving away from the membrane during O 2 offloading. Thus, even if D Hb were zero, the empty tetramers would pile up—at least mathematically—at the inner surface of the membrane, still maintaining a gradient for HbO 2 tetramers to diffuse toward the membrane. At D Hb = 0, the small decrease in MMM- k HbO2 would reflect absent diffusion of free Hb into the depth of the cytosol, and thus decreased formation of new HbO 2 . ℓ EUF Figure 11 shows that MMM- k HbO2 has only weak dependence on ℓ EUF . Starting at x = 100% of the standard parameter value (i.e., ℓ EUF = 1 μm), decreasing ℓ EUF to as low as 90% of control (i.e., 0.1 μm) could raise SC- k HbO2 by <25%. Could an increase in ℓ EUF account for our physiological data? By itself, a ∼3× increase in ℓ EUF would be needed to replicate the SC- k HbO2 decrease that we see with AQP1-KO/Ctrl (see downward dashed arrow), and a ∼20× increase would be needed to replicate the RhAG-KO. No amount of ℓ EUF increase, alone, could account for our other physiological data. Ø Sphere Theoretical dependence of k HbO2 on Øsphere The main part of Figure 12 and its inset (with a greatly expanded y-axis) show that MMM- k HbO2 is rather sensitive to changes in the diameter of the equivalent sphere because of the impact of O 2 diffusion distance. The tails of the downward arrows are at the intersections of the blue MMM- k HbO2 curve and their respective observed SC- k HbO2 values (i.e., horizontal dashed lines): Arrow #1 Our largest Ø Sphere was for RhAG-KO/Ctrl, namely 2.18 μm ( Table 2 , data row 5), which is ∼7.9% greater than the value of 2.02 μm for WT/Ctrl. According to the curve in Figure 12 (see leftmost downward arrow), the percent increase in Ø Sphere would have had to have been nearly 3× greater—about 20%—to account, by itself, for the observed decrease in SC- k HbO2 . Arrow #2 Our second largest computed Ø Sphere was for dKO/Ctrl, namely 2.14 μm ( Table 2 , data row 5), which is only ∼5.9% greater than the WT/Ctrl value. According to our sensitivity analysis, the percent increase in Ø Sphere would have to have risen by nearly 7×—by somewhat more than 40% (i.e., x ≅ 142%; 2 nd downward arrow)—in order to account for the observed decrease Recall that the analyses in Figure 5 —which take into account not just Ø Sphere but all genotype-specific data—predict only a ∼4% decrease for RhAG-KO/Ctrl (vs. observed ∼17%), and a ∼4% to ∼6% decrease for dKO/Ctrl (vs. observed ∼30%). Arrows #3, #4, #5 At greater increases in Ø Sphere , the curve flattens, so that one would have to invoke a ∼100% increase (i.e., x ≅ 200%; 3 rd downward arrow) to account for the dKO/DIDS data, more than a 140% increase (i.e., x > 240%; 4 th downward arrow) to account for the WT/pCMBS data, and nearly a 300% increase (i.e., x > 394%; rightmost downward arrow) to account for the dKO/pCMBS results. All of these values are unrealistic. We conclude that, by themselves, changes in Ø Sphere cannot account for our physiological data. Tendency of MCHC and MCV effects to cancel As already noted, in RBCs from KO strains, MCHC (a surrogate for [Hb Total ] i ; see above 36 ) tends to fall, thereby speeding O 2 offloading. On the other hand, as we just saw here, MCV (related to Ø Sphere ) tends to rise, thereby slowing O 2 offloading. These two effects tend to cancel, which is a reason that we see relative stability among the predicted MMM- k HbO2 values among genotypes in Figure 5 . Accommodation for non-BCDs The inputs to the MMM simulation include four observed experimental parameters: (1) MCV and (2) MCH (from which we compute MCHC), both of which come from automated hematology ( paper #1 37 ); (3) Ø Major , from imaging flow cytometry ( paper #2 38 ); and (4) k HbO2→Hb+O2 , from stopped-flow absorbance spectroscopy on lysed RBCs ( paper #1 39 ). These values reflect all cells in the blood sample. An additional input is an estimate of P M,O2 . The output of the simulation is k HbO2 . Because the blood comprises both the dominant biconcave disks and minority non-BCDs, the paired values P M,O2 (input) and k HbO2 (output) represent hybrid values that to varying extents (depending on the sample) conflate BCDs and nBCDs. Our goal in this section is to obtain a ( P M,O2 , k HbO2 ) pair that estimates as accurately as possible the values for BCDs. Provisional estimates of P M,O2 and MMM-k HbO2 (steps #13 and #14 & 14′ in Figure 1 ) Table 4 is an expanded version of Table 3 . The first two data rows of Table 4 lists the prevalences of BCDs and nBCDs from paper #2 40 ; we consider these in the next sub-section. View this table: View inline View popup Table 4. Detailed parameter values predicted by macroscopic mathematical modeling As noted in Methods, we chose our provisional P M,O2 for WT/Ctrl cells ( Table 4 , data row 3) to match the P M,CO2 value of 0.15 cm s −1 that Endeward et al ( Endeward et al ., 2006 ) obtained for the membrane CO 2 permeability of human RBCs. MMM simulations based on P M,O2 = 0.15 cm s −1 yield a provisional MMM- k HbO2 of 3.99 s −1 (row 4), a value that is rather close to the highest of the three shape-corrected stopped-flow values (i.e., ∼4.09 s −1 ; row 14 in Table 4 ) for three separate WT/Ctrl data sets that we employed in paper #1 41 . For the seven columns to the right of “WT/Ctrl” in Table 4 , we chose the provisional P M,O2 values in iterative processes in which each P M,O2 generated a corresponding provisional MMM- k HbO2 that approximately matched the experimentally determined HC- k HbO2 value. Note that all ( P M,O2 , k HbO2 ) pairs lie on the sigmoidal k HbO2 vs. log( P M,O2 ) curve in Figure 6 . nBCD prevalence (steps #15 & 15′ in Figure 1 ) Recall that the first two data rows of Table 4 lists the prevalences of BCDs and nBCDs. The values for WT/Ctrl and dKO/Ctrl are around 2%, and may overestimate the actual values in SF experiments because of the extra time needed to prepare the samples for IFC analyses (compared to SF experiments; discussed in paper #2 42 ). The nBCD prevalence values for the AQP1-KO/Ctrl and RhAG-KO/Ctrl groups are interpolations of the WT/Ctrl and dKO/Ctrl values. The nBCD prevalence values are ∼6-fold higher for the WT/pCMBS cells than the corresponding WT/Ctrl cells. The dKO/pCMBS values are only ∼2-fold greater than the corresponding dKO/Ctrl cells, probably reflecting the partial absence of pCMBS targets after the elimination of AQP1 and Rh Cx . The largest nBCD prevalence comes with WT/DIDS cells, at ∼41%; the figure falls to ∼21% in the dKO/DIDS group. These large percentages probably reflect the long incubation with DIDS (1 hour), compounded—as noted above—by the extra time needed for executing the IFC studies. In the next four sections, we present a novel, 2-step approach for deconvoluting the HC- k HbO2 data on a mixture of BCDs and nBCDs to extract an estimate of the k HbO2 of BCDs, and thereby arrive at a P M,O2 that we assume to apply to both BCDs and nBCDs. First linear-combination analysis of k HbO2 values (step #16 in Figure 1 ) Using WT/Ctrl cells to illustrate our approach, we begin by making the (incorrect) assumption that the provisional MMM- k HbO2 of ∼3.99 s −1 ( Table 4 , row 4) represents only BCDs (note that row 5 repeats row 4). This allows us to construct an idealized time course of HbO 2 desaturation for BCDs ( Figure 13 A , green curve/furthest to the left). In parallel, knowing the measured Ø Major of nBCDs, and assuming them to be spheres with the same P M,O2 as the BCDs (i.e., 0.15 cm s −1 ), we construct another idealized desaturation time course, but this time for nBCDs ( Figure 13 A , blue curve/furthest to the right). We then linearly combine these two curves in proportion to the prevalence of their respective cells. Because the weighting is ∼98.59% BCD (rows 5: LC-MMM- k HbO2 = 3.99 s −1 ) and 1.41% nBCD (rows 6: LC-MMM- k HbO2 = 1.83 s −1 ), the curve ( Figure 13 A , black) that describes the mixture of BCDs+nBCDs (rows 7: LC-MMM- k HbO2 = 3.95 s −1 ) lies barely to the right of the green BCD curve. Download figure Open in new tab Figure 13. First linear-combination analysis of the rate constant for deoxygenation of HbO 2 , k HbO2 A, WT/Ctrl. The green curve is the computed decay of HbSat (hemoglobin saturation) for a biconcave disk (BCD). The blue curve is the computed decay for a non-BCD. The two sigmoid curves in the inset illustrate how, given a value for P M,O2 , we compute the corresponding MMM-HbO 2 . The black curve, barely visible to the right of the green curve, is the linear combination of the green and blue curves, weighted for the respective prevalences of BCDs and nBCDs, respectively. The horizontal dashed curve represents the value of HbSat after 1 time constant for the mixture. The vertical dashed curve represents the corresponding time. B, dKO/pCMBS. Similar to panel A except here the calculations are for a dKO cell treated with pCMBS. Figure 13B illustrates a similar analysis for dKO/pCMBS cells, at the opposite extreme of our experimentally observed spectrum of HC- k HbO2 values ( Table 4 , rows 13–22). Here, reflecting the low P M,O2 , all three curves show a slower offloading of O 2 from RBCs than in Figure 13 A . Moreover, because the nBCD prevalence is ∼4-fold higher than for the WT/Ctrl cells in Figure 13 A , the separation between the green (BCDs) and black (mixture) is more discernable. Rebalancing of k HbO2 values (steps #17 & 17′ in Figure 1 ) Recall that in row 7 of Table 4 , k HbO2 for the WT/Ctrl mixture is only ∼3.95 s −1 , which is ∼1% lower than our provisional estimate (∼3.99 s −1 ), yet much higher than the k HbO2 for the nBCDs (1.83 s −1 ). The goal of the present step is to rebalance the three k HbO2 values so that the value for the mixture matches the original provisional k HbO2 as closely as possible. We make two assumptions: The term (MMM- k HbO2/Mixture – MMM- k HbO2/BCDs )/(MMM- k HbO2/BCDs )) is maintained from the first linear combination throughout the rebalancing. That is, the rebalanced MMM- k HbO2/BCDs exceeds the rebalanced MMM- k HbO2/Mixture by the same fraction (∼1.02% for WT/Ctrl) as it did during the first linear-combination step. The ratio (MMM- k HbO2/nBCDs )/(MMM- k HbO2/BCDs ) remains similarly fixed from the first linear combination through the rebalancing. For WT/Ctrl, this is ∼45.9%. These provisos ensure that, after rebalancing, MMM- k HbO2/Mixture is the same, within rounding errors, as our original provisional k HbO2 , which, after all, really encompassed both BCDs and nBCDs. Thus, after rebalancing, MMM- k HbO2/BCDs is slightly elevated (∼4.04 s −1 for WT/Ctrl). Achieving this higher k HbO2 requires that our corresponding rebalanced P M,O2 be increased from the provisional value—our next step. Rebalancing P M,O2 values (steps #18 – #20 in Figure 1 ) Armed with the rebalanced MMM- k HbO2 (∼4.04 s −1 for WT/Ctrl in Table 4 , row 8), we compute the rebalanced P M,O2 from an inverted/linearized version of Figure 6 , shown in Figure 14 . The result in the case of WT/Ctrl is a value (about “0.1546” cm s −1 ) that is slightly higher than our provisional P M,O2 of “0.1500” cm s −1 . Download figure Open in new tab Figure 14. Inverted sigmoid plot This figure is analogous to Figure 5 except that here we plot P M,O2 on the y-axis, which is now linear, and the rate constant for deoxygenation of HbO 2 , k HbO2 , on the x-axis. The upward-sloping curve is a best fit created from a 5 th -order polynomial. This is a convenient tool for interpolating P M,O2 values that correspond to a particular k HbO2 value as part of the rebalancing of P M,O2 . The upward arrow illustrates how, for a wild-type BCD under control conditions (no drugs), we identify the point on the curve that corresponds to the P M,O2 indicated by the leftward-pointing arrow. Shape-correction factor (F SC ; steps #21 and #22 in Figure 1 ) For a given data genotype/treatment, F SC is the ratio (rebalanced MMM- k HbO2 )/(provisional MMM- k HbO2 ), ∼1.0102 in the case of WT/Ctrl cells. Multiplying an HC- k HbO2 value (∼4.05 s −1 for the first set of WT/Ctrl data) by F SC yields an estimate of SC- k HbO2 (∼4.09 s −1 in this example). For dKO/pCMBS cells, with a higher nBCD prevalence than WT/Ctrl cells, F SC is correspondingly higher, ∼1.0376. The remainder of Table 4 , following the F SC values, shows five pairs of reference values (HC- k HbO2 and SC- k HbO2 ). The first set pertains to three data sets for control experiments: AQP1-KO/Ctrl, RhAG-KO/Ctrl, and dKO/Ctrl. The second set pertains to WT/pCMBS; the third, to WT/DIDS. The fourth and fifth sets are reference values for dKO/pCMBS and dKO/DIDS, respectively. The last two rows of Table 4 show the percent decreases in HC- k HbO2 and SC- k HbO2 , based on shape-corrected values, for each genotype/treatment, relative to WT/Ctrl. These are the same values listed above in the bottom two rows of Table 3 . Discussion To test whether the decreases in k HbO2 observed in paper #1 correspond to decreases in P M,O2 and, if so, to provide estimates of P M,O2 from measured k HbO2 values, we developed a three-dimensional reaction-diffusion model of O 2 efflux from an RBC. In this model, we simplify the RBC by assuming that it is a perfectly symmetric sphere, the diameter of which equals the computed thickness of an RBC. Note that the thickness of the biconcave disk near its edge (∼2 μm)—not the major diameter (∼6.8 μm for mice)—is the dimension of the RBC (when it is a BCD) that is critically important for gas exchange, as pointed out by several authors ( Roughton, 1932 ; Ponder, 1948 ; Forster, 1964 ; Endeward, 2012 ). Our approach, which has the advantage of geometric (and thus computational) simplicity, ignores the dimpled center of the BCD, which contains relatively little hemoglobin. Our macroscopic mathematical model describes how the concentrations of O 2 , HbO 2 and Hb change in time and space, when O 2 diffuses out of the cytosol, across the plasma membrane, throughout the EUF and finally to the bECF. For our initial estimate of P M,O2 in our mathematical simulations, we assume that P M,O2 in mouse RBCs has the same value as measured by Endeward et al . (2006 b ) for the CO 2 permeability of human RBCs. Even though our MMM model is relatively simple and based only on first principles, our assumptions as well as the physiological data that inform the model are sufficiently robust that our model makes physiologically meaningful predictions for a WT mouse. One strength of this study is the use of the genotype-specific flow-cytometry and hematology data that we generated in the two accompanying papers to establish internally consistent parameter values. Krogh-Erlang Equation In his pioneering modeling work on O 2 delivery from capillary to tissue, August Krogh (1919) introduced a now-famous equation—derived by his mathematician-colleague Mr. K Erlang—that played a major role in elucidating the fundamental principles of how [O 2 ] changes both longitudinally along the capillary and radially away from it. This equation reflects a highly simplified view of the system, which of course was necessary at that time, given the limitations in available computational algorithms and computing power. The analysis by Kreuzer (1982) lists 15 assumptions—some of which he and others had noted previously—that are implicit in the derivation of the Krogh-Erlang equation. Assumption #12 states that “the diffusion coefficient [of O 2 ] is the same throughout the tissue.” Assumption #7 (previously noted by Hess, 1930 )—really a corollary of #12—states that “the capillary wall does not present any resistance to oxygen diffusion.” Another corollary of #12 is that no cell membrane, including that of the RBC, offers resistance to O 2 diffusion. We examine implications of this implicit assumption in the section b elow . How others concluded that the plasma membrane offers no resistance to O2 diffusion After Krogh’s landmark paper, most practitioners in the RBC field adopted the aforementioned implicit assumption that membranes offer no resistance to O 2 diffusion. However, a series of studies from Roughton and his colleagues led to the suggestion that the plasma membrane of RBCs offers significant resistance to the movement of gases, including CO 2 and O 2 ( Blank & Roughton, 1960 ; Gibson et al ., 1955 ; Forster, 1964 ; Roughton, 1964 ; Nicolson & Roughton, 1951 ; Lawson et al ., 1965 ). However, later studies questioned this possibility and attributed the observed resistance to the extracellular unconvected fluid layer 43 that surrounds the cells after flow stops in the SF apparatus ( Kutchai, 1975 ; Coin & Olson, 1979 ; Huxley & Kutchai, 1981 ; Vandegriff & Olson, 1984 b ; Holland et al ., 1985 ; Hughes & Bates, 2003 ). It is true that studies that focus on O 2 influx across RBCs are intrinsically challenging because this influx depletes O 2 at the surface of the RBC, and the EUF of that surrounds the RBC offers “resistance” to the replenishment of O 2 from the bECF ( Coin & Olson, 1979 ; Huxley & Kutchai, 1981 ; Holland et al ., 1985 ; Hughes & Bates, 2003 ). Moreover, as pointed out by Vandegriff and Olson, the low solubility of O 2 in water enhances the effect of EUFs in experiments of O 2 influx ( Vandegriff & Olson, 1984 b ). Nevertheless, it is possible that the implicit assumption of past authors—namely, that the plasma membrane offers no resistance of O 2 diffusion (i.e., R M,O2 —or R M for short—is 0)—may have led them to dismiss Roughton hypothesis prematurely. The total resistance ( R Total ) to O 2 diffusion from its original location in the cytosol to bECF is the sum of the effective resistance of the intracellular fluid ( R ICF ), R M , and the resistance of the extracellular unconvected fluid ( R EUF ). If one implicitly assumes that R M ≅ 0, then the sum ( R M + R EUF ) reduces to R EUF , with the result that one computes erroneously high values for the EUF thickness. In the process, one concludes that the large EUF explains the experimental data without the need of invoking membrane resistance—a classic bootstrap maneuver. Because Vandegriff and Olson recognized the challenge of studying O 2 influx, they investigated O 2 efflux. They reduced the effects of the EUF by adding large amounts of sodium dithionite (NDT) in the suspending medium ( Vandegriff & Olson, 1984 b ). The NDT near the outer side of the plasma membrane annihilates the O 2 that has just left the RBC, thereby minimizing the build-up of O 2 and minimizing the EUF. Viewed somewhat differently, the NDT maximizes the outwardly directed O 2 gradient and therefore the rate of O 2 efflux ( Holland et al ., 1985 ). Vandegriff and Olson used a reaction-diffusion model of an RBC, idealized as a cylindrical disk, to analyze the time course of their O 2 release data. They concluded that their mathematical model—which lacked a term analogous to R M —could simulate their data without the need to include a membrane resistance to the movement of O 2 . Instead, they accounted for their O 2 -release data in the presence of NDT in the extracellular medium by including in the model an EUF, the width of which increased with time from the initial value of 1 μm to ∼30 μm (computed from an equation in ref. Holland et al ., 1985 ). In summary, because they implicitly assumed R M = 0, these authors invoked a very large R EUF , and used this result to conclude that R M ≅ 0. It is worth noting that at least 3 sets of authors have performed physicochemical experiments in which they concluded that lipid mono/bilayers can offer substantial resistance to CO 2 or O 2 ( Blank & Roughton, 1960 ; Strutwolf et al ., 2001 ; Ivanov et al ., 2004 ). Recent observations on D O 2 in the cytosol of RBCs In their 2020 paper, Richardson et al conclude that the D O2 in the cytosol of RBCs is far lower than previously thought ( Richardson et al ., 2020 ). The generally accepted value for D O2 in RBC cytosol has been 5.09 × 10 −6 cm 2 / s −1 at 10 °C ( Clark et al ., 1985 ), which is ∼38% of the value in water (i.e., 13.313 × 10 -6 cm 2 s −1 at 10 °C). However, Richardson et al conclude that the actual D O2 value in RBC cytosol is <0.7 ×10 -6 cm 2 s −1 at 23 °C; we calculate that this value is < ∼3.5% of the value in water at 23 °C (i.e., 20 × 10 -6 cm 2 s −1 ; see Han & Bartels, 1996 ). Thus, the D O2 in cytoplasm predicted by Richardson et al should be 70/2000 = 1/29 of that in water. Our calculation of the D O 2 of RBC cytosol (excluding membrane) at 10 °C To apply the work of Richardson et al to our own, we must recompute their D O2 for a temperature of 10 °C. If we divide the aforementioned figure of 13.313 × 10 -6 cm 2 s −1 by 29, we arrive at a predicted value of D O2 (based on the work of Richardson et al) of 0.45907 × 10 −6 cm 2 s −1 . We are concerned that this D O2 value is an underestimate because it is only ∼10% of the previously accepted value, a point recently made by Al-Samir et al . (2025) . An inappropriately low estimate for ( D O2 ) i would impact us in two ways: (1) We would attribute to total diffusive resistance 44 an ICF component that in fact is due to a combination of both the membrane and EUF layer. (2) Our standard k HbO2 value for WT/Ctrl would fall from 3.99 s −1 —which comports very well with our physiological data—to only 2.61 s −1 . Therefore, we decided to compromise by averaging the Richardson value of 0.45907 × 10 −6 cm 2 s −1 with the generally accepted value of 5.09 × 10 −6 cm 2 s −1 ( Clark et al ., 1985 ), and thereby arrive at a D O2 in RBC cytoplasm of 2.7745 × 10 −6 cm 2 s −1 . Importance of values used in modeling Thickness of EUF (ℓ EUF ) Several groups— Lawson et al . (1965) , Coin and Olson (1979) , Vandegriff and Olson (1984 b ) , Yamaguchi et al . (1985) , and Holland et al . (1985) —have emphasized the importance of employing a sufficiently high [NDT] o to minimize the EUF layer in stopped-flow experiments. Lawson et al . (1965) concluded that the final NDT after mixing must be at least 10 mM, although this low value was later challenged on the basis of the “packet model” used in their data analysis. Coin and Olson (1979) showed that the “packet model” greatly overestimated the power of NDT, and concluded (see their figure 9 ) that an NDT of 25 mM is required to maximize k HbO2 . Vandegriff and Olson (1984 b ) similarly discount the “packet model” and conclude that a final [NDT] of at least 25 mM is required. Yamaguchi et al . (1985) , used a final [NDT] of 40 mM to maximize k HbO2 . In preliminary work, we systematically reduced [NDT] and noted a small k HbO2 falloff between [NDT] Final values of 25 and 20 mM, and progressively steeper declines below 20 mM. Therefore, we chose a final [NDT] of 25 mM. We note that commercial supplies of NDT are not reliable, and we routine discard the majority of NDT bottles because of infiltration of air and H 2 O vapor (see paper #1 45 ); NDT from such bottles routinely leads to lower k HbO2 values. Despite the use of NDT in our SF experiments—which virtually abolishes the EUF ( Holland et al ., 1985 )—in our mathematical model we make the conservative assumption that the EUF has a thickness of 1 μm and that no NDT is present in this layer. We simulate the annihilation of O 2 by NDT by assuming that the bECF contains no O 2 during the duration of the simulations. In Figure 11 , we examine the dependence of k HbO2 on ℓ EUF . When we increase ℓ EUF , the simulated k HbO2 asymptotically falls to a value that approximates the biological value that we observe with RBCs from an RhAG-KO/Ctrl mouse. Thus, no increase in ℓ EUF can account for our data with RBCs from dKO mice, or RBCs treated with pCMBS or DIDS. Lowering ℓ EUF to values smaller than 1 μm causes steep increases in the simulated k HbO2 (i.e., decreases in the resistance of the EUF to O 2 diffusion, R EUF ). Starting from such elevated k HbO2 values, the model can generate lower k HbO2 values (e.g., 3.99 s −1 ) by increasing R M , that is, by decreasing the permeability of the plasma membrane to O 2 so that the sum ( R EUF + R M ) remains the same. In other words, if the EUF is thinner than 1 μm—which is likely to be the case in the presence of high [NDT]—the membrane must make a greater contribution to the overall diffusive resistance of O 2 . Calculation of the contribution of the plasma membrane to total diffusive resistance to O2 Let us assume that we can treat the diffusion of O 2 from the ICF of the RBC to the bECF like an electrical circuit (see Figure 2 A ), where, as in the preceding equation, R Total is the sum R ICF + R M + R EUF . In the next 3 sub-sections, we will compute each of the resistances on the right side of the equation, and then we will compute the fraction of R Total that R M represents. See Table 1 for parameter values used in simulations of RBCs from WT/Ctrl mice. Calculate R ICF We start by computing—at the time t 37 (i.e., 0.2503 s), when HbSat has fallen to 1/e of its initial value—the parallel fluxes of free O 2 ( J O2 ) and HbO 2 ( J HbO2 ) from the position of the average O 2 and HbO 2 molecules in the ICF. From these two fluxes, we then compute the total O 2 flux ( J Total,O2 ). First, we determine the number of O 2 molecules in each of the 101 shells ( i = 1 to 101) that make up the model RBC, as well as the distance (ℓ) from the center of each shell to the inner surface of the plasma membrane (iM). For a WT cell with no inhibitors, we find that the average distance of an O 2 molecule from the iM is 2.792×10 -5 cm. For HbO 2 , a similar analysis yields an average distance of 2.560×10 -5 cm. Because these distances are so similar, we used the average distance for O 2 and HbO 2 , which is 2.68×10 -5 cm, which corresponds to shell i = k = 76 from the center of the sphere. We call this average distance ℓ k . Thus, the flux of free O 2 from ℓ k to the iM is: where D O2 is the diffusion constant for O 2 in the ICF (see Table 1 for value), and where [O 2 ] k = 0.01528 mM and [O 2 ] iM = 0.01007 mM at t 37 . For HbO 2 , where D HbO2 is the diffusion constant for HbO 2 in the ICF (see Table 1 for value), and where [HbO 2 ] k = 6.869 mM and [HbO 2 ] iM = 6.744 mM at t 37 . Thus, the total O 2 flux from ℓ k to the iM is The effective resistance to the diffusion of O 2 is 46 Calculate R M We calculate the resistance of the plasma membrane to O 2 diffusion as the reciprocal of the “true” microscopic permeability of the plasma membrane to O 2 , which is 0.15 cm s −1 (see Table 1 for value). Thus, Calculate R EUF We calculate the resistance of the EUF to O 2 diffusion as where ℓ EUF is the thickness of the EUF and D O2 is the diffusion constant for O 2 in the EUF (see Table 1 for values). Compute the contribution of R M to R Total We calculate the contribution of R M to R Total as Note that if ℓ EUF is in fact <1 μm (i.e., R EUF < 7.511 s cm −1 ), the contribution of R M to total resistance would be greater than ∼33%. Note also that if our compromise estimate for ( D O2 ) i is in fact too low (i.e., if we are overestimating R ICF ), this consideration would also increase the contribution of R M to R Total . Potential contribution of channels to R M and R total In reaching the above conclusion that R M represents at least ∼33% of R Total for WT/Ctrl RBCs under the conditions of our experiments, we did not consider inhibitors or KO strains. Our observed 30% decrease in SC- k HbO2 caused by the dKO/Ctrl corresponds to a ∼55% decrease in P M,O2 (see Figure 6 ), so that R M is now (6.666 s cm −1 )/(1–0.55), which represents ∼52% of R Total . The 78% decrease in k HbO2 caused by dKO/pCMBS corresponds to a ∼91% decrease in P M,O2 , so that R M now rises to (6.666 s cm −1 )/(0.55), so that R M now is ∼84% of R Total —even with our conservative estimates of R ICF (i.e., intracellular D O2 ) and R EUF (i.e., ℓ EUF ). Viewed differently, for WT/Ctrl RBCs under the conditions of our experiments, the reason that R M makes only a ∼33% contribution to R Total is that the O 2 channels make the membrane leaky. It is not clear how drugs confined to the outside of the membrane or the genetic deletion of membrane proteins could produce the observed decreases of k HbO2 in any way other than decreasing O 2 egress through protein channels. Moreover, an analysis of 8 key parameters (e.g., ℓ EUF , cell thickness) based on mathematical simulations shows that no reasonable change in any could explain our k HbO2 data ( Figure 8 through Figure 12 ). Uniqueness of the sigmoidal k HbO2 vs. log( P M,O 2 ) curve Our development of the sigmoid curve in Figure 6 is novel. The curve describes all possible combinations of P M,O2 and k HbO2 for a particular set of conditions. The detailed shape of the curve depends on experimental conditions (e.g., temperature) and MMM-simulation parameters (e.g., ℓ EUF ) as summarized in Table 1 . Beyond these values, the shape of the sigmoid curve is potentially unique for each species or strain of animal—or individual humans with unique genetics and medical histories—because each will have a characteristic set of genotype-specific values for MCV and MCH (and thus MCHC), Ø Major , and k HbO2→Hb +O2 (see Table 2 ). Thus, with different experimental conditions and strains/species, we expect to see major differences in the maximal-possible k HbO2 at the far right of the curve, which represents a membrane made of a film of water. Accommodation for nBCDs Our accommodation for nBCDs is also novel, founded on our MMM-simulations with various algebraic manipulations, together with a meticulous set of collected data. For RBCs from our WT and KO strains—studied in the absence of drugs—the prevalence of nBCDs is small and thus the corrections are very minor. In the presence of pCMBS the corrections become larger, and for DIDS, still larger. The accuracy of the corrections likely increases as nBCD prevalence falls, and would improve with a more detailed knowledge of the surface-to-volume ratio. Additional Information Competing interests The authors declare that they have no competing interests. Author contributions R.O., P.Z. & W.F.B. designed the study; R.O. developed and implemented the computational model, and performed all numerical simulations; R.O., P.Z. and W.F. B. analyzed data. R.O., P.Z. & W.F.B. prepared the figures; R.O., P.Z., F.J.M., & W.F.B. wrote the manuscript. Acknowledgements We thank Jean-Pierre Cartron for the gift of RhAG-KO mice. We thank him and Gerolf Gros and for extremely helpful discussions at the outset of the project. We also thank Pawel Swietach for comments on an earlier version of the manuscript. We thank Seong-Ki Lee for developing an improved approach for genotyping the RhAG-KO mice. We thank systems analyst Dale Huffman for using MATLAB to generate some of the figures. The authors gratefully acknowledge Daniela Calvetti and Erkki Somersalo for having developed the engine of an earlier version of the CO 2 /pH reaction-diffusion model of an oocyte, which in part served as the starting point for the RBC model. We thank Thomas Radford for organizing the husbandry of the mouse colonies; Gerald Babcock for his role as laboratory manager; James W. Jacobberger and Philip G. Woost of the CWRU Flow Cytometry and Imaging Microscopy Core (FCIMC) for their assistance with flow cytometry; and Daniela Schlatzer of the CWRU Center for Proteomics and Bioinformatics for their assistance with mass spectrometry. This work was supported by Office of Naval Research (ONR) grant N00014-11-1-0889, N00014-14-1-0716, and N00014-15-1-2060; a Multidisciplinary University Research Initiative (MURI) grant N00014-16-1-2535 from the DoD, NIH grant multi-scale modeling grant 5U01GM111251 (to WFB), and NIH grant R01HL160857 (to WFB). R.O. and the modeling were supported in part by NIH grant K01-DK107787. W.F.B. gratefully acknowledges the support of the Myers/Scarpa endowed chair. Funder Information Declared Office of Naval Research , N00014-11-1-0889 , N00014-14-1-0716 , N00014-15-1-2060 , N00014-16-1-2535 National Institutes of Health National Institute of General Medical Sciences , 5U01-GM111251 National Institutes of Health National Heart Lung and Blood Institute , R01-HL160857 National Institutes of Health National Institute of Diabetes and Digestive and Kidney Diseases , K01-DK107787 Footnotes Minor revisions to Abstract text. Revised text of Results and Discussion New Figure 1 is added to demonstrate the workflow and interaction between the experiments and simulations between the 3 papers; the present paper #3 and associated paper #1 and paper #2. Paper #1: https://doi.org/10.1101/2025.03.05.639948 Paper #2: https://doi.org/10.1101/2025.03.05.639962 V1_Fig.1 becomes V2_Fig2 New Figure 3 to present the conversion of biconcave disk geometry into torus geometry. V1_Fig.1 becomes V2_Fig4 and now also displays two new panels. Panel C: Concentration of the sum of free O 2 and HbO 2 monomers, integrated over the entire volume of cytoplasm. Panel F: Transmembrane O 2 flux ( J M,O2 ). V1_Fig.3 becomes V2_Fig3 and has a new Panel B: displaying MMM- k HbO2 values predicted from simulations k HbO2 --> Hb+O 2 assumed to be specific for each genotype V1_Fig.4 becomes V2_Fig6 V1_Fig.5 becomes V2 Fig7: Fig7F now has an inset showing a magnification of the first 1 ms of the simulated J M,O2 V1_Fig.6 becomes V2_Fig8 and the symbol key is updated with revised nomenclature V1_Fig.7 becomes V2_Fig9 and the symbol key is updated with revised nomenclature V1_Fig.8 becomes V2_Fig10 and the symbol key is updated with revised nomenclature V1_Fig.9 becomes V2_Fig12 and the symbol key is updated with revised nomenclature V1_Fig.10 becomes V2_Fig12. The symbol key is updated with revised nomenclature, and the position of the inset is moved. V1_Fig.11 becomes V2_Fig13. Added annotations for a biconcave disk (BCD) and non-BCD for computed decay of HbSat (hemoglobin saturation). Also added annotation for the linear combination of the green and blue curves, "Mixture". V1_Fig.12 becomes V2_Fig14. The cross-references to Paper #1 and Paper #2, the two supporting preprints, are updated. ↵ 1 As a shorthand, we will refer to the three papers as “paper #1”, “paper #2”, and “paper #3”. ↵ 2 This manuscript was first published as a preprint: Occhipinti R, Zhao P, Moss FJ & Boron WF (2025). Role of channels in the O₂ permeability of murine red blood cells. III. Mathematical modeling and simulations. bioRxiv. https://www.biorxiv.org/content/10.1101/2025.03.05.639964 ↵ 3 When we use “Hb”—as is standard practice in the literature—without using a subscript designation “T” for tetramer or “M” for monomer, we mean the mass of hemoglobin (in grams), both deoxyhemoglobin (Hb) and oxyhemoglobin (HbO2). Thus, [HbTotal]i is the total intracellular Hb concentration—both Hb and HbO2—measured in g/dl. ↵ 4 See Methods>Mathematical modeling and simulations>Calculation of RBC thickness based on the geometry of a torus. The “>” symbol separates heading levels. It is understood that the reference is to a location in the present paper (i.e., paper #3) unless “See” is followed by “Paper #1” or “Paper #2”. ↵ 5 Not to be confused with r Torus, which we use below to describe the minor radius of a torus. ↵ 6 See Paper #1>Methods>Determination of rate constant k HbO2→Hb+O2 (workflow #11) ↵ 7 These include immature RBCs (i.e., nucleated cells, reticulocytes, and spherocytes) as determined by imaging flow cytometry in paper #2. ↵ 8 See Paper #2>table 3 ↵ 9 See Paper #2>table 4 ↵ 10 Here we use “Total” to mean all Hb tetramers, whether they bind O2 (i.e., HbTO2) or not (i.e., HbT). Thus, [HbT,Total] = [HbTO2] + [HbT]. ↵ 11 These hematology data pertain to the subset of mice used in the imaging flow cytometry and MMM studies. ↵ 12 Here we use “Total” to mean all Hb monomers, whether they bind O2 (i.e., HbMO2) or not (i.e., Hbm). Thus, [HbM,Total] = [HbMO2] + [HbM]. ↵ 13 This is the “permeability” computed on the basis of concentrations immediately adjacent to the two sides of the membrane (i.e., corresponding to R Sphere– and R Sphere+). Physiologists usually define permeability operationally, based on the respective concentrations in the bulk intra- and extracellular fluids (see Boron, 2010 ). ↵ 14 P M,S is the general description for membrane permeability to solute S . When S is O2, elsewhere in the three papers, we refer to the permeability as P M,O 2 . ↵ 15 See Paper #1>Results>Effect of pCMBS or DIDS on O2 offloading from RBCs>Stopped-flow Approach ↵ 16 See (a) Paper #1>Methods>Physiological Solutions … & … (b)..>Stopped-flow absorbance spectroscopy (workflow #1) ↵ 17 See Discussion>Recent observations on D O 2 in the cytosol of RBC ↵ 18 See Paper #1>Methods>Calculation of raw- k HbO2 (workflow #3) ↵ 19 See Methods>Mathematical modeling and simulations>Model formulation ↵ 20 See Paper #1>Results>Effect of genetic deletions on O2 offloading from RBCs ↵ 21 See Paper #1>figure 3 d , dark-green curve ↵ 22 See Methods>Mathematical modeling and simulations>Simulation of time course of HbO2 deoxygenation, and estimation of k HbO2 ↵ 23 See Results>Accommodation for non-BCDs ↵ 24 See Paper #1>figure 5 b ↵ 25 See Paper #2>figure 8 b–c , “Ctrl” bars ↵ 26 See Paper #1>figure 5 b ↵ 27 See Results>Mathematical simulations exploring the predicted sensitivity of k HbO2 to eight key kinetic and geometric parameters ↵ 28 See Discussion>Uniqueness of the sigmoidal k HbO2 vs. log( P M,O 2 ) curve ↵ 29 See Paper #2>table 5 ↵ 30 See Results>Accommodation for non-BCDs ↵ 31 See Paper #2>Results>Proteomics ↵ 32 See Paper #1>figure 5 d , light purple bar ↵ 33 See Paper #2>table 4 ↵ 34 See Paper #2>table 6 ↵ 35 See Methods>Mathematical modeling and simulations>Parameter values ↵ 36 See Results>Mathematical simulations exploring the predicted sensitivity …>[HbTotal]i ↵ 37 See Paper #1>Results>Hematological and related parameters >Automated hematology ↵ 38 See Paper #2>Results>Morphometry>Imaging flow cytometry ↵ 39 See Paper #1>Results>Hematological and related parameters>Hb kinetics ( k HbO2→Hb+O2) ↵ 40 See Paper #2>table 3 ↵ 41 See Paper #1>Results>Mathematical simulations>Summary ↵ 42 See Paper #2>Discussion>Morphometry>Imaging flow cytometry…>nBCD (poikilocyte) abundance >Time dependence ↵ 43 We avoid using the classic term “unstirred layer” because it is defined operationally; it has meaning only in the steady state, in principle differs for each solute, and is not easily implemented in MMM (see Figure 2 A ). ↵ 44 Note that this equation includes all diffusive terms but does not include the reaction term for the dissociation of HbO2 ↵ 45 See Paper #1>Discussion>Implications of the macroscopic mathematical model>Basic features of the model>Simulated k HbO2 for WT/Ctrl RBCs ↵ 46 Apparent deviations in computed values are due to rounding errors References Adair GS , Bock AV & Field H ( 1925 ). The hemoglobin system VI. the oxygen dissociation curve of hemoglobin . J Biol Chem 63 , 529 – 545 . OpenUrl FREE Full Text ↵ Al-Samir S , Kyriazi D , Yool AJ , Moser I , Kyriazi K , Gros G , Tsiavaliaris G & Endeward V ( 2025 ). Aquaporin-1 acts as an O₂ channel. The permeability of human and mouse red cell membranes for oxygen . Am J Physiol Cell Physiol ; DOI: 10.1152/ajpcell.00858.2024 . OpenUrl CrossRef ↵ Blank M & Roughton FJW ( 1960 ). The permeability of monolayers to carbon dioxide . Trans Faraday Soc 56 , 1832 – 1841 . OpenUrl CrossRef ↵ Boron WF ( 2010 ). Sharpey-Schafer lecture: Gas channels . Exp Physiol 95 , 1107 – 1130 . OpenUrl CrossRef PubMed Web of Science ↵ Boron WF , Endeward V , Gros G , Musa-Aziz R & Pohl P ( 2011 ). Intrinsic CO₂ permeability of cell membranes and potential biological relevance of CO₂ channels . ChemPhysChem 12 , 1017 – 1019 . OpenUrl CrossRef PubMed Web of Science ↵ Clark A , Federspiel WJ , Clark PA & Cokelet GR ( 1985 ). Oxygen delivery from red cells . Biophys J 47 , 171 – 181 . OpenUrl CrossRef PubMed Web of Science ↵ Coin JT & Olson JS ( 1979 ). The rate of oxygen uptake by human red blood cells . J Biol Chem 254 , 1178 – 1190 . OpenUrl Abstract / FREE Full Text ↵ Cooper GJ & Boron WF ( 1998 ). Effect of pCMBS on CO₂ permeability of Xenopus oocytes expressing aquaporin 1 or its C189S mutant . Am J Physiol 275 , C1481 – 1486 . OpenUrl ↵ Cooper GJ , Occhipinti R & Boron WF ( 2015 ). CrossTalk proposal: Physiological CO₂ exchange can depend on membrane channels . J Physiol ; DOI: 10.1113/JP270059 . OpenUrl CrossRef ↵ Deonikar P , Abu-Soud HM & Kavdia M ( 2014 ). Computational analysis of nitric oxide biotransport to red blood cell in the presence of free hemoglobin and NO donor . Microvasc Res 95 , 15 – 25 . OpenUrl CrossRef PubMed ↵ Deonikar P & Kavdia M ( 2013 ). Contribution of membrane permeability and unstirred layer diffusion to nitric oxide-red blood cell interaction . J Theor Biol 317 , 321 – 330 . OpenUrl CrossRef PubMed ↵ Endeward V ( 2012 ). The rate of the deoxygenation reaction limits myoglobin- and hemoglobin-facilitated O₂ diffusion in cells . J Appl Physiol 112 , 1466 – 1473 . OpenUrl CrossRef PubMed ↵ Endeward V , Al-Samir S , Itel F & Gros G ( 2014 ). How does carbon dioxide permeate cell membranes? A discussion of concepts, results and methods . Front Physiol 4 , 382 . OpenUrl CrossRef PubMed ↵ Endeward V , Musa-Aziz R , Cooper GJ , Chen L-M , Pelletier MF , Virkki LV , Supuran CT , King LS , Boron WF & Gros G ( 2006 ). Evidence that aquaporin 1 is a major pathway for CO₂ transport across the human erythrocyte membrane . FASEB J 20 , 1974 – 1981 . OpenUrl CrossRef PubMed Web of Science ↵ Fick A ( 1855 ). V. On liquid diffusion . Lond Edinb Dubl Phil Mag 10 , 30 – 39 . OpenUrl ↵ Forster RE ( 1964 ). Rate of gas uptake by red cells . In Handbook of Physiology, pp. 827–837 . American Physiological Society . Forstner H & Gnaiger E ( 1983 ). Appendix A. Calculation of equilibrium oxygen concentration. In Polarographic oxygen sensors: aquatic and physiological applications . Springer-Verlag , Berlin Heidelberg New York . ↵ Geers C & Gros G ( 2000 ). Carbon dioxide transport and carbonic anhydrase in blood and muscle . Physiol Rev 80 , 681 – 715 . OpenUrl CrossRef PubMed Web of Science ↵ Gibson QH , Kreuzer F , Meda E & Roughton FJ ( 1955 ). The kinetics of human haemoglobin in solution and in the red cell at 37 degrees C . J Physiol (Lond ) 129 , 65 – 89 . OpenUrl CrossRef PubMed Web of Science ↵ Graham T ( 1829 ). Notice of the singular inflation of a bladder. Quart J Sci II , 88 – 89 . ↵ Graham T ( 1833 ). XXVII. On the law of the diffusion of gases . Lond Edinb Dubl Phil Mag 2 , 175 – 190 . OpenUrl ↵ Han P & Bartels DM ( 1996 ). Temperature Dependence of Oxygen Diffusion in H₂O and D₂O . J Phys Chem 100 , 5597 – 5602 . OpenUrl CrossRef Web of Science ↵ Hartridge H & Roughton FJ ( 1927 ). The rate of distribution of dissolved gases between the red blood corpuscle and its fluid environment: Part I . Preliminary experiments on the rate of uptake of oxygen and carbon monoxide by sheep’s corpuscles. J Physiol (Lond ) 62 , 232 – 242 . OpenUrl PubMed ↵ Hess WR ( 1930 ). Die regulierung des Blutkreislaufes: gleichzeitig ein Beitrag zur Physiologie des vegetativen Nervensystemes . Thieme . ↵ Holland RA , Shibata H , Scheid P & Piiper J ( 1985 ). Kinetics of O₂ uptake and release by red cells in stopped-flow apparatus: effects of unstirred layer . Respir Physiol 59 , 71 – 91 . OpenUrl CrossRef PubMed Web of Science ↵ Huffman DE , Occhipinti R , Somersalo E , Calvetti D & Boron WF ( 2025 ). A graphical user interface for “Role of channels in the O2 permeability of murine red blood cells III. Mathematical modeling.” Available at : doi: 10.5281/zenodo.14932553 . OpenUrl CrossRef ↵ Hughes JMB & Bates DV ( 2003 ). Historical review: the carbon monoxide diffusing capacity (DLCO) and its membrane (DM) and red cell (Theta . Vc) components. Respir Physiol Neurobiol 138 , 115 – 142 . OpenUrl CrossRef PubMed ↵ Huxley VH & Kutchai H ( 1981 ). The effect of the red cell membrane and a diffusion boundary layer on the rate of oxygen uptake by human erythrocytes . J Physiol (Lond ) 316 , 75 – 83 . OpenUrl CrossRef PubMed Web of Science ↵ Ivanov II , Fedorov GE , Gus’kova RA , Ivanov KI & Rubin AB ( 2004 ). Permeability of lipid membranes to dioxygen . Biochem Biophys Res Commun 322 , 746 – 750 . OpenUrl CrossRef PubMed ↵ Kreuzer F ( 1982 ). Oxygen supply to tissues: the Krogh model and its assumptions . Experientia 38 , 1415 – 1426 . OpenUrl CrossRef PubMed Web of Science ↵ Krogh A ( 1919 ). The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissue . J Physiol (Lond ) 52 , 409 – 415 . OpenUrl CrossRef PubMed Web of Science ↵ Kutchai H ( 1975 ). Role of the red cell membrane in oxygen uptake . Respir Physiol 23 , 121 – 132 . OpenUrl CrossRef PubMed Web of Science ↵ Lawson WH , Holland RA & Forster RE ( 1965 ). Effect of temperature on deoxygenation rate of human red cells . J Appl Physiol 20 , 912 – 918 . OpenUrl CrossRef PubMed ↵ Michenkova M , Taki S , Blosser MC , Hwang HJ , Kowatz T , Moss FJ , Occhipinti R , Qin X , Sen S , Shinn E , Wang D-K , Zeise B , Zhao P , Malmstadt N , Vahedi-Faridi A , Tajkhorshid E & Boron WF ( 2021 ). Carbon dioxide transport across membranes . Interface Focus 11 , 2 . OpenUrl ↵ Mitchell JK ( 1830 ). On the penetrativeness of fluids . Am J Med Sci 7 , 36 – 67 . OpenUrl CrossRef ↵ Mitchell JK ( 1833 ). Art. IX. On the penetration of gases . Am J Med Sci 13 , 100 – 112 . OpenUrl CrossRef ↵ Moll W ( 1968 ). The influence of hemoglobin diffusion on oxygen uptake and release by red cells . Respir Physiol 6 , 1 – 15 . OpenUrl CrossRef PubMed Web of Science ↵ Moss FJ , Zhao P , Salameh AI , Taki S , Wass AB , Jacobberger JW , Huffman DE , Meyerson HJ , Occhipinti R & Boron WF ( 2025 ). Role of channels in the O₂ permeability of murine red blood cells II. Morphological and proteomic studies . bioRxiv ; DOI: 10.1101/2025.03.05.639962 . OpenUrl Abstract / FREE Full Text ↵ Nakhoul NL , Davis BA , Romero MF & Boron WF ( 1998 ). Effect of expressing the water channel aquaporin-1 on the CO₂ permeability of Xenopus oocytes . Am J Physiol 274 , C543 – 548 . OpenUrl ↵ Nicolson P & Roughton FJW ( 1951 ). A theoretical study of the influence of diffusion and chemical reaction velocity on the rate of exchange of carbon monoxide and oxygen between the red blood corpuscle and the surrounding fluid . Proc R Soc Lond, B, Biol Sci 138 , 241 – 264 . OpenUrl CrossRef ↵ Overton E ( 1895 ). Über die osmotischen Eigenschaften der Zelle in ihrer Bedeutung für die Toxikologie und Pharmacologie . Z Phys Chem 22 , 189 – 209 . OpenUrl ↵ Overton E ( 1899 ). Ueber die allgemeinen osmotischen Eigenschaften der Zelle, ihre vermutlichen Ursachen und ihre Bedeutung für die Physiologie . Vierteljahrsschr Naturforsch Ges Zürich 44 , 88 – 135 . OpenUrl ↵ Ponder E ( 1948 ). Hemolysis and Related Phenomena . New York : Grune & Stratton . Available at: http://hdl.handle.net/2027/wu.89047892849 . ↵ Popel AS ( 1989 ). Theory of oxygen transport to tissue . Crit Rev Biomed Eng 17 , 257 – 321 . OpenUrl PubMed Web of Science ↵ Richardson SL , Hulikova A , Proven M , Hipkiss R , Akanni M , Roy NBA & Swietach P ( 2020 ). Single-cell O₂ exchange imaging shows that cytoplasmic diffusion is a dominant barrier to efficient gas transport in red blood cells . PNAS 117 , 10067 – 10078 . OpenUrl Abstract / FREE Full Text Roughton FJ , Adair GS , Barcroft J , Goldschmidt G , Herkel W , Hill RM , Keys AB & Ray GB ( 1936 ). The thermochemistry of the oxygen-haemoglobin reaction: Comparison of the heat as measured directly on purified haemoglobin with that calculated indirectly by the Van’t Hoff Isochore . Biochem J 30 , 2117 – 2133 . OpenUrl FREE Full Text ↵ Roughton FJW ( 1932 ). Diffusion and chemical reaction velocity as joint factors in determining the rate of uptake of oxygen and carbon monoxide by the red blood corpuscle . Proc R Soc Lond B 111 , 1 – 36 . OpenUrl CrossRef ↵ Roughton FJW ( 1964 ). Transport of oxygen and carbon dioxide . In Handbook of Physiology, pp. 767–825 . American Physiological Society . ↵ Somersalo E , Occhipinti R , Boron WF & Calvetti D ( 2012 ). A reaction-diffusion model of CO 2 influx into an oocyte . J Theor Biol 309 , 185 – 203 . OpenUrl CrossRef PubMed Web of Science ↵ Stannett V ( 1978 ). The transport of gases in synthetic polymeric membranes—an historic perspective . J Membrane Sci 3 , 97 – 115 . OpenUrl CrossRef ↵ Strutwolf J , Zhang J , Barker AL & Unwin PR ( 2001 ). Effect of phospholipids on the kinetics of dioxygen transfer across a 1,2-dichloroethane/water interface . Phys Chem Chem Phys 3 , 5553 – 5558 . OpenUrl CrossRef ↵ Vandegriff KD & Olson JS ( 1984 a ). A quantitative description in three dimensions of oxygen uptake by human red blood cells . Biophys J 45 , 825 – 835 . OpenUrl CrossRef PubMed Web of Science ↵ Vandegriff KD & Olson JS ( 1984 b ). The kinetics of O 2 release by human red blood cells in the presence of external sodium dithionite . J Biol Chem 259 , 12609 – 12618 . OpenUrl Abstract / FREE Full Text ↵ Waisbren SJ , Geibel JP , Modlin IM & Boron WF ( 1994 ). Unusual permeability properties of gastric gland cells . Nature 368 , 332 – 335 . OpenUrl CrossRef PubMed ↵ Williams JB & Kutchai H ( 1986 ). Use of a membrane-bound fluorophore to characterize diffusion boundary layers around human erythrocytes . Biophys J 49 , 453 – 458 . OpenUrl CrossRef PubMed ↵ Wroblewski S ( 1879 ). On the nature of the absorption of gases . Nature 21 , 190 – 192 . OpenUrl CrossRef ↵ Yamaguchi K , Nguyen-Phu D , Scheid P & Piiper J ( 1985 ). Kinetics of O2 uptake and release by human erythrocytes studied by a stopped-flow technique . J Appl Physiol (1985) 58 , 1215 – 1224 . OpenUrl CrossRef PubMed Web of Science ↵ Yap EW & Hellums JD ( 1987 ). Use of Adair Four-Step Kinetics in Mathematical Simulation of Oxygen Transport in the Microcirculation. In Oxygen Transport to Tissue IX, Advances in Experimental Medicine and Biology , pp. 193 – 207 . Springer, Boston, MA. Available at: https://link.springer.com/chapter/10.1007/978-1-4684-7433-6_22 [Accessed May 18, 2018]. ↵ Zhao P , Moss FJ , Occhipinti R , Geyer RR , Huffman DE , Meyerson HJ & Boron W F ( 2025 ). Role of channels in the O₂ permeability of murine red blood cells. I. Stopped-flow and hematological studies . bioRxiv ; DOI: 10.1101/2025.03.05.639948 . 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