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Mathematical Models Estimate Cell Membrane Permeability

By Erkki Somersalo

Gas transport across cell membranes is a key factor that sustains life. As oxygen (O2) molecules journey from inhaled air to cell respiration, they must first pass through the alveolar membrane to reach the capillaries. The molecules next enter the erythrocytes, where hemoglobin binds them for transport via blood flow. Bound O2 is then released into the plasma prior to reaching the tissue. After crossing the capillary walls and reaching the extracellular space, this O2 passes both the cell membrane and—once inside the cell—the mitochondrial outer membrane before finally partaking in oxidative phosphorylation: the main mechanism that produces the ATP that maintains vital functions in an organism. 

Figure 1. Mechanism of pH control across the cell membrane based on gas transport across the membrane. Figure courtesy of Daniela Calvetti and Erkki Somersalo.
Gas transport also plays a central role in the regulation of pH, which in turn affects a number of vital functions like O2 transport by erythrocytes. The regulation of intra- and extracellular pH depends on sophisticated mechanisms that maintain the balance of the proton concentration. The passage of carbon dioxide across the cell membrane effectively acts as a shuttle for protons and controls pH balance (see Figure 1).

Understanding cell membranes’ permeability to gases such as O2 and carbon dioxide (CO2) is an important component of cell physiology. In addition to simple diffusion through cell membranes, specified gas channels that facilitate travel through the membranes help explain the passage of gases [2, 3]. Most notable among the putative gas passage facilitators are aquaporins — proteins that serve as channels for water molecules through the phospholipid bilayer. This discovery earned Peter Agre the 2003 Nobel Prize in Chemistry

How can we measure permeability? One possibility is to do so in a roundabout way via pH control by CO2 transport. Researchers routinely measure pH on cell surfaces with special pH electrodes. We can therefore write a mathematical model that expresses pH’s dependency on the CO2 concentration near the membrane, which involves membrane permeability to CO2 (see Figure 2). Estimation of the membrane permeability from the measured pH is then reduced to the solution of the corresponding inverse problem.

Figure 2. Surface pH is measured via a liquid-membrane pH-sensitive electrode. The pH is recorded in the domain under the electrode tip. Figure courtesy of Daniela Calvetti and Erkki Somersalo.
Fortunately, writing a computational model for the dynamic evolution of pH in a protocol wherein a cell is placed in a CO2-rich environment is rather straightforward. But this is where the trouble begins. Comparison of model predictions with measured pH curves indicates that while the model qualitatively predicts the pH behavior, the computer values are much lower than the actual measurements. One culprit is the enzyme carbonic anhydrase (CA), which speeds up the hydration of CO2 and forms carbonic acid that dissociates to bicarbonate and proton. Although adding membrane-bound CA to the model does help [4], model predictions are still significantly different from measured data; better ideas are thus needed. Could the reason behind the discrepancy be related to the measurement apparatus?

Quantum physics has taught us that the observer is always part of nature, and one cannot observe nature without interfering in it. The same principle may also be true at the cell-level scale. When the pH electrode is pushed against the cell membrane, it creates a tiny pocket with its own microenvironment (see Figure 3). Therefore, while the forward model may be accurate in the context of free cell membranes, this is no longer true for the microenvironment in which the pH electrode takes its readings. Once we assess the reaction-diffusion model for the electrode-induced microenvironment using finite elements—and state-of-the-art multiscale solvers to calculate predictions for the extremely stiff and large model—the model pH predictions are in quantitative agreement with the measured data [1]. Yet now another problem materializes.

Figure 3. Pushing the electrode against the cell membrane causes a dimple, and the partial clamping by the rim of the electrode tip may create a pocket with its own chemical dynamics. Figure courtesy of Daniela Calvetti and Erkki Somersalo.
To solve the inverse problem that estimates membrane permeability, we must repeatedly solve the forward model — particularly if Bayesian statistical methods like Markov chain Monte Carlo (MCMC) are used. Unfortunately, the fine-scale model that captures the electrode microenvironment depends on several new unknown parameters that require estimation. Moreover, the problem’s multiscale nature means that each run of the forward model requires hours of computing time. The solution is to find a fast approximate model (or a proxy) that effectively emulates the detailed, large-scale solver but requires significantly less time to solve. We used ideas from compartment models (see Figure 3) to develop an effective one-dimensional model and estimate the unknown parameters using MCMC methods. Properly addressing model discrepancy that stems from model reduction is both the next challenge and topic of our current research.


Erkki Somersalo presented this research during a minisymposium presentation at the 2021 SIAM Conference on Computational Science and Engineering, which took place virtually in March 2021.

References
[1] Calvetti, D., Prezioso, J., Occhipinti, R., Boron, W.F., & Somersalo, E. (2020). Computational model of electrode-induced microenvironmental effects on pH measurements near a cell membrane. Multiscale Model. Sim., 18(2), 1053-1075.
[2] Endeward, V., Musa-Aziz, R., Cooper, G.J., Chen, L.M., Pelletier, M.F., Virkki, L.V., …, Gros, G. (2006). Evidence that aquaporin 1 is a major pathway for CO2 transport across the human erythrocyte membrane. FASEB J., 20(12), 1974-1981.
[3] Nakhoul, N.L., Davis, B.A., Romero, M.F., & Boron, W.F. (1998). Effect of expressing the water channel aquaporin-1 on the CO2 permeability of Xenopus oocytes. Amer. J. Physiol., 274(2), C543-C548.
[4] Somersalo, E., Occhipinti, R., Boron, W.F., & Calvetti, D. (2012). A reaction-diffusion model of CO2 influx into an oocyte. J. Theor. Biol., 309, 185-203.

Erkki Somersalo is a professor of applied mathematics at Case Western Reserve University. His areas of interest include computational inverse problems and Bayesian methods, uncertainty quantification, biomedical imaging, and mathematical modeling in the life sciences. 
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