# Diffusion coefficient of cells in blood?

What's the diffusion coefficient of white cells in blood? Is it well defined, or are cells too large and few as to be treated as particles in this context?

P.S. I have tried to look this up, but what I find inevitably is about diffusion coefficients of molecules inside cells, which is not what I want.

Cells absolutely can be considered as diffusing objects. However, the origin of the "diffusion" can be very different than for, say, a bead in water. The reason is that the thermal motion that creates the diffusion of a micron-sized bead can be much less important for a (large) cell. For instance, the diffusion coefficient due to thermal forces of a sphere in a fluid with viscosity $$eta$$ is $$D_{therm} = frac{k_B T}{6 pi eta R}$$ where $$R$$ is the cell radius (this is the "Stokes-Einstein" result). For a white blood cell (say 10 microns in radius), this would - assuming blood is at least as viscous as water - lead to thermal diffusion coefficients around 1 microns$$^2$$/hour- i.e. we would need many many hours for a cell to travel its own diameter. This result neglects a bunch of complexities - blood is not perfectly viscous, there is a really high density of red blood cells in blood, etc… but the basic result is that the thermal buffeting that drives the diffusion of a colloidal bead is not enough to drive a cell.

However, cells are not passive beads! They crawl, both randomly, and (as cancerconnector notes), in response to signals like chemical gradients. If a cell crawls with a speed $$v$$ and maintains its orientation over a time $$au$$, it's reasonable to describe its motion (over times much longer than $$au$$ as being diffusive with an effective diffusion coefficient $$D_{motility} = frac{1}{2} v^2 au$$ which you can get mostly from dimensional analysis, or more rigorous models. For white blood cells, $$v$$ is on the order of microns/minute - at least when measured on slides - it might be different in the body. $$au$$ is a few minutes, so $$D_{motility}$$ is on the order of many microns$$^2$$/minute - orders of magnitude faster than mere thermal diffusion.

A good introduction to modeling of this question is probably chapters 9-11 of Leah Edelstein Keshet's "Mathematical Models in Biology" and she cites some classic Lauffenberger papers on macrophage response, etc. You may have access to this online: http://dx.doi.org/10.1137/1.9780898719147

Also, note that everything I've said so far neglects blood flow - if this is important in your problem (which will depend on the context), things can get more complex (have to include flow in your diffusion equation). But using diffusion equations to model the spread of randomly motile cells is a very common approach, and there is a lot of literature on it!

This is a difficult question because white blood cells do exhibit chemo and haptotactic responses, and also because of their large size - though cells are often modeled as moving by diffusion (cancer cells in particular, see: "Modelling biological invasions: Individual to population scales at interfaces" as one of MANY examples). One could model their movement in a dish, and come up with a diffusion coefficient, but I'm not sure what the utility of this would be. Why do you want to know? Is there another underlying question?

## Tracer diffusion coefficients of oxyhemoglobin a and oxyhemoglobin s in blood cells as determined by pulsed field gradient NMR

It is demonstrated that tracer diffusion coefficients can be determined for oxyhemoglobin A (HbA-O2) and oxyhemoglobin S (HbS-O2) in intact blood cells by means of pulsed field gradient NMR (PFG-NMR). This is possible because the method discriminates between both rapidly moving water molecules and molecules having small proton transverse relaxation times (T2). The results indicate that only hemoglobin molecules contribute to the echo signals when large field gradients are used. The dependence of the measured diffusion coefficients on osmolarity and pH are attributed to changes in hemoglobin concentration resulting from changes in cell volume.

## Membrane fluidity of blood cells

Plasma membranes are fluid structures and the maintenance of fluidity is a prerequisite for function, viability, growth and reproduction of cells. Membrane fluidity is the reciprocal of membrane microviscosity, which in turn is inversely proportional to rotational and lateral diffusion rates of membrane components. In the absence of constraints most lipids and unrestrained integral proteins freely diffuse in the plane of the membrane with high diffusion coefficients. The fluid mosaic model of plasma membrane structure is essentially still valid but this model is by its nature a macroscopic one. At present, attention is focused on molecular structural details of protein-lipid interactions and on the static and dynamic structure of membrane proteins. Highly potent new macroscopic and microscopic methods have been developed to measure translational diffusion of membrane lipids and proteins. The microscopic methods can reveal diffusion via encounters between labeled molecules. Fluorescence anisotropy measurements are the most widely used techniques in biological research. The use of different permeant and non-permeant fluorophores have contributed much to a better understanding of the changes in the ordered states and motional freedom of the membrane phospholipids in different cells during development, aging and physiological functions as well as in pathological conditions. The application of fluorophores with non-random distribution have shed light on the asymmetrical changes between the outer and inner domain of the lipid bilayer and on the dynamics of 'flip-flop' in signal transduction. Membrane fluidity was shown to have a decisive role in the efficiency of ligand binding, in the outcome of direct cell to cell contacts and in the modulation of the activity of membrane enzymes. Cell filtrability reflects whole cell viscosity that can not always be correlated with the fine changes in membrane fluidity. Cell viscosity depends inter alia on the size and shape of the cells as well as on membrane rigidity. In contrast to this, membrane fluidity is only dependent on the freedom of mobility of the membrane constituents. Increased release of free radicals and reactive oxygen specie (ROS) affect membrane fluidity, cellular Ca2+ homeostasis, induce lipid peroxidation and finally cell death. Investigation of membrane fluidity proved to be a useful and sensitive additional method to obtain a better insight into the mechanisms by which different compounds, drugs and contact with foreign surfaces are affecting cellular functions. The measurements of membrane fluidity may gain more widespread use for monitoring the safety and efficacy of these actions. During the last few years, changes in membrane fluidity of blood cells have been reported during development and aging and as a result of physiological cell functions. Membrane fluidity changes have been described in thrombocythaemia, hyperlipidaemia, hypercholesterolaemia, hypertension, diabetes mellitus, obesity, septic conditions and in allergic and burnt patients, in alcoholics, in Alzheimer's disease and in schizophrenia. A short summary is given on red cell membrane fluidity changes in a Hungarian triosephosphate isomerase (TPI)-deficient family, reflecting how the very subtle changes in membrane fluidity can help to establish underlying biological differences between the clinical phenotypes of a severe enzyme (TPI) deficiency caused by the defect of a single gene in two brothers one with and one without neurological symptoms.

## MATERIALS AND METHODS

The Supplemental Material includes details of the experimental protocols. In brief, we performed confocal imaging and FCS measurements using an LSM 510 Meta/Confocor 2 apparatus (Carl Zeiss, Jena, Germany) with standard configurations. FCS has been reviewed previously (Elson, 2001 Rigler and Elson, 2001 Marguet et al., 2006 Haustein and Schwille, 2007). We microinjected the micelles using an InjectMan NI2 with FemtoJet pump from Eppendorf, typically mounted on an Axiovert 200M microscope (Carl Zeiss) fitted with a long working distance 40× phase contrast objective.

We obtained similar results with three different types of Bodipy TMR-phosphatidylinositol 4,5-biphosphate [PI(4,5)P2], by using three different types of micelles, as discussed in Supplemental Material. One major advantage of FCS over fluorescence recovery after photobleaching (FRAP) is that the fluorescent PIP2 we introduce into the membrane does not perturb significantly the endogenous level of PIP2 in the plasma membrane. Specifically, the illuminated (∼0.2-μm radius) portion of the plasma membrane contains ∼10 5 lipids and ∼10 3 PIP2, and our technique incorporated 1–100 fluorescent PIP2 into this area.

We made most of our measurements on Rat1 fibroblasts or HEK293 epithelial cells, but we also report data from FCS measurements on REF52, Cos1, NIH3T3 fibroblasts, and FRT epithelial cells. The diffusion coefficient of PIP2 on the inner leaflet did not differ significantly between the cell types or lines, and we pooled all data. All measurements were made at 25°C.

## Contents

Diffusion is of fundamental importance in many disciplines of physics, chemistry, and biology. Some example applications of diffusion:

to produce solid materials (powder metallurgy, production of ceramics) design design in chemical industry can be diffused (e.g., with carbon or nitrogen) to modify its properties during production of semiconductors.

Diffusion is part of the transport phenomena. Of mass transport mechanisms, molecular diffusion is known as a slower one.

### Biology Edit

In cell biology, diffusion is a main form of transport for necessary materials such as amino acids within cells. [1] Diffusion of solvents, such as water, through a semipermeable membrane is classified as osmosis.

Metabolism and respiration rely in part upon diffusion in addition to bulk or active processes. For example, in the alveoli of mammalian lungs, due to differences in partial pressures across the alveolar-capillary membrane, oxygen diffuses into the blood and carbon dioxide diffuses out. Lungs contain a large surface area to facilitate this gas exchange process.

Fundamentally, two types of diffusion are distinguished:

• Tracer diffusion and Self-diffusion, which is a spontaneous mixing of molecules taking place in the absence of concentration (or chemical potential) gradient. This type of diffusion can be followed using isotopic tracers, hence the name. The tracer diffusion is usually assumed to be identical to self-diffusion (assuming no significant isotopic effect). This diffusion can take place under equilibrium. An excellent method for the measurement of self-diffusion coefficients is pulsed field gradient (PFG) NMR, where no isotopic tracers are needed. In a so-called NMR spin echo experiment this technique uses the nuclear spin precession phase, allowing to distinguish chemically and physically completely identical species e.g. in the liquid phase, as for example water molecules within liquid water. The self-diffusion coefficient of water has been experimentally determined with high accuracy and thus serves often as a reference value for measurements on other liquids. The self-diffusion coefficient of neat water is: 2.299·10 −9 m²·s −1 at 25 °C and 1.261·10 −9 m²·s −1 at 4 °C. [2]
• Chemical diffusion occurs in a presence of concentration (or chemical potential) gradient and it results in net transport of mass. This is the process described by the diffusion equation. This diffusion is always a non-equilibrium process, increases the system entropy, and brings the system closer to equilibrium.

The diffusion coefficients for these two types of diffusion are generally different because the diffusion coefficient for chemical diffusion is binary and it includes the effects due to the correlation of the movement of the different diffusing species.

Because chemical diffusion is a net transport process, the system in which it takes place is not an equilibrium system (i.e. it is not at rest yet). Many results in classical thermodynamics are not easily applied to non-equilibrium systems. However, there sometimes occur so-called quasi-steady states, where the diffusion process does not change in time, where classical results may locally apply. As the name suggests, this process is a not a true equilibrium since the system is still evolving.

Non-equilibrium fluid systems can be successfully modeled with Landau-Lifshitz fluctuating hydrodynamics. In this theoretical framework, diffusion is due to fluctuations whose dimensions range from the molecular scale to the macroscopic scale. [3]

Chemical diffusion increases the entropy of a system, i.e. diffusion is a spontaneous and irreversible process. Particles can spread out by diffusion, but will not spontaneously re-order themselves (absent changes to the system, assuming no creation of new chemical bonds, and absent external forces acting on the particle).

Collective diffusion is the diffusion of a large number of particles, most often within a solvent.

Contrary to brownian motion, which is the diffusion of a single particle, interactions between particles may have to be considered, unless the particles form an ideal mix with their solvent (ideal mix conditions correspond to the case where the interactions between the solvent and particles are identical to the interactions between particles and the interactions between solvent molecules in this case, the particles do not interact when inside the solvent).

In case of an ideal mix, the particle diffusion equation holds true and the diffusion coefficient D the speed of diffusion in the particle diffusion equation is independent of particle concentration. In other cases, resulting interactions between particles within the solvent will account for the following effects:

• the diffusion coefficient D in the particle diffusion equation becomes dependent of concentration. For an attractive interaction between particles, the diffusion coefficient tends to decrease as concentration increases. For a repulsive interaction between particles, the diffusion coefficient tends to increase as concentration increases.
• In the case of an attractive interaction between particles, particles exhibit a tendency to coalesce and form clusters if their concentration lies above a certain threshold. This is equivalent to a precipitation chemical reaction (and if the considered diffusing particles are chemical molecules in solution, then it is a precipitation).

Transport of material in stagnant fluid or across streamlines of a fluid in a laminar flow occurs by molecular diffusion. Two adjacent compartments separated by a partition, containing pure gases A or B may be envisaged. Random movement of all molecules occurs so that after a period molecules are found remote from their original positions. If the partition is removed, some molecules of A move towards the region occupied by B, their number depends on the number of molecules at the region considered. Concurrently, molecules of B diffuse toward regimens formerly occupied by pure A. Finally, complete mixing occurs. Before this point in time, a gradual variation in the concentration of A occurs along an axis, designated x, which joins the original compartments. This variation, expressed mathematically as -dCA/dx, where CA is the concentration of A. The negative sign arises because the concentration of A decreases as the distance x increases. Similarly, the variation in the concentration of gas B is -dCB/dx. The rate of diffusion of A, NA, depend on concentration gradient and the average velocity with which the molecules of A moves in the x direction. This relationship is expressed by Fick's Law

where D is the diffusivity of A through B, proportional to the average molecular velocity and, therefore dependent on the temperature and pressure of gases. The rate of diffusion NA, is usually expressed as the number of moles diffusing across unit area in unit time. As with the basic equation of heat transfer, this indicates that the rate of force is directly proportional to the driving force, which is the concentration gradient.

This basic equation applies to a number of situations. Restricting discussion exclusively to steady state conditions, in which neither dCA/dx or dCB/dx change with time, equimolecular counterdiffusion is considered first.

If no bulk flow occurs in an element of length dx, the rates of diffusion of two ideal gases (of similar molar volume) A and B must be equal and opposite, that is N A = − N B > .

The partial pressure of A changes by dPA over the distance dx. Similarly, the partial pressure of B changes dPB. As there is no difference in total pressure across the element (no bulk flow), we have

For an ideal gas the partial pressure is related to the molar concentration by the relation

where nA is the number of moles of gas A in a volume V. As the molar concentration CA is equal to nA/ V therefore

where DAB is the diffusivity of A in B. Similarly,

Considering that dPA/dx=-dPB/dx, it therefore proves that DAB=DBA=D. If the partial pressure of A at x1 is PA1 and x2 is PA2, integration of above equation,

## Evaluating diffusion-dependent processes in muscle

### Aerobic metabolism

Measurements of D in muscle as described above, along with microscopic analyses of diffusion distances and measurements of the rates of metabolic processes, have been used to quantitatively evaluate the reaction–diffusion processes in skeletal muscle shown in Fig. 1. By far, the most widely studied is aerobic metabolism, which depends on the diffusion of O2 to the mitochondria and the subsequent diffusion of ATP to sites of utilization in the fiber (Fig. 1A,B). A comprehensive review of the literature on the role of diffusion in aerobic metabolism is beyond the scope of this paper, so we will briefly focus on a few examples and some of the general conclusions.

The pioneering work of August Krogh and A. V. Hill provided equations that are still used to describe concentration profiles of O2 (Krogh, 1919) and high-energy phosphate molecules (Hill, 1965) in muscle. Mainwood and Rakusan (Mainwood and Rakusan, 1982) applied these equations to show that clustering of mitochondria near capillaries and the presence of a near-equilibrium CK reaction led to a smaller decrease in PO2 across the cell and less steep gradients for PCr, ATP and ADP, and therefore helped preserve the free energy of ATP hydrolysis, ΔG, across the fiber. This provided the first quantitative demonstration that the distribution of mitochondria in skeletal muscle is influenced by diffusion constraints (see below). A number of more elaborate mathematical models of O2 flux and metabolism have been developed for skeletal muscle (e.g. Federspiel, 1986 Groebe, 1995 Hoofd and Egginton, 1997 Piiper, 2000 Lai et al., 2007 Dash et al., 2008), and there are a number of reviews of aspects of O2 and aerobic substrate transport to mitochondria in muscle (e.g. Hoppeler and Weibel, 1998 Wagner, 2000 Suarez, 2003 Weibel and Hoppeler, 2004). Some conclusions from past work are that (1) control of O2 flux to, and usage by, mitochondria is shared among the various steps in the O2 cascade, (2) a substantial decrease in PO2 occurs between the capillary and the fiber, and (3) intracellular O2 gradients may be present. Thus, the rate of O2 diffusion into and across the fiber to the mitochondria may influence muscle structure and function. The intracellular O2 concentration gradients that are necessary for diffusion control of aerobic metabolism are notoriously difficult to demonstrate experimentally. However, spatial gradients in the redox state of isolated frog skeletal muscle fibers suggest that O2 gradients may influence rates of oxidative phosphorylation even under conditions of high extracellular PO2 (Hogan et al., 2005). Further, measurements of O2 consumption rates and force production in isolated frog skeletal muscle fibers and rat myocardial trabeculae suggest that maximal respiration rates cannot be attained in vivo because O2 diffusive flux is insufficient to prevent anoxia in the fiber core (van der Laarse et al., 2005).

The wide interest in intracellular O2 transport has prompted many studies of Mb, as it is the final mediator of O2 flux to the mitochondria. Mb is a small intracellular oxygen-binding heme protein that is found in aerobic fibers, and is thought to function primarily in temporal buffering of PO2 and in facilitated transport of O2 to mitochondria (reviewed in Conley et al., 2000 Jurgens et al., 2000 Wittenberg and Wittenberg, 2003 Ordway and Garry, 2004). While its role as a temporal buffer is generally accepted, the importance of the related facilitated diffusion function has been questioned based on reaction–diffusion analyses that incorporated the relatively low D for Mb in muscle fibers (e.g. Jurgens et al., 2000 Lin et al., 2007). However, Mb-facilitated diffusion becomes more important in skeletal muscle at low PO2 (Lin et al., 2007). Compensatory responses in the cardiovascular system that enhance O2 delivery in Mb knock-out mice first reported by Gödecke and colleagues seem to support an O2 transport role for Mb (Gödecke et al., 1999). However, Mb also has nitric oxide (NO) oxygenase activity (Flögel et al., 2001), and it has been proposed that the ‘naturally occurring genetic knockout’ of Mb and Hb in Antarctic icefishes leads to cardiovascular compensations that stem from high levels of NO, rather than from reduced O2 delivery per se (Sidell and O'Brien, 2006). Thus, while a facilitated diffusion function is an unavoidable consequence of the free diffusion of Mb and rapid reversible binding to O2, the relative importance of this transport role remains the source of debate.

The diffusion of ATP from the mitochondria to cellular ATPases has also been a subject of contention. Bessman and Geiger (Bessman and Geiger, 1981) originally proposed the ‘PCr shuttle’ to explain ATP delivery from the mitochondria to cellular ATPases (Fig. 1B), where the bulk of ATP-equivalent transport occurred via PCr diffusion, rather than directly as ATP. Central tenets of this idea as it has evolved are that the mitochondrial outer membrane is a barrier for ATP/ADP diffusion, but not for PCr/Cr diffusion (although this notion has been disputed) (see Kongas et al., 2004), and that ATP produced in the mitochondria passes directly from the adenine nucleotide translocator in the inner membrane to the mitochondrial form of CK. Meyer and colleagues provided an opposing view using a simple facilitated diffusion model (akin to that used to assess Mb function) to evaluate ATP-equivalent diffusive flux (Meyer et al., 1984) and concluded that (1) most ATP-equivalent diffusion should occur in the form of PCr because of its higher concentration and higher D, and (2) there are minimal concentration gradients of high energy phosphates in muscle. There has been a great deal of experimental and modeling work since these early studies that has characterized so-called phosphotransfer networks like the CK system with respect to enzyme localization, channeling of substrates, and restricted diffusion in mammalian skeletal and cardiac muscle, and a number of contrasting reviews are available (e.g. Walliman et al., 1992 Dzeja and Terzic, 2003 Saks et al., 2008 Beard and Kushmerick, 2009). While the function of the CK system is largely viewed through the lens of the mammalian cardiomyocyte, Ellington provides a review of the evolution and function of the entire family of phosphagen kinases that are found in animals (Ellington, 2001).

The role of diffusion in responses of fish muscle aerobic metabolism to cold has been the focus of study because both catalytic and diffusive processes are slower at lower temperature. Muscle from cold-acclimated or -adapted fishes typically has a high mitochondrial volume density and high lipid composition, compared with that in warm water species. The high mitochondrial content is thought to not only offset the reduced reaction rates at colder temperatures but also shorten diffusion distances between mitochondria (e.g. Johnston, 1982 Egginton and Sidell, 1989) (reviewed in Sidell, 1998). Hubley and colleagues used a reaction–diffusion model of high-energy phosphates to evaluate metabolism as a function of temperature in red and white goldfish muscle, and concluded that diffusion constraints of these molecules were not the primary cause of mitochondrial proliferation in the cold (Hubley et al., 1997). However, the higher lipid content in cold water fishes, including the lipid membranes in the abundant mitochondria, likely aid diffusion of O2 because of its higher solubility in lipids than in the aqueous cytosol (Sidell, 1998). Further, the mitochondria in hemoglobin-free icefishes are larger, have a higher lipid:protein ratio, and do not have a higher catalytic capacity than in warm water species, indicating that mitochondrial volume increases may serve to promote O2 diffusion in this group rather than to maintain catalytic capacity in the cold (O'Brien and Mueller, 2010). Egginton and colleagues used a reaction–diffusion model to evaluate O2 gradients in aerobic muscle from Antarctic, sub-Antarctic and Mediterranean species of fishes (Egginton et al., 2002). Temperature had a large influence on the extent of gradients as expected, but the PO2 in the core of the fiber was inversely related to fiber size across species and temperature regimes, indicating that diffusion distance is a critical parameter constraining aerobic design in some fish muscle.

### Ca 2+ cycling

Muscle contraction entails the release of Ca 2+ from the terminal cisternae during muscle activation and diffusion of Ca 2+ to troponin C (TnC), where it binds and promotes actin–myosin interactions. Relaxation of muscle requires the reuptake of Ca 2+ into the SR, which is catalyzed by the SR/endoplasmic reticulum (ER) Ca 2+ ATPase (SERCA) (Fig. 1C). PA is a small soluble protein that reversibly binds Ca 2+ and is present in some muscles, with greater quantities in fast-twitch fibers. PA therefore might be expected to serve a facilitated diffusion role. However, the binding of Ca 2+ by PA is too slow to promote equilibrium of [Ca 2+ ], [PA] and [PACa 2+ ] during a series of rapid contractions, and PA probably serves as a slow-acting temporal buffer which binds the excess Ca 2+ that accumulates during consecutive contractions (Permyokov, 2006).

Cannell and Allen first evaluated Ca 2+ cycling as a reaction–diffusion process in frog skeletal muscle, and these authors generated a mathematical simulation that compared well to experimental measurements of Ca 2+ transients (Cannell and Allen, 1984). A principal finding was that substantial gradients in Ca 2+ appear to exist over the sarcomere. Baylor and Hollingworth performed a similar analysis in frog skeletal muscle where they included the influence of Ca 2+ binding to ATP (Baylor and Hollingworth, 1998). In addition to finding Ca 2+ gradients that persisted for tens of milliseconds after release from the SR, they found that ATP serves to facilitate Ca 2+ diffusion as ATP occurs in high concentrations and has a relatively high D. This allowed for a more uniform distribution across the sarcomere of Ca 2+ that was bound to TnC, which may lead to a more unified contractile response. Baylor and Hollingworth also noted that the facilitated diffusion role played by ATP makes it a temporal buffer as well (Baylor and Hollingworth, 1998). That is, ATP binding and release of Ca 2+ caused the free Ca 2+ transient to be broader and of a lower magnitude than if ATP was absent or immobilized. This model was more recently applied to mammalian skeletal muscle, where it differed from the model for frog muscle in that the Ca 2+ release sites were offset by 0.5 μm from the sarcomere Z-line, based on morphological studies (Baylor and Hollingworth, 2007). At a common temperature, positioning of the Ca 2+ release sites near the middle of the thin filaments, as seen in mammals, had the advantage of promoting a more uniform distribution of Ca 2+ in the TnC binding sites. This would seem to indicate that mammalian SR structure has been subjected to selective pressure to offset diffusion constraints that otherwise may limit contractile function.

Groenendaal and colleagues (Groenendaal et al., 2008) adapted the model of Baylor and Hollingworth (Baylor and Hollingworth, 1998) to explicitly examine the differences between mammalian and frog muscle at 35°C. Again, the models implied that steep concentration gradients existed for Ca 2+ during contractions, and that [Ca 2+ ] was 5-fold higher in the region of TnC than in other regions of the sarcomere in mouse and frog muscle. In addition, Groenendaal and colleagues showed that [Ca 2+ ] was high in the region of the mitochondria, and they speculated that this arrangement facilitated Ca 2+ activation of oxidative phosphorylation and helped balance ATP demand during contraction (Groenendaal et al., 2008), although it should be noted that their simulations analyzed fast-twitch fibers that rely primarily on anaerobic fuels (PCr, glucose) to power rapid contractions.

### Nuclear function

Nuclei are associated with the transcription, translation and transport of a variety of molecules ranging in size from small metabolites to large complexes such as polysomes (Fig. 1D). Skeletal muscle fibers are multinucleated cells with the nuclei typically located at the periphery of the fiber, and each nucleus serves a volume of cytoplasm known as the myonuclear domain. The myonuclear domain has often been considered to be conserved in skeletal muscle, including during increases or decreases in fiber size (e.g. Allen et al., 1995 Roy et al., 1999 Bruusgaard et al., 2003 Bruusgaard et al., 2006 Brack et al., 2005), although this notion does not seem to be generally applicable and remains the source of debate (reviewed in Gundersen and Bruusgaard, 2008). Nevertheless, the size of the myonuclear domain presumably is regulated to ensure sufficient transcriptional capacity as well as limited distances over which nuclear substrates and products must travel to reach sites of action. There is evidence that transport within the myonuclear domain governs nuclear distribution at the sarcolemma, as mathematical analyses indicate that myonuclei in mammalian skeletal muscle have a uniform distribution at the sarcolemma (rather than a random distribution), which minimizes transport distances within the domain. This suggests that there are through-space signals that influence nuclear positioning (Bruusgaard et al., 2003 Bruusgaard et al., 2006).

A 3-dimensional reconstruction of the microtubule array (orange) in and around nuclei (blue) from white skeletal muscle of the smooth butterfly ray (Gymnura micrura). The microtubule network may be involved in the positioning of nuclei and trafficking of nuclear products in fibers (see text).

A 3-dimensional reconstruction of the microtubule array (orange) in and around nuclei (blue) from white skeletal muscle of the smooth butterfly ray (Gymnura micrura). The microtubule network may be involved in the positioning of nuclei and trafficking of nuclear products in fibers (see text).

The capacity for movement of nuclear products appears to be highly constrained in skeletal muscle, and proteins tend to remain in the vicinity of the nucleus from which they originated (Hall and Ralston, 1989 Pavlath et al., 1989 Ono et al., 1994). The distribution of mRNA and messenger ribonucleoprotein particles (mRNPs) in skeletal muscle is also consistent with greatly hindered diffusion, and it has been suggested that these large complexes are excluded from the myofibrillar space (Russell and Dix, 1992). However, Gauthier and Mason-Savas found ribosomes (and possibly polysomes) within the thick and thin filament lattice (Gauthier and Mason-Savas, 1993), suggesting that these large complexes have access to this region and may promote localized translation. It is also possible that diffusion constraints may be overcome in part by the extensive microtubule array in skeletal muscle, which is closely associated with the nuclei (Bruusgaard et al., 2006) and may be used in the trafficking of mRNA and proteins. A dynamic, anti-parallel microtubule network develops in growing myotubes, and it has been shown that myosin can be transported along these filaments by motor proteins (Pizon et al., 2005). Scholz and colleagues demonstrated that in fully differentiated cardiac myocytes microtubule-based transport is essential for the movement of mRNPs from the nucleus to sites of translation, and that disruption of the microtubule system inhibits protein synthesis (Scholz et al., 2008). To our knowledge, cytoskeleton-based transport of proteins or mRNA has not been demonstrated in mature skeletal muscle fibers. However, the microtubule network is closely associated with nuclei in mature skeletal muscle from mammals (e.g. Bruusgaard et al., 2006), crustaceans and fishes (Fig. 4), and it appears to be well suited for both positioning of nuclei (perhaps playing a role in controlling myonuclear domain size) and transporting nuclear products.

† These authors contributed equally to this study.

‡ Present address: ISIS Pulsed Neutron and Muon Facility, Science and Technology Facilities Council, Rutherford Appleton Laboratory, Harwell Science and Innovation Campus, Oxon OX11 0QX, UK.

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## Directed vs. Undirected Movement

In this module, we&rsquore going to talk about movement of materials over very short distances. Along the way, we&rsquoll also find that this discussion can be broadened to address topics as varied as macrophages hunting down viruses, and the differences between rhinos and amoebas.

First, however, we need to describe two different types of movement. Living in the macroscopic world as we do, it is natural to think only of "directed" movement. When we move somewhere, we usually move with a purpose! We are used to standing in line or driving along a road. Or we might think of blood being pumped through a circulatory system, or air being sucked in and pushed out of our lungs. Even though the paths followed by blood and air are complicated, they are still examples of materials flowing in some organized, directed way.

However, this sort of organized movement (especially within a cell or body) is a relatively modern invention. True, vertebrates, insects, and plants all do it. But early life forms didn&rsquot. Single-celled organisms like bacteria don&rsquot have circulatory systems, and after completing this entire module, you'll see why. Single-celled organisms were the earliest forms of life and still represent the lion&rsquos share of modern biodiversity. We're going to focus our module on the type of movement that these biological pioneers relied and still rely upon.

So, early life forms didn&rsquot actively suck in, circulate, or spit out material. How then did they accomplish getting necessary substances (like nutrients or oxygen) to all parts of their "bodies"? The answer is that small particles, like gas molecules and ions, move around and bounce off of each other constantly, in an undirected way. You can think of the movement of each individual particle as being essentially random &ndash like balls bouncing around in a pinball table, or people milling around at a party.

In fact, one of the most prominent modern physicists, Richard Feynman, said that if all scientific knowledge was destroyed and humans could pass on only one sentence to the next generation, it should begin with:

&ldquoAll things are made of atoms &ndash little particles that move around in perpetual motion&rdquo

So, when we think about movement, we now know that there are two main types of movement -- directed and undirected -- and our focus is on undirected movement.

## Introduction

The principal biological function of red blood cells (RBCs) is to exchange large volumes of O2 and CO2 during their brief (typically <1 s) transit through the microvasculature 1 . The speed of gas turnover by RBCs is therefore a measure of their physiological fitness, and highly-conserved biological adaptations are expected to relate to faster gas exchange. The steps in the gas exchange cascade include gas permeation across the cell membrane and binding onto hemoglobin. Additionally in the case of CO2, gas molecules are converted reversibly to HCO3 − ions (which are then transported by anion exchanger 1, AE1) plus H + ions (which are buffered by hemoglobin). These processes are coupled together by cytoplasmic diffusion. According to the prevailing consensus, efficiency of gas exchange is strongly dependent on protein-facilitated membrane transport, including AE1-assisted HCO3 − transport and aquaporin1-assisted gas permeation 2,3 .

A conserved and characteristic feature of human and animal RBCs is their flattened shape. The physiological relevance of this geometry has largely been attributed to the mechanical benefits it offers to circulating blood. RBC flattening allows microvasculature to co-evolve smaller luminal diameters and therefore maximize capillary density. In the case of human RBCs, the biconcave shape supports laminar flow and shear-thinning 4 , minimizes platelet scatter and atherogenic risk 5 , permits cells to squeeze through microvasculature 6,7 and is the lowest energy-level that the cell returns to following deformation in capillaries 8,9,10,11 . Elliptical or otherwise flattened RBCs of many non-human species may also manifest, to some degree, these mechanical benefits. Recently, cell geometry has been proposed as a stringent criterion by which splenic inter-endothelial slits select healthy RBCs for continued circulation 12 , but it is unclear if similar criteria apply in non-human species. These aforementioned mechanical considerations do not, however, offer a unifying explanation for the wide range of RBC thicknesses observed naturally in different animal species, and its inverse relationship with mean corpuscular hemoglobin concentration (MCHC) 13,14,15 . We postulate that RBC thickness and its relationship with MCHC may, instead, relate more closely to the efficiency of gas turnover.

Theoretically, the flattened RBC form may facilitate gas exchange in two ways. Firstly, it increases the cell’s surface area/volume ratio (ρ), which allows faster membrane transport. Secondly, it collapses the path-length for cytoplasmic diffusion, which reduces the time delays associated with intracellular gas transport. In the case of human RBCs (major radius, r, of 4 μm), adapting the shape from a hypothetical sphere (volume 268 fl surface area 200 μm 2 ) to a flattened form (volume 90 fl surface area 136 μm 2 ) doubles ρ and hence transmembrane flux 13,14,15 . The significance of this effect can be evaluated by considering the slowest membrane transport process, AE1-facilitated HCO3 − transport, which has an apparent permeability constant 3 (Pm,HCO3) of 18 μm/s and can be ascribed a time constant equal to 1/(ρ × Pm,HCO3). The AE1-related time constant for the spherical geometry is 0.074 s, which is reduced to 0.037 s for the flattened shape, yet both are compatible with equilibration during capillary transit. The biconcave shape of human RBCs reduces the mean cytoplasmic path-length to 0.9 μm (equal to the cell’s average half-thickness, h), which shortens intracellular diffusion time delays by a factor of seven, compared to a spherical RBC variant (r 2 /6 ÷ h 2 /2). However, this seven-fold acceleration will not translate into a meaningful improvement to gas exchange efficiency if gas diffusion coefficients are high, as they are in water. Paradoxically, gas diffusivity inside gas-carrying RBCs has not been measured, but is intuitively assumed to be rapid. Indeed, O2 and CO2 diffusivity measurements in hemoglobin solutions and hemolysates (

10 3 μm 2 /s) appear to support this assertion 2,14,16 . At this magnitude of diffusivity, cytoplasmic path-length and cell shape are not expected to be critical for determining the efficiency of gas exchange. In summary, our current understanding of RBC physiology predicts only a modest advantage of the flattened RBC shape for gas exchange, but this inference is based on an indirect characterization of gas transport inside the RBC.

In this study, we revisited the notion that gases diffuse rapidly in RBC cytoplasm. The required experimental approach must be capable of resolving diffusion on the scale of a single intact cell, and exclude contributions from permeation across extracellular unstirred layers and the surface membrane. To meet these criteria, we evaluated CO2 diffusivity (DCO2) in intact RBCs by measuring the ability of CO2/HCO3 − to facilitate cytoplasmic H + diffusion 17,18,19 . This measurement method is based on the observation that cytoplasmic H + ions are heavily buffered 20 , and therefore diffuse only as fast as the buffers there are bound to (with essentially no free ion movement) 21,22 . Thus, by determining the CO2/HCO3 − -dependent component of buffer-facilitated H + diffusion 17,18,23 , it is possible to quantify DCO2. Our measurements on human RBCs demonstrate substantially restricted CO2 diffusivity, to 5% of the rate in water, which is an order of magnitude slower than estimates made previously in cell-free hemoglobin solutions. By imaging osmotically-swollen human RBCs (to dilute MCHC) and RBCs from species with different hemoglobin concentrations, we demonstrate that DCO2 decreases sharply with MCHC, consistent with hemoglobin-imposed tortuosity to small-molecule diffusion. We conclude that diffusion across cytoplasm is a hitherto unrecognized rate-limiting step for gas exchange which imposes a critical limit on the RBC thickness, beyond which physiological function becomes inefficient and incomplete. In particular, the full manifestation of the Bohr Effect 24 (i.e. the process by which hemoglobin releases O2 at acidic vascular beds) is attainable only in RBCs that are adequately thin because the underlying H + trigger is transmitted intracellularly by slow CO2/HCO3 − diffusion. Highly restricted cytoplasmic diffusivity can explain the inverse RBC thickness/MCHC relationship observed amongst different animal species, and highlights the potential vulnerability of gas exchange efficiency in diseases that involve a change in RBC shape, such as in spherocytosis.

## Author summary

Urticaria is a common skin disease but the mechanism underlying wheal formation is not well understood. Our mathematical model suggests that not only the self-activation of histamine production via mast cells, but also self-inhibition of histamine dynamics plays a critical role in generating the wide-spread wheal patterns observed in urticaria this has not been previously considered in medicine. The study findings may increase the understanding of the pathogenesis of urticaria and may aid decision-making for appropriate treatments. It may also open an entirely new avenue for mathematical approaches to analyze various skin diseases with geometric eruptions and predict the effectiveness of treatments through in silico experiments.

Citation: Seirin-Lee S, Yanase Y, Takahagi S, Hide M (2020) A single reaction-diffusion equation for the multifarious eruptions of urticaria. PLoS Comput Biol 16(1): e1007590. https://doi.org/10.1371/journal.pcbi.1007590

Editor: Roeland M.H. Merks, Mathematical Institute and the Institue for Biology, Leiden, NETHERLANDS

Received: April 2, 2019 Accepted: December 8, 2019 Published: January 15, 2020

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

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

Funding: This work was supported by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology (JSPS), Japan (JP16K17643, JP19H01805, and JP17KK0094 to SSL https://www.jsps.go.jp) and by the JST PRESTO program, Japan to SSL (JPMJPR16E2 https://www.jst.go.jp). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

## MATERIALS AND METHODS

### Finite element approximations and simulation details of Laplace–Beltrami equations

We first derived a Laplace–Beltrami equation (Eq. 5 in Supplemental Text S1) for tubular surfaces. We would like to emphasize that complementary approaches for deriving the equation of continuity can be found elsewhere (Marsden et al., 1984 Frankel, 2011). To solve this equation for a tubular surface, we have developed a univariate FEM solver (Oden, 2006). This FEM solver is designed for tubular diffusion under symmetric conditions, that is, all prescribed solutions are independent of the angle about the tube’s center axis. Consequently the data depend only on the position along the tube’s center axis. Under such symmetric conditions, the tubular diffusion has only a single degree of freedom and is modeled in a univariate setting. The symmetric tubule diffusion using defined virtual coordinates reduces to the form utk[A(x)ux]x = 0 for specific forms of A(x) determined by the tube’s geometry. This is then solved using our 1-D FEM solver. The numerical solution is computed by semidiscrete methods. First, the uniform mesh and approximating spline space are user specified. As the basis of the symmetric solver, we use normalized B-splines Sd r (Δ) (Schumaker, 2015). This reduces the computation time and, unlike other methods such as nodal basis elements, avoids artifacts such as negativity while smoothing the splines. We then use the De’Castlejeau algorithm for the evaluation of the B-splines without having to construct individual basis spline in the Sd r (Δ) space. Then a Galerkin procedure is implemented. The integration is accomplished through a Gaussian quadrature exact up to polynomials of degree 11. For example, when the approximation is conducted in any of the recommended spaces S2 0 ,S2 1 ,S3 2 (Δ), the quadrature is exact. The remaining temporal part is then handled using Matlab’s ODE45 solver. All our simulations were performed using Matlab R2014/R2015a on Windows computers. A more general two-dimensional code capable of handling even asymmetric boundary conditions is described in Supplemental Text S2. Finally, as a note, we do remind readers that numerical approximation of this model is entirely a different problem to solve and this is independent from the model’s theoretical justification and derivation. We have included ways to improve numerical solutions and avoid artifacts due to approximation in Supplemental Text S7.