Do Skeletal Dynamics Mediate Sugar Uptake and Transport in Human Erythrocytes?

Abstract

We explore, herein, the hypothesis that transport of molecules or ions into erythrocytes may be affected and directly stimulated by the dynamics of the spectrin/actin skeleton. Skeleton/actin motions are driven by thermal fluctuations that may be influenced by ATP hydrolysis as well as by structural alterations of the junctional complexes that connect the skeleton to the cell’s lipid membrane. Specifically, we focus on the uptake of glucose into erythrocytes via glucose transporter 1 and on the kinetics of glucose disassociation at the endofacial side of glucose transporter 1. We argue that glucose disassociation is affected by both hydrodynamic forces induced by the actin/spectrin skeleton and by probable contact of the swinging 37-nm-long F-actin protofilament with glucose, an effect we dub the “stickball effect.” Our hypothesis and results are interpreted within the framework of the kinetic measurements and compartmental kinetic models of Carruthers and co-workers; these experimental results and models describe glucose disassociation as the “slow step” (i.e., rate-limiting step) in the uptake process. Our hypothesis is further supported by direct simulations of skeleton-enhanced transport using our molecular-based models for the actin/spectrin skeleton as well as by experimental measurements of glucose uptake into cells subject to shear deformations, which demonstrate the hydrodynamic effects of advection. Our simulations have, in fact, previously demonstrated enhanced skeletal dynamics in cells in shear deformations, as they occur naturally within the skeleton, which is an effect also supported by experimental observations.

Introduction

Material exchange between cells and the environment is achieved via transport through the phospholipid cell membrane, principally through transmembrane protein transporters. The process involves several kinetic steps, including binding to transporters at the exterior side of the cell, translocation through the transporter, and disassociation on the endoplasmic side and eventual entry into the cell interior (1). A particularly interesting case is the uptake of glucose (Glc) into erythrocytes, for which Carruthers and co-workers (2, 3, 4) demonstrate that compared with the first two steps, the third step may be the slowest. Glc is constrained by hydrogen bonding to the transporter and by clustering so that the disassociation step ultimately dominates the kinetics of the transport process. Herein, we explore a novel, to our knowledge, hypothesis that disassociation is strongly affected and directly stimulated by the dynamics of the red blood cell (RBC) skeleton and in particular by the dynamics of the actin protofilament that is the hub of the spectrin skeleton’s junctional complex located and bound to glucose transporter 1 (GLUT1) (5, 6, 7). Our proposed mechanism occurs via 1) advection due to fluid flow driven by skeleton/actin dynamics, 2) probable contact of F-actin protofilaments with Glc, and 3) the creation of open molecular regions, allowing escape of Glc from the endoplasmic space of the GLUT1 transporter. Our hypothesis is supported by various experimental observations that couple to our own experimental and theoretical findings, as reviewed below.

Biological transport phenomena involve contributions from both diffusion and advection. Particularly relevant here is the role of fluctuations of the cytoskeleton in the release of materials from the inner side of the lipid membrane of an erythrocyte before they enter the cell. This idea stems from our previous simulations, which are summarized below (8, 9). According to these studies, there are significant advection effects around the cytoskeleton of an erythrocyte because of its thermally activated fluctuations. Specifically, our numerical results demonstrate that due to the mechanical design of the junctional complex (JC), the F-actin protofilaments (as shown in Fig. 1) undergo large oscillations covering a large area and act like brooms or mixers in the vicinity of the lipid membrane. Moreover, fluctuations of these protofilaments are enhanced when the membrane is under shear deformation, even we note by the modest deformations that are a natural feature of the cell’s biconcave shape (10). This behavior is consistent with the experimental observations of Lee and Discher (11), which measured accentuated fluctuations of actin-attached beads in the cell membrane with different deformations. Further, our preliminary experimental measurements suggest that Glc uptake into erythrocytes is indeed influenced by the cell’s hydrodynamic activity, which is a finding confirmed by a correlation with the hydrodynamic capillary number. Inspired by these findings, we conducted preliminary simulations that prove that fluctuations of actin can indeed expedite transport of ions and molecules. Based on these observations, we explore herein the hypothesis that the interaction between bound molecules and the F-actin protofilament through advection, direct contact, and molecular movement is an important mechanism in the transport process of Glc or other molecules near the cell membrane. In the Results and Discussion, we use our results to provide a possible explanation of a certain dysfunction in Glc uptake in Chinese type 2 diabetes, which is consistent with the structural defects of GLUT1 suggested by Hu et al (12).

Figure 1.

Molecular structure of a human erythrocyte junctional complex (JC) (after Lux 2015 (6)). Note that both GLUT1 (the glucose transporter) and band 3 (the anion transporter) are associated with glycophorin C (GlyC), which binds actin. Thus, GLUT1 and band 3 are colocated at the JC. To see this figure in color, go online.

Materials and Methods

Background on Glc uptake into erythrocytes

Carruthers and co-workers (2, 3, 4, 13, 14) have postulated that net sugar (Glc) transport in erythrocytes is a multistep process that involves cytosolic binding to sites located just beneath the membrane/skeleton and near, and possibly associated with, the sugar transporter (GLUT1). For radio-labeled sugar uptake, they initially fit their experimental data to a rate equation of the form,

$$c\left(t\right)=c_0+c_1\left(1-{\text{e}}^{-k_{1}t}\right)+c_2\left(1-{\text{e}}^{-k_{2}t}\right),$$ (1)

i.e., a biexponential form. Here, $c\left(t\right)$ and $c_0$ are the counts in the cells at times t and $t=0$, respectively. $c_1$ and $c_2$ are equilibrium counts in their two postulated compartments (viz., compartment 2, the bulk cytosol, and compartment 1, which was proposed as a sugar-binding complex located near GLUT1). In effect, their results and model concept were that sugar uptake involves at least two kinetic steps, viz., 1) protein (GLUT1)-mediated transport across the membrane (translocation) and 2) reversible, saturable binding within compartment 1, which releases sugar to the bulk of the cytosol, which is compartment 2. They found that the results were in fact consistent with the view that the faster compartment comprised ∼66% of the cell’s water space, leaving 33% for the slower compartment.

Experiments were conducted on normal RBCs and RBC ghosts, which showed reduced uptake kinetics. In addition, Glc uptake was measured using α-toxin permeabilized red cells and cells treated with Ca⁺⁺ + A23187 ionophore. This allows rapid entry of Glc bypassing the GLUT1 transporter. The results revealed a number of notable features, as summarized below.

• 1)The kinetics are reduced by increases in viscosity, as accomplished by the use of glycerol or sucrose as described by the enzyme model of Blacklow et al (15). This demonstrated that the kinetics are limited by diffusion or combined diffusion-advection. In fact, Cloherty et al. found that the second-order rate constant for 3-O-methylglucose (3OMG) fell monotonically with increasing medium viscosity using a glycerol concentration in the range 0–30% (v/v).
• 2)The results for uptake in ghosts showed that although the kinetics are slower, they still retained a biexponential form. This suggested a role of ATP in enhancing the kinetics and suggests that Glc binding to hemoglobin, either bound to the membrane or in solution, does not constitute the series barrier, i.e., compartment 1.
• 3)The results for the permeabilized cells indicated that the series barrier is associated with or at least located near GLUT1. This followed from the fact that bypassing GLUT1, and therefore the binding site, resulted in the likewise bypassing of the slower of the kinetic steps. The permeabilized cells indeed filled with Glc within 5 s. These results also discounted the notion that the series barrier is an “unstirred layer” because bypassing the GLUT1 transporter and hence the series barrier resulted in rapid sugar entry into the cell. Care was taken to be confident that at least the major structural elements remain in position within the cell membrane in the permeabilized cells.
• 4)Interestingly, ATP depletion (e.g., by ghost formation) slightly slows the more rapid uptake step (presumably translocation) but significantly slows the shunting step associated with compartment 1 releasing sugar to compartment 2. Normal uptake was, however, restored in ghosts upon inclusion of Mg $\cdot$ ATP during ghost resealing. Cloherty et al. (2) offer two potential reasons for such ATP dependence: ATP-hydrolysis-dependent reversal of Ca⁺⁺ inhibition of sugar transport (16) and allosteric-hydrolysis-independent modulation of transport resulting from nucleotide binding to GLUT1 (17). Indeed, herein we hypothesize that a role of ATP is to enhance membrane-skeleton dynamics that, in turn, causes an advection contribution to release Glc from the shunt site to the bulk cytosol.
• 5)Cells whose cytoskeletons were disrupted by combined calcium ion and ionophore A23187 showed significantly reduced sugar uptake kinetics. Cloherty et al. (2) suggested this effect may have been related to ATP depletion promoted by Ca loading using ionophore (18), but below we propose an alternate cause for the effect of reduced uptake kinetics with disrupted skeletons or disrupted skeleton attachments to the cell membrane; namely, a reduced contribution from advection.
• 6)Cloherty et al. (2) also noted that unlike the anion transporter (band 3), GLUT1 is anisotropically distributed across the membrane. Thus, it may be that both binding of Glc and its release may be enhanced by closely spaced GLUT1s.

The compartmental transport scheme is illustrated in Fig. 2, which depicts the erythrocyte membrane-skeleton assembly.

Figure 2.

Schematic of the erythrocyte membrane and attached spectrin skeleton. Note that the major sites of attachment, including JCs, of the skeleton-to-membrane are the glucose transporter, GLUT1, and the anion exchanger band 3. Associated proteins are also depicted and labeled. Compartment 1, the glucose binding site, is conceptually illustrated as being either structurally or functionally associated with the endofacial side of GLUT1; it is colored in a faint yellow. To see this figure in color, go online.

Dynamics of the RBC skeleton and advection

We have created a multiscale modeling approach for the viscoelastic response of erythrocytes (8, 9, 19, 20), which is sketched in Fig. 3. The current approach contains three models at different length scales; in the complete cell level (referred to as Level III), the membrane is modeled as two layers of continuum shells using the finite element method. The skeleton-bilayer interactions are depicted as a “slide” in the lateral (i.e., in-plane) directions caused by the mobility of the skeleton-bilayer integral attachment proteins (viz., band 3 and GLUT1) and normal contact forces induced by deformation. At Level II, the constitutive laws of the inner layer (the cytoskeleton) are obtained from a molecular-detailed model of the JC, the repeating unit in the cytoskeleton. The mechanical properties of the spectrin, including its folding and/or unfolding reactions, are obtained with a stress-strain model based on a thermally activated/force affected Arrhenius rate equation (Level I). These three models are coupled through an information-passing algorithm, in which predictions of Level I and Level II models are summarized as constitutive laws and employed in Level II and Level III models, respectively. On the other hand, the three-dimensional configuration and deformation predicted by the Level III model will be utilized by the Level II model to determine mesoscale mechanics and mechanical loads at the protein-to-protein and protein-to-lipid linkages, so that the possibility of mechanically induced skeleton deformation and remodeling can be studied. We then combine this multiscale model with a boundary-element model based on Stokes flow to create a flow-cell interaction model. The following are all incorporated into the model: viscosity effects (including the viscosities of the fluids inside and outside of the cell), the viscosity of the lipid bilayer, the highly nonlinear rate- and temperature-dependent entropic elasticity of the protein skeleton, and the viscous connectivity between the two, enabled by the mobility of the transmembrane proteins within the lipid bilayer. We also note that the fluid-solid interactions between the skeleton and fluid impart a viscous nature to the skeletal motion.

Figure 3.

Schematics of the three levels of our multiscale framework. The three levels (i.e., models) are connected through an information-passing algorithm: the Level I model computes the constitutive relations of the Sp (as in (8, 9)) that are then used in the Level II model. Similarly, the Level II model creates the constitutive response (e.g., areal and shear stiffness) of the cytoskeleton that is used in the Level III cell model. To see this figure in color, go online.

We note that the three levels are not run simultaneously. The stress-strain relation and the constitutive laws are precalculated and stored. As for the typical timescale at each level, it may be said that Level I has none, whereas Level II is in the microsecond range and Level III is in the range of seconds.

Our Level II model is based on state-of-the-art understanding of the molecular architecture of the JC (the repeating unit in a cytoskeleton). In this model, the JC is depicted as a three-dimensional structure with an actin protofilament at its center and six surrounding spectrin (Sp) dimers (see Fig. 4). This is essentially different from other models of the RBC membrane in which the skeleton is represented as a two-dimensional network. Further, in our model, the junctions between the Sp and the actin protofilament, Sp-bilayer/actin-bilayer interactions, and thermal effects are all accurately described; details for this are found in our previous reports, e.g., (8, 9). Subsequently, this comprehensive model leads to important findings about the mechanics and dynamics of this biological structure that, to our knowledge, have never been illustrated before. Most notably, it was found that under shear deformations, the JC becomes bistable with two equilibrium states, characterized by different orientations of the actin protofilament (Fig. 4). Moreover, under the effect of thermal fluctuations, the JC keeps on switching back-and-forth between these two equilibrium states (8, 9). During such mode-switching events, a large sweeping motion of the protofilament is recorded (Fig. 5). This, together with the thermal fluctuation of the F-actin itself, leads to significant hydrodynamic flow that can drive advection. Moreover, we note that the actin protofilament is relatively long (with respect to the dimension of the JC), and because its motion effectively sweeps over the entire JC area, it has a high probability of directly impacting any molecule emerging from the endoplasmic side of GLUT1.

Figure 4.

Bifurcation of the JC with shear deformations. The parameter λ (see (8, 9) for details) measures the amount of shear deformation of the skeleton, during which the skeleton is stretched by a factor of λ in one direction and compressed by a factor of $1/\lambda$ in the orthogonal direction.

Figure 5.

A mode-switching event in thermal fluctuations when the JC is bistable.

For later reference, we note that typical actin vibrational and mode switch frequencies, as shown in Figs. 4 and 5, are between 10⁶ and 2 × 10⁶ s⁻¹ and between 10⁴ and 5 × 10⁴ s⁻¹, respectively. Further details about vibrational motions and in particular about mode switching are given in our previous reports (8, 9). Here, we emphasize that we believe that the onset of mode switching is important in that it leads to a greatly expanded areal coverage of the JC/GLUT1 region by the vibrating actin protofilament. This is highlighted by the discussion of Fig. 13.

Figure 13.

Thermal fluctuation of the actin protofilament illustrated by combinations of its snapshots at various instants. The small bead represents the initial location of the glucose. The pointed end of the protofilament is shown in blue. To see this figure in color, go online.

Also, we note that the “mode switching” induced by even quite small shear deformations does not require imposition of significant shear flow as used. Indeed, our simulations (10) demonstrate that without any external loading, the prestress in the skeleton causes a shear deformation larger than 1.2 in the rim area of the cell, which is above the threshold of shear deformation to trigger bifurcation and mode switching (see Fig. 5). Additionally, cell deformation induced by even modest shear flow or cell transport can increase the shear in other areas (8, 9, 21) and create mode switching there.

It is noteworthy that the dynamic effects we describe here and analyze in the novel, to our knowledge, context of transport are unquestionable, as further simulation and experimental evidence can be found in (22, 23, 24).

Example: advection of ions

To assess the potential of skeleton dynamics driving advection-enhanced solute transport, we performed a simulation of such an effect on a divalent ion such as Ca⁺⁺ bound to a membrane and/or skeleton region because of its simple electrostatic field. The field was modeled via a Gouy-Chapman double layer assuming either 1:1 or 2:2 electrolytes with molarity concentration in the range 0.01–0.1 (25). This was designed to serve as a simple scenario to model modest interactions between the electrostatic membrane and ions. A sketch is contained in the inset at the upper left corner of Fig. 6.

Figure 6.

Time histories of the penetration distance of a calcium cation into the cell in different conditions. The inset shows a cation within the field of a Gouy-Chapman electrical double layer at a bilipid membrane. To see this figure in color, go online.

Analysis via solving for the electric field within the double layer yields the force,

$$f_z=\kappa {\psi }_s{z}_{i}e{\text{e}}^{-\kappa z}=-\kappa {\psi }_{s}e{\text{e}}^{-\kappa z},$$ (2)

where ${\psi }_s$ is the surface charge on the membrane, κ is the Debye length, and e is the electronic charge. Representative values for these variables are as follows. Note that we are considering strictly the ions of valance $z_i=+2$ (thus, the negative sign in the latter portion of Eq. 2). The values used in this example are as follows:

$$\begin{array}{cc}\kappa ={10}^9{\text{m}}^{-1},& {\kappa }^{-1}={10}^{-9}{\text{m}}^{,}\\ {\psi }_s=-50\text{mV}=-50\times {10}^{-3}\text{V}& \\ e=1.602\times {10}^{-19}\text{C}.& \end{array}$$ (3)

These values are taken from Alberts et al. (26) for ${\psi }_s$ and from Masliyah and Bhattacharjee (25) for κ.

By using the molecular-detailed Level II model, we simulated the motion of a calcium cation whose diameter is 0.3 nm near the RBC membrane with or without the advection effect. For simplicity, in this preliminary modeling study we only consider the fluctuations of the cytoskeleton (especially that of the actin protofilament). The lipid bilayer is assumed to be rigid; membrane fluctuations would serve to enhance advective flow and will be included in our proposed simulations. The motion of the cation is determined by a combination of Browning motion (predicted through the Langevin equation), the electrostatic force, and the hydrodynamic interactions between the cation and the cytoskeleton (advection). Because the cation moves freely in the horizontal (i.e., parallel to the membrane) directions, we simplify the skeleton as a doubly periodic network of JCs with a period of 60 nm in each direction. At any instant, the dynamics of the cation is significantly affected by the close-by JC only because hydrodynamic interactions decay rapidly with distance. The shear deformation λ of the JCs is 1.5. The cation is originally located at the center of one JC right above the bilayer. In Fig. 6, we plot trajectories of the cation without advection and with advection. It is seen that with advection, it is much easier for the cell to escape the restraining effect of the electrostatic field. This implies that advection plays a key role in the process.

Hypothesis and its basis

Based on the above, we hypothesize that because the binding site is located near the sugar transporter and because these are the regions of significant skeleton activity, advection due to skeleton/bilayer-driven fluid flow and the stickball effect play a role in Glc uptake via enhancing the rate of Glc disassociation with the binding site. Additional bases for the above hypothesis include the following.

• 1)The effects of ATP are consistent with the above hypothesis because it is known that ATP enhances membrane/skeleton fluctuations (27, 28, 29). Hence, ATP depletion is expected to reduce the proposed advection effect by reducing the energetics driving skeleton dynamics.
• 2)The effects of calcium ions and ionophore A23187 are known to include skeleton disruption that, in turn, leads to loss of cell membrane and deformability (30, 31), and therefore also in cell rheological properties, all of which we are able to assess within our simulations. This too will reduce skeleton dynamics and, in turn, the influence of advection.
• 3)The kinetics of uptake are directly affected by viscosity, as are the predicted kinetics and effectiveness of our mechanisms. As our dynamic analysis would consistently forecast, we note that Cloherty et al. (2) found that the rate of 3OMG uptake fell monotonically with increasing viscosity. Reductions in viscosity will lead to accentuated skeletal dynamics and increased kinetics, as we demonstrate below. This also applies to the kinetics of diffusion and advection.
• 4)A most favorable correlation of our proposed mechanisms with experimental observations involves the timescales for Glc uptake in the vicinity of GLUT1. As developed in Kinetic Analysis Relating Our Experiments with Our Simulations and Glc Disassociation and Its Activation Length, we find that the timescales of our proposed advection/contact effects are quite consistent with measured $k_{\text{cat}}$ rates, especially at physiological temperatures such ${37}^{\circ }$C. In Kinetic Analysis Relating Our Experiments with Our Simulations, in fact, we extract an estimate for an activation distance from an experiment that compares quite favorably with a model estimate based on our binding picture.
• 5)Our experiments, as summarized in Results and Discussion, also show that when the shear rate is fixed, an increase in the viscosity of the surrounding fluid enhances Glc uptake. This is consistent with the fact that cell deformation (including deformation of the skeleton) and the corresponding actin fluctuations are determined by the capillary number defined as $\eta R\dot{\gamma }/\mu$, where η is the viscosity of the surrounding fluid, R is the effective radius of the cell, $\dot{\gamma }$ is the shear rate, and μ is the shear stiffness of the cell.

Kinetic analysis relating our experiments with our simulations

Our simulations of advection-enhanced Glc uptake also provide additional quantitative insight into the nature of the kinetic barrier identified by Cloherty et al. (2), which provides quantitative verification of our hypothesis. For example, an approximate illustration of the kinetic steps of Glc uptake along with a schematic sketch of the energy landscape for the release of Glc from the binding site (2) into the cytosol (3) is shown in Fig. 7, a and b, respectively. Release from the binding site is influenced by the hydrodynamic forces $f_h$, as illustrated. To explain the concept by example, we term $f_h^{\left(w\right)}$ and $f_h^{\left(wo\right)}$ as forces with and without advection, respectively.

Figure 7.

(a) Compartment $1\to 2$ kinetic model for glucose uptake and (b) force-affected energy landscape. To see this figure in color, go online.

The general form of the rate constant for the release step would be $k_2\propto \text{exp}\left(-\Delta G/kT\right)$, with the activation free energy represented by $\Delta G=\Delta G^{\circ }-f_h\delta \overrightarrow{x}$. $\delta \overrightarrow{x}$ is the activation distance over which the forces act. Hence we find,

$$\frac{k_2^{\left(w\right)}}{k_2^{\left(wo\right)}}=\text{exp}\left\{\left(f_h^{\left(w\right)}-f_h^{\left(wo\right)}\right)\delta \overrightarrow{x}/kT\right\}\text{or}$$ (4)

$$\delta \overrightarrow{x}=\frac{kT}{f_h^{\left(w\right)}-f_h^{\left(wo\right)}}\text{ln}\left(\frac{k_2^{\left(w\right)}}{k_2^{\left(wo\right)}}\right).$$ (5)

Advection forces acting on Glc are found by our preliminary simulations as discussed in Example: Advection of Ions to be in the range of $f_h\sim 2-4$pN. For intact cells and those disrupted by combined calcium and A23187, Cloherty et al. (2) found that $k_2=0.56$ and $k_2=0.054{\text{min}}^{-1}$. We assume for now, again by way of example, that these values represent uptake with and without advection, respectively. This would place the activation length in the range of $\delta \overrightarrow{x}\sim 2-4$nm, which is in the range of 2–3 times the diameter of a Glc molecule.

Hydrodynamic forces acting on Glc obtained from our simulations, as described in Example: Advection of Ions, can be used to create a consistent kinetic framework via the analysis of Cloherty et al. (2) and our added thermally activated rate analysis, which can be further enhanced and based on both our experimental and theoretical findings. Toward this end, advection forces should be simulated by incorporating the changes made to the skeleton and/or the membrane or medium viscosity. Determination of a consistent value for $\delta \overrightarrow{x}$ will thus allow an estimate of changes to $\Delta G$ (i.e., $k_2$) and, in turn, to the uptake kinetics, as measured in our parallel experiments. This is a pathway to establishing a consistent estimate for $\delta \overrightarrow{x}$, as it was the successful pathway that was utilized—for example, by Zhu and Asaro—(9) to determine the activation lengths for Sp unfolding and refolding used in our multiscale erythrocyte models.

Glc disassociation and its activation length

We begin by recalling the structure of the JC of a human erythrocyte’s membrane, as shown in Fig. 1 (5, 6, 7), with specific focus on the membrane-to-skeleton connections. Fig. 1 shows an illustration of the molecular structure of the two Sp-skeleton pinning complexes in the human erythrocyte (viz., band 3) and the JC centered at glycophorin C (GlyC). Most important for our current concerns are the associations of both the GLUT1 and the anion transporter (band 3) with the GlyC that binds the actin to which the Sp skeleton is attached. We also note that the 37-nm-long actin protofilament, whose dynamics are the cause of both hydrodynamic and mechanical forces on Glc, is also colocated with the JC/GLUT1 complex. Associated proteins such as ankyrin, 4.1, and 4.2 are also rendered in Fig. 1.

The significance of these complexes, especially the JC complex, is that transport of ions and molecules may be challenged or assisted and possibly even mediated by the dense structure and dynamics of the erythrocyte’s skeleton. To our knowledge, this is a novel concept that may have far-reaching consequences with regards to understanding the underlying causes (i.e., mechanisms) of a legion of blood-related diseases. We next look at the structure of GLUT1 within the JC complex (32). We first realize that the transmembrane proteins extend a considerable distance into the cytoplasm from the lipid membrane.

Consider again the structure of GLUT1 with a bound Glc molecule, as shown now in Fig. 8.

Figure 8.

Glucose bound to a Lys residue on loop 6-7 of GLUT1 (edited from Salas-Burgos et al. (32)). Inset at upper left shows that thermal fluctuations will initially accelerate the glucose-Lys complex until the loop is sufficiently stretched. Note that when stretched, tension T develops in the loop. Note that all dimensions shown are in Å. To see this figure in color, go online.

In this case, Glc is not bound to a helix of GLUT1 but to a loop of GLUT1.

According to Cunningham et al. (33), there is a docking site (binding site) for Glc on loop 6-7, connecting GLUT1 helices 6 and 7, as shown in Fig. 8; see also the upper left inset. This loop contains a charged lysine (Lys) residue. Thus, the binding is taken to be an electrostatic charge-charge or charge-dipole bond; such bond forces fall off more slowly with separation than dipole-dipole interactions. Moreover, we realize that loop 6-7 is more flexible than a helix until it is sufficiently stretched. Loop stretching induces tension T, as indicated in the upper left inset to Fig. 8.

Consider the yellow-shaded region surrounding Glc and the section of the loop to which it is bound (i.e., Lys), as illustrated in the inset at the upper left. Thermal fluctuations in this region create forces on the Glc-Lys complex; hence, this complex is accelerated until the loop is sufficiently stretched and stops. At that point, the Glc may be accelerated away from the loop. In other words, at first the “thermal force” is partitioned between the Glc and Lys (in some manner) until the loop is stretched and the developed tension, T, ceases to move downward with the bound Glc.

In reality, this is a continuous process whereby the net force on the Lys is decreased because of the increasing tension, T, in the stretched loop. However, at some point, the net force between the glc and Lys will exceed the electrostatic bonding force, and the Glc will separate from the Lys. The process can be visualized as follows. Fig. 9 a depicts Glc electrostatically bound to a Lys located on loop 6-7 of GLUT1; the Lys-Glc binding force is of magnitude $f_d$. Fig. 9, b and c illustrate two idealized stages in the time after a significant thermal fluctuation on the Glc-Lys complex. We assume in Fig. 9 b that a total thermal force of $2f$ has been imposed: f on Glc and f on Lys. At that point, Glc is bound to Lys with an electrostatic force $f_d$. The complex is accelerated with zero tension on the loop, i.e., zero tension force on Lys. These forces accelerate the complex, which causes loop 6-7 to stretch and exert a backforce on Lys. This causes Lys to decelerate, whereas Glc continues to move until the force on it exceeds the electrostatic binding force $f_d$. This causes Glc to disassociate from Lys (and hence from GLUT1) and enter the cytoplasm.

Figure 9.

Breakaway of glucose from the Lys on loop 6-7 of GLUT1. (a) A glucose bound to Lys with loop 6-7, which is initially under zero tension, is shown; the magnitude of the binding force is $f_d$. (b) Imposition of thermal fluctuation and forces f on the Lys-glucose complex are shown. This accelerates the complex and leads to stretching and a back-force on Lys, causing it to decelerate, as in (c). (c) The net force on Lys decays to zero, and the net force on glucose causes rupture on the Lys-glucose bond. Note, all dimensions shown are in Å. To see this figure in color, go online.

We note that this conceptual process will provide an energy landscape for the disassociation process and to an activation distance $\delta \overrightarrow{x}$ because Glc must travel a finite distance in a finite time under imposed hydrodynamic and mechanical forces to create conditions for debonding from Lys.

For perspective, we note that the length of the actin protofilament, as sketched in Figs. 1 and 10, is ∼37 nm. This underscores the notion that, given its motion, actin may indeed sweep the entire binding site region. In this regard, we also note that although the particular shape drawn below in Fig. 10 is idealized, it remains that other viable conceived shapes would invariably be spanned by a 37-nm-long filament as well.

Figure 10.

Idealization of loop 6-7; the shape of the loop as indicated is only schematic for modeling purposes. The loop is part of the GLUT1 transporter, as shown. The number of residues (abbreviated as “res”) in each section of loop 6-7 is indicated as, e.g., 18res., and the approximate length of each section is in nanometers. The “linear” dimensions of $4\times 11$nm are only meant to convey the approximate area of the loop. The Lys-glucose bond is conceived as a charge-dipole bond. F-actin protofilaments are ∼35 nm long with diameters in the range of 7–9 nm [?]. To see this figure in color, go online.

Still, another perspective is gained by examining Fig. 10, which shows Glc bound to loop 6-7 within the stochastic flow field of actin. A schematic of the actin protofilament is shown to illustrate the span of actin. It is important to note, however, that actin is not centered at GLUT1; rather, it is centered on GlyC. Indeed, actin would contact Glc more toward its tips, i.e., in a stickball style.

A simple implementation

The random flow field driven by skeletal-actin motion together with forces arising from thermal fluctuations impose forces on both Glc and the Lys residue residing on loop 6-7; these forces are pictured in Fig. 10. As loop 6-7 is stretched, it imposes a back-force on Lys (but not on Glc). Sufficient back-force will decelerate Lys and cause the net force on Glc to increase until it reaches the binding value $f_d$, which is also depicted in the idealized sketch of Fig. 11. At that stage, Glc may disassociate from Lys. A simple implementation of this concept is discussed next.

Figure 11.

Idealization of the loop 6-7 binding model of Glc to Lys. The Glu-Lys complex is subjected to a combined thermal-hydrodynamic force $2f$, assumed for simplicity to be equally partitioned between Glc and Lys. As loop 6-7 is stretched, it imposes a back-force on Lys; the spring-like response of loop 6-7 is represented by $k_{\ell }$. When loop 6-7 is critically stretched, Lys is decelerated, and Glc may break away. The charge-dipole Glc-Lys bond geometry is sketched in the upper right inset. To see this figure in color, go online.

Fig. 11 depicts a simplified picture of the basic elements of the binding model of Fig. 10. In it, Glc and Lys are electrostatically bound with a charge-dipole force of magnitude $f_d$.

The Glc-Lys complex is subjected to a combined thermal-hydrodynamic force $2f$, assuming an equal partition between Glc and Lys. The maximal net force on Glc occurs when the net force on Lys is zero. The back-force on Lys is given by

$$f_b={\mathcal{F}}_b\left(\delta \zeta \right)\mathrm{⇝}{f}_b\approx k_{\ell }\delta \zeta ,$$ (6)

where ${\mathcal{F}}_b(\cdot )$ is derived via an appropriate entropic force versus stretch model for the loop 6-7 polypeptide, and $\delta \zeta$ is the distance stretched from a reference position where $f_b=0$; if we linearize the first version of Eq. 6, we can use its second version. As Lys and Glc attract each other with a force $f_d$, the equations of motion for them are

$$\begin{array}{c}f+f_d-k_{\ell }{x}_{\text{Lys}}=D_{\text{Lys}}\frac{\text{d}{x}_{\text{Lys}}}{\text{d}t}+m_{\text{Lys}}\frac{{\text{d}}^2{x}_{Lys}}{\text{d}{t}^2},\\ f-f_d=D_{\text{Glc}}\frac{\text{d}{x}_{\text{Glc}}}{\text{d}t}+m_{\text{Glc}}\frac{{\text{d}}^2{x}_{Glc}}{\text{d}{t}^2},\end{array}$$ (7)

respectively, where $D_{\text{Lys}}$ and $D_{\text{Glc}}$ are the viscous drag coefficients of Lys and Glc. We note that within the context of the assumed Stoke’s flow envisioned here, the inertial terms were found to be negligible compared to the viscous drag forces (see (34)). Moreover, for simplicity, let $D_{\text{Lys}}\approx D_{\text{Glc}}=D$ and define $\delta x\equiv x_{\text{Glc}}-x_{\text{Lys}}$; at rest, $\delta x$ is the separation distance d between Glc and Lys, as in Table 1. Then, because $x_{\text{Lys}}\approx x_{\text{Glc}}$ until they separate,

$$D\frac{\text{d}\delta x}{\text{d}t}=-2f_d+k_{\ell }{x}_{\text{Lys}}.$$ (8)

Only if $-2f_d+k_{\ell }{x}_{Lys}>0$ does $\delta \overline{x}$ increase in a positive sense, i.e., only then does Glc separate from Lys because in such case $f_d$ falls off rapidly, as shown below. This defines a critical distance (i.e., an activation distance $\delta \overrightarrow{x}$), as in Kinetic Analysis Relating Our Experiments with Our Simulations, such that $\delta \overrightarrow{x}=2f_d/k_{\ell }$. Hence, breakaway occurs when $x_{Lys}\ge \delta \overrightarrow{x}$.

Table 1.

Force $f_d$ versus Dipole-Charge Separation

Distance, d (nm)	Force, $f_d$ (pN)
0.4	19.00
0.5	12.19
0.6	8.45
0.7	6.19
0.9	3.68
1.0	2.94
1.5	1.21
2.0	0.63

Analysis of $k_{\ell }$

Let loop 6-7 be idealized as a circle, as in Fig. 12, with perimeter p, and let it be distorted to an elliptical shape as shown below; assume that $a\ge b$. Note that the perimeter of an ellipse is given as

$$p=4a{\int }_0^{\pi /2}\sqrt{1-{\mathrm{ϵ}}^2{\text{cos}}^2\theta }d\theta ,\mathrm{ϵ}=\sqrt{1-b^2/a^2},a\ge b.$$ (9)

An approximation is given as

$$p=2a\left\{2+\left(\pi -2\right){\left(\frac{b}{a}\right)}^{1.4546}\right\}.$$ (10)

Then, the equation is given as

$$\delta p=\frac{\partial p}{\partial a}\delta a+\frac{\partial p}{\partial b}\delta b.$$ (11)

Now, use Eq. 10 and make the following additional assumptions to obtain a simple estimate. Let $a=b$ (assume an initial circle) and assume the distortion is area preserving; this second assumption yields $\delta b=-\delta a$. This leads to

$$\delta p\approx 2\left(1.067\right)\delta a\approx 2\delta a.$$ (12)

Next, using the results of the modeling of polypeptide tensile moduli, we will take the modulus of loop 6-7 to be (35) $E\sim 1-1.5$ pN/nm. Hence, we find that the force versus displacement relation of our assumed linear loop 6-7 spring is

$$f\approx 2-3\left(\text{pN}/\text{nm}\right)\delta a\left(\text{nm}\right),$$ (13)

and hence, $k_{\ell }\sim 2-3\text{pN}/\text{nm}$; in the following simulations, we choose $k_{\ell }=2.5\text{pN}/\text{nm}$.

Figure 12.

Idealization of the loop 6-7 binding model of Glc to Lys. The Glc-Lys complex is subjected to a combined thermal-hydrodynamic force $2f$, assumed for simplicity purposes to be equally partitioned between Glc and Lys. As loop 6-7 is stretched, it imposes a back-force on Lys; the spring-like response of loop 6-7 is represented by $k_{\ell }$. When loop 6-7 is critically stretched, Lys is decelerated, and Glc may break away. To see this figure in color, go online.

Simulations of the effects of skeletal dynamics

To perform specific simulations, we next considered the charge-dipole bond strength $f_d$ versus Lys-Glc distance d. The interaction potential between a point charge e located at ${\mathbf{r}}_i$ and a dipole $e{\mathbf{p}}_j$ (where ${\mathbf{p}}_j$ is the dipole vector defined in Fig. 11) is

$$V=-\frac{e^2}{4\pi \mathrm{ϵ}{r}_{ji}}\left[1-\frac{1}{{\left(1+2{\mathbf{r}}_{ji}·{\mathbf{p}}_j/r_{ji}^2+p_j^2/r_{ij}^2\right)}^{1/2}}\right],$$ (14)

where e is the electronic constant. E is the permitivity, hereby chosen as 6.95 × 10⁻¹⁰ F/m. $r_{ji}=\left|{\mathbf{r}}_{ji}\right|$ and $p_j=\left|{\mathbf{p}}_j\right|$.

Accordingly, the charge-dipole binding force $f_d$ is computed, as listed in Table 1; the geometry that defines the separation distance d and the dipole vector ${\mathbf{p}}_j$ is illustrated in the inset in Fig. 11.

We note that the magnitude of $f_d$ is a strong function of d and scales approximately as $f_d\sim 1/d^2$. Steric effects, however, limit the closeness that Glc may share with Lys. Here, we estimate the steric offset to be on the order of the diameter of Glc but slightly less, viz., $d=0.5-1.0\text{nm}$. This puts the range of $f_d$ to be $2.94\le f_d\le 12.9\text{pN}$, and the range for $\delta \overrightarrow{x}$ is $2.35\le \delta \overrightarrow{x}\le 9.75\text{nm}$. We note that this range compares reasonably with the activation distance range $\delta \overrightarrow{x}\approx 2-4\text{nm}$found in Kinetic Analysis Relating Our Experiments with Our Simulations.

We then conducted systematic simulations by using the mesoscale (Level II) model reported in (8, 9) to document the time it takes for a Glc molecule to be released from the Lys. Toward this end, we invoke the Langevin equation and model the dynamics of a Glc molecule. Because the inertia effect is negligible, we have,

$$D\frac{\text{d}{\mathbf{x}}_{\text{Glc}}}{\text{d}t}={\mathbf{F}}_{k_{\ell }}+{\mathbf{F}}_{adv}+{\mathbf{F}}_{stk}+{\mathbf{F}}_{\zeta },$$ (15)

where D is the drag coefficient, which is $3\pi \eta \delta /\left(1-\left(9\right)/\left(32\right)\delta /z\right)$ in the directions parallel with the bilayer and $3\pi \eta \delta /\left(1-\left(9\right)/\left(16\right)\delta /z\right)$ in the direction perpendicular to the bilayer ($\eta =6\times {10}^{-3}$ Pa s is the dynamic viscosity of the cytosol. z is the distance to the bilayer. δ is the diameter of the particle. To account for the fact that Glc moves together with Lys, we choose $\delta =2$nm). ${\mathbf{F}}_{k_{\ell }}$ represents the restoring effect from loop 6-7. ${\mathbf{F}}_{adv}$ is the advection force imposed by the F-actin, obtained as $\left(D\right)/\left(8\pi \eta \right){\int }_{actin}\left(1\right)/\left(r^3\right)\left[r^2\tilde{\mathbf{f}}+\left(\mathbf{r}·\tilde{\mathbf{f}}\right)\mathbf{r}\right]\text{d}\mathbf{x}\text{'}$, where $\mathbf{r}\equiv {\mathbf{x}}_{\text{Glc}}-\mathbf{x}\text{'}$, $r\equiv \left|\mathbf{r}\right|$, and $\tilde{\mathbf{f}}$ is the density of Stokeslet on the actin obtained in our Level II model. ${\mathbf{F}}_{stk}$ is the stickball effect estimated by modeling the direct contact between the Glc and the actin, which was modeled as a linear-spring force when the two touched each other. The component $F_{{\zeta }_i}$ (i = 1, 2, or 3) of the Brownian force ${\mathbf{F}}_{\zeta }$ is a Gaussian variable with zero mean and variance $\langle F_{{\zeta }_i}\left(t\right)F_{{\zeta }_j}\left(t^{\prime }\right)\rangle =2D{k}_{B}T{\delta }_{i,j}\delta \left(t-t^{\prime }\right)$.

Three different scenarios were considered: 1) solely thermally activated motion (i.e., diffusion) without advection or Glc contact with the actin protofilament; 2) stochastic advection driven by the actin protofilament; and 3) both advection and direct contact (stickball effect).

To elaborate, in these simulations only one JC is considered. The lipid bilayer is simplified as a wall that limits the motion of the actin protofilament to the upper half space. We further assume that the JC undergoes a shear deformation of λ = 1.5 (the same deformation as the one considered in Fig. 6). The initial location of the Glc is 7 nm from the center of the JC and 1 nm above the bilayer, as shown in Fig. 13. Each simulation is conducted until the displacement of the Glc reaches the activation length. The time it takes to achieve this is recorded as the release time. If it does not occur before 0.1 s, the case is tagged as “stuck.” To account for the stochastic nature of the problem, for each set of parameters, 1000 simulations are conducted to find the average value of the release times, which is dubbed ${\tau }_i$, where i = “free,” “adv,” or “adv/st” for scenarios 1, 2, and 3, respectively.

Given the dimensions of the F-actin protofilament and the low-pitch angle of F-actin, as qualitatively shown in Fig. 13 and quantitatively plotted in Fig. 4, we add that contact between actin and Glc has quite a high probability of occurring. The human erythrocyte’s F-Actin is indeed a model candidate for a batting championship title.

The results show several interesting outcomes that include, inter alia, a definite effect of advection on reducing the release time and a most definite stickball effect at all values of d and hence $\delta \overrightarrow{x}$. For example, examination of Table 2 shows the following trends. At the larger values of d, say $0.9\le d\le 1.0\text{nm}$, the effect of advection alone is seen to reduce release times by only 3–5%. The stickball effect, however, is proportionately much larger and tends to lead to ${\tau }_{\text{adv}/\text{st}}\approx 0.5{\tau }_{\text{free}}$. When d is in the range of $0.7\le d\le 0.8\text{nm}$, the effects of both advection and mechanical contact become even larger. Finally, when $d\to 0.5\text{nm}$, the binding force is so large that diffusion alone is insufficient to release Glc at any measurable times, but the stickball effect remains effective. We view these results as preliminary, yet nonetheless informative. Future theoretical study coupled with focused experimental data and observed phenomenology are required to refine our model of both Glc binding and clustering, and accordingly, their key parameters. For further perspective, we later compare the simulated release times with data on measured Glc uptake rates and use such comparisons to illustrate the value of our concept.

Table 2.

Release Times ${\tau }_i$ of Glucose versus d

d (nm)	$\delta \overrightarrow{x}$ (nm)	${\tau }_{\text{free}}$ (s)	${\tau }_{\text{adv}}$ (s)	${\tau }_{\text{adv}/\text{st}}$ (s)
1.0	2.35	$9.71\times {10}^{-8}$	$9.22\times {10}^{-8}$	$5.05\times {10}^{-8}$
0.9	2.94	$1.93\times {10}^{-7}$	$1.86\times {10}^{-7}$	$1.09\times {10}^{-7}$
0.8	3.75	$5.45\times {10}^{-7}$	$1.86\times {10}^{-7}$	$2.45\times {10}^{-7}$
0.7	4.95	$5.08\times {10}^{-6}$	$4.53\times {10}^{-6}$	$3.69\times {10}^{-7}$
0.6	6.76	$5.34\times {10}^{-4}$	$4.68\times {10}^{-4}$	$8.79\times {10}^{-7}$
0.5	9.75	stuck	stuck	$8.53\times {10}^{-6}$

Results and Discussion

We have demonstrated herein that Glc uptake can indeed be affected or even mediated by the dynamics of the human erythrocyte skeleton. In this, we have focused on the influence of hydrodynamic forces and the direct mechanical contact of F-actin and Glc (i.e., the stickball effect) that primarily arise from the motion of the F-actin protofilament located at the Sp JC, as illustrated in Figs. 1, 2, 8, and 12. We have, additionally, noted that skeletal dynamics may contribute by creating open molecular space, allowing egress of trapped Glc from the endoplasmic space of GLUT1. This latter effect can be appreciated by noting how such space is opened up during mode transitions, as depicted by snapshots in the third column of Fig. 13. Hence, molecules trapped in compartment 1 of Fig. 2 may find release, but on timescales that are a factor of 20–40 larger than the ${\tau }_i$ of Table 2 (see Fig. 5).

Still another interesting question concerns the range of values forecasted for Glc release times ${\tau }_i$, as listed in Table 2, compared to the kinetics of Glc transit through GLUT1. For such perspective, we consider the data obtained by Hu et al. (12) in their study of Glc transport in patients with Chinese type 2 diabetes. They measured an approximate 25% reduction in Glc uptake in type 2 diabetic patients as compared to their “control healthy” group. This was accompanied by a 30% increase in their estimated uptake activation energy. They defined the initial rate of Glc concentration as $\text{d}{C}_s/\text{d}t=V_0$ and found that $V_0=40.56\pm 9.04$ mM/(min-50 μL) and $V_0=31.36\pm 3.67$(min-50 μL) in healthy and diabetic patients, respectively. Their estimate for the number of cells in their study was $\sim 3\times {10}^8$ cells/mL. The estimated number of GLUT1 copies per cell is here taken as $2\times {10}^5$ (36). If these numbers are used, we estimate the Glc transit rate (i.e., number of molecules/s) to be of order $\mathcal{O}\left({10}^5{\text{s}}^{-1}\right)$, which would place the average transit time of a Glc molecule through GLUT1 to be of order $\mathcal{O}\left({10}^{-5}s\right)$. On the other hand, data from Lowe and Walmsely (37) and Cloherty et al. (3) find values for $V_{\text{max}}\sim 1.8\text{M}/\left(\text{L of cell water }-\text{ min}\right)$. Taking the volume of cell water at $\sim 66\text{fL}$ and the number of copies of GLUT1 at $1.5\times {10}^5/\text{cell}$, we find the Glc passage rate per GLUT1 to be $\mathcal{O}\left({10}^4{\text{s}}^{-1}\right)$. When either of these estimates are compared to the ${\tau }_i$ of Table 2, we conclude that the skeletal dynamic effects we have postulated and demonstrated will indeed act on relevant Glc transit timescales and should affect Glc uptake significantly. We take this observation a step further using the Hu et al. (12) study.

Hu et al. (12) postulate that decreased Glc entry through erythrocyte membranes is due to “the GLUT1 change in structure.” They speculate that such changes may occur at the “outer domain” of GLUT1 (12). Alternatively, we speculate that such reductions may easily be caused by any one of many effects that interfere with skeletal/actin dynamics, which can include, inter alia, 1) cell aging that leads to a disruption of the Sp/actin JC (38); 2) disruption of actin stability (39); and 3) a host of factors that affect skeletal stability and its connectivity with the membrane (see (19, 20)). Thus, our postulate and our preliminary results presented herein create, to our knowledge, an entirely novel pathway for understanding the underlying causes of disease motifs, and therefore, a novel pathway for their potential remedies.

Advection caused by skeletal dynamics may have another quite general effect on intracellular transport that relates to the concept of unstirred layers that often enter into modeling intracellular diffusion, especially within the endofacial region (1). Of course, any such assessment needs to address the fact that endofacial regions are quite occluded with various matter (40), as we have emphasized throughout. Hence, the idea that F-actin filaments “stir” an open fluid medium needs to be analyzed in more detail and vis-à-vis the particular geometries presented by various cells. Nonetheless, the hydrodynamic activity we have revealed and analyzed suggests that such layers, if existent, may be the result of molecular crowding and that skeletal dynamics does tend to drive mixing. Still another broader question concerns the role(s) of intramolecular dynamics on transport, especially within the “crowded” endofacial cellular space. Recent studies have focused on analysis of non-Fickian transport within such crowded spaces (37, 41, 42, 43, 44), but without specific attention to intracellular molecular dynamics driving advection and/or direct mechanical action, as in our stickball effect. Our results and hypothesis call for focused attention on such effects.

As noted above, Carruthers and co-workers (3, 13) have found that 3OMG uptake is monophasic in the absence of cytoplasmic ATP but biphasic when ATP is present. Biphasic exchange is observed as the rapid filling of a large compartment comprising ∼66% of the cytoplasmic space. As this occurs at both lower and higher Glc concentrations, viz., $\sim 20\mu \text{M}$ and $\sim 20\text{mM}$, respectively, they questioned that the slow step was simply associated with Glc binding to GLUT1 because the concentration is such that $\left[GLUT1\right]\mathrm{\lesssim }20\mu \text{M}$. The results described herein, however, suggest a somewhat augmented scenario based on skeletal dynamics, which could go as follows. As Glc “slides” (33), or “tumbles” via docking sites through GLUT1 in serial form, binding to GLUT1 is probable as we describe, but we show that skeletal dynamics is quite effective, especially in the presence of ATP, in its release. Thus, the lower concentration of GLUT1, i.e., $\left[Glut1\right]<\left[Glc\right]$, does not deny Glc-GLUT1 binding as an important kinetic step. Skeletal dynamics is indeed an important player in the total kinetic process. We also call attention to the contribution of what we have dubbed “opening up of molecular space,” which is associated with the mode switching described in Fig. 4. It is noteworthy that this occurs on a timescale of order $\mathcal{O}\left({10}^{-5}-{10}^{-4}\text{s}\right)$, which is on the order of the Glc passage rates described above.

We further note that except for the symporter GLUT13, all GLUT transporters are uniporters that facilitate passive diffusion (45). They share a similar structure to that of GLUT1 and comprise transmembrane helices with a binding site located in the central region of the transporter delineated by residues in the N- and C-domains (46). A substantial endofacial cytosolic linker joins the N- and C-domains and may play a role in transport function by securing the closure of the inward gate (46, 47). Additionally, GLUT1 is responsible for cellular Glc transport in erythrocytes, smooth muscle, astrocytes, and endothelial cells of blood tissue barriers (45). Therefore, the model postulated here may apply more generally to other transporters subjected to hydrodynamic forces or shear deformations. The concept of skeleton-enhanced transport may indeed provide important insight into the mechanistic understanding of sugar transport in a wide variety of processes.

Advection-enhanced transport may also be driven by hydrodynamics caused by thermal fluctuations of the lipid bilayer itself (along with the skeleton). Within the framework of Stokes flow, as used herein, hydrodynamic flow fields have been analyzed as part of the study of membrane motion; this has been done for microrheology measurements (e.g., (48)) in which fluid-structure interactions are required to determine the membrane motions within a fluid medium. Of course, in our full cell model, as shown in Fig. 3, this is also accounted for. However, such effects are far less important than those caused by the skeletal motions and do not in any way account for the important effects involving mechanical contact, i.e., our stickball effect, which we have shown to be predominant especially when mode switching occurs, such as in shear deformation.

Clearly, what is required for progress are focused experimental studies that specifically explore correlations between altered skeletal dynamics and Glc uptake kinetics. Such studies could include, inter alia, measuring uptake kinetics with disruption of the skeleton and/or the JC, disruption of actin stability, and possibly subjection of cells to imposed shear deformation. Note that all such influences have a predictable impact on skeletal dynamics. Data obtained from such studies allow for the advancement of the simple models for Glc binding and/or entrapment that we have used herein. As for imposed shear deformation, however, given the predicted effects on skeletal dynamics, e.g., as driven by imposed shear flows, such experiments may involve a complex path to interpreting the effect on GLUT 1 rates (i.e., $k_{\text{cat}}$). For example, we have discussed that such shear leads to higher influx of Ca⁺⁺ (49, 50), and this in turn leads to activation of Ca ATPase and the interaction of Ca-calmodulin complexes that can disrupt skeletal/membrane linkages and hence adversely affect skeletal dynamics (51, 52, 53). Ca-calmodulin complexes are reported to down-regulate the ability of adducin to bind actin and stimulate Sp-actin binding (53). Moreover, such kinetic experiments involving input of energy via substantial imposed shear flow will require precise temperature control given the magnitude of the activation energies for GLUT1 transport, as reported by Lowe and Walmsey (37). Hence, a less ambiguous path might be to explore biochemical means such as those noted above.

In regard to biochemical perturbations on membrane/skeleton dynamics, we note the studies of Fowler et al (39, 54, 55, 56, 57). They report that actin stability directly affects the biomechanical properties of the erythrocyte membrane/skeleton. Specifically, they studied actin stability and connectivity to the skeleton via treatment with disrupting drugs such as cytochalasin-D (CytoD), latrunculin-A, and jasplakinolide and demonstrated that these drugs lead to significant alteration in mechanical response. For example, CytoD inhibits actin subunit association and disassociation at barbed ends, and latrunculin-A destabilizes and depolymerizes dynamic actin filaments by binding to and sequestering actin monomers. Hence, studies of Glc uptake with or without imposed shear would be of potential value to assess the role of the skeletal/actin dynamics we propose. Such studies could be extended to other members of the cytochalasin family as well as to phallodin (58, 59). Of course, such studies should account for any collateral effects such as alterations in medium or cytosol viscosity, as we have discussed above. Interestingly, Cooper et al. (58) have already reported the apparent effects of CytoD binding to actin on cell deformability.

The influences of membrane dynamics coupled with skeletal dynamics were briefly discussed above and should be explored in further detail. Iglesias-Fernandez et al. (46) have indeed initiated an important such step via molecular dynamics simulations of the effects of temperature on the coupling of the membrane to GLUT1. They show that as the membrane temperature decreases and approaches the gel phase, the “size of cavities and tunnels traversing the transporter” is reduced, as is the net Glc transport rate. We add that these same changes in membrane behavior with temperature will have an attenuating effect as well on skeletal dynamics, and hence we suggest that an extension of these simulation models should be pursued that includes the dynamic skeleton and especially focuses on the dynamics of the loop 6-7 linker (47).

Finally, with regard to such biochemical effects, we mention that Jung et al. (60) reported that cytochalasin-B induced “rather drastic changes in the exchange kinetics” of hydrogen exchange within and around GLUT1 and attributed this to a possible “significant perturbation in the protein (GLUT1).” We may speculate that they were recording effects of altered membrane/skeletal dynamics along the lines we hypothesize.

Supporting experimental results

Our additional recent experimental results suggest that erythrocyte membranes cannot be described as passive transport structures (61). Support for this concept has been provided by the presence of tropomyosin and myosin in the erythrocyte membrane (62). In the case of erythrocytes, their glycolytic system provides sufficient amounts of ATP for contractile and transport ATPases and kinases. Previous studies have shown that erythrocytes, ATPases, and kinases consume ∼25% of the ATP, and the rest is used by the contractile enzymatic system of Sp-dependent ATPases (63). These experimental studies extended the behavior of erythrocytes under mechanical stress.

We note that although the timescales of the experiments we describe below in Supporting Experimental Results do not directly relate to the kinetics of GLUT1, the results do reveal hydrodynamic influence via correlations with the capillary number that provides perspective, as discussed below and in Hypothesis and Its Basis. The results are consistent with those of Kodi$\stackrel{⌣}{c}$ek et al. (50), who suggest a metabolic Ca²⁺ ATPase influence associated with skeletal reconstruction under shear flow. As reviewed by Bogdanova et al. (51), Ca⁺⁺ uptake is indeed enhanced by shear deformations, as noted in experiments imposing shear flow, and this is accompanied by activation of Ca ATPase, which is important for maintaining skeletal/membrane integrity under enhanced dynamics caused by imposed shearing (49). However, as discussed above, we note that the resulting interaction of calcium-calmodulin complexes with the skeleton’s elements can disrupt skeletal connections and affect actin as well.

Methods and results

Fresh, heparinized blood was collected from healthy volunteers and centrifuged, and plasma and buffy coat were discarded. The erythrocytes were then washed four times to remove any remaining white cells or platelets. RBCs were washed and suspended at 50% hematocrit in fresh plasma (1.2 cP). The Glc concentration in those aliquots was initially set at 12 mM by adding Glc solution. The aliquots of the RBCs at 37°C were maintained at staged condition or subjected to shear rates of 1, 10, and 100 s⁻¹ in a computerized cone-plate (4 cm diameter and 2° cone cell) rheometer AR-G2 (TA Instruments, New Castle, DE). Samples (50 μL) were taken immediately after adding Glc and after applying mechanical shear for 30, 60, 120, 180, and 240 min. Plasma was separated from the samples by centrifugation, and the Glc concentration in the plasma was determined using an YSI 2300 STAT Plus (Yellow Springs Instrument, Yellow Springs, OH). Glc consumption was calculated from a linear regression of Glc concentration versus time. As demonstrated in Fig. 14, these experiments confirm that mechanical stress strongly enhances the consumption of Glc by RBCs.

Figure 14.

(Left) Time course of glucose utilization by erythrocytes suspended in plasma at different shear rates. (Middle) The erythrocyte glucose consumption is shown at different shear rates, calculated based on the slopes of the glucose concentration using least-square linear regression. (Right) The absolute values of erythrocyte glucose consumption at different shear rates are shown. To see this figure in color, go online.

Similarly, washed RBCs were suspended in plasma with increased viscosity (1.8 or 2.4 cP) by the addition of dextran 70k Da (Pharmacia, Uppsala, Sweden). It is seen that in shear flows, higher viscosity of external fluid leads to more Glc consumption (Fig. 15).

Figure 15.

Effects of plasma viscosity on erythrocyte glucose consumption at different shear rates. These experiments confirm that mechanical stress strongly enhances the consumption of glucose by RBCs under physiological conditions. The results show a good hydrodynamic correlation with the capillary number defined as $\eta R\dot{\gamma }/\mu$; here, η is viscosity, R is the effective cell radius, $\dot{\gamma }$ is the shearing rate, and μ is the cell stiffness.

We have found that similar trends exist at far shorter timescales, as discussed in the following section with regard to Fig. 16.

Figure 16.

(a) Glucose uptake measured at $2.3\pm {0.2}^{\circ }$C vs. time. (b) Temperature is over 250 s. (c) Uptake is shown over first the 30 s, showing linearity. To see this figure in color, go online.

Supporting experimental results: short timescales

Glc uptake was measured as a function of time at a temperature of $2.3\pm {0.2}^{\circ }$C, with and without an imposed shear rate of $\dot{\gamma }=100s^{-1}$; the results are shown in Fig. 16.

As noted above, care was taken to control temperature to at least $\pm {0.2}^{\circ }$C for time periods up to 250 s (see Fig. 16 b); this meant that $k_{\text{cat}}$ estimates would be at least in the range of $\pm 5\text{\%}$. For $t<30$ s, the uptake versus time response was sufficiently linear to calculate estimates of $k_{\text{cat}}$ (see Fig. 16 c). These lead to estimates of $k_{\text{cat}}^{\text{w}}=132\pm 6.6$ and $k_{\text{cat}}^{\text{wo}}=99\pm 4.95{\text{min}}^{-1}$, with(w) and without(wo) imposed shear, respectively. This represented an approximately $33\text{\%}$ increase in uptake rate at ${2.3}^{\circ }$ at $\dot{\gamma }=100s^{-1}$. We mention that these rates are quite consistent with those reported by Lowe and Walmsley at $0^{\circ }$C (37); indeed, as above, we used their measured activation energies between $0-{20}^{\circ }\text{C}$ to estimate the temperature dependence of $k_{\text{cat}}$. Here again, we note that precise assessment of these results awaits further detail of the timescales of the Ca⁺⁺-Ca ATPase-Ca calmodulin process versus $k_{\text{cat}}$ rates at short times (see (64)). We speculate, however, that our results are consistent with the view that our proposed mechanisms are robust enough to maintain high $k_{\text{cat}}$ rates even with the possibility of existent disruptions to the skeleton.

Methods at shorter timescales

Net uptake of 20 mM extracellular 3OMG was measured in cells that were initially free of intracellular sugar. Blood was collected via venupuncture from four healthy volunteers into syringes containing citrate-phosphate dextrose. RBCs were washed three times with ice-cold HEPES wash buffer (pH 7.4) with 150 mM KCl, 5 mM, and 0.5 mM EDTA via centrifugation (2600 × g at 4°C for 5 min). All ascorbic acid was consumed with ascorbate oxidase (15 U/mL), and incubation was conducted at room temperature for 30 min. RBCs were depleted of intracellular sugar by incubation at room temperature for 1 h postdilution in 20 volumes of HEPES wash buffer. Uptake measurements of 3OMG at 20 mM were completed four times via mixing 100 μL of a 50% suspension of RBCs with 400 μL of 25 mM 3OMG solution. Reactions were stopped by the addition of 1 mL of ice-cold stop solution containing 10 μM cytochalasin-B and 100 μM phloretin in HEPES wash buffer. The solution was centrifugated at 10,000 × g for 2 min and washed once with the stop solution. Then, the pellet was lysed in 0.2 mL of 5% perchloric acid and centrifuged at 10,000 × g for 2 min, and 100 μL of the clear supernatant was assayed to determine the amount of labeled 3OMG.

Conclusions

We demonstrate herein that hydrodynamic forces and the mechanical contact that the skeleton makes with Glc resulting from skeletal/actin dynamics significantly affect the kinetics of Glc release from the GLUT1 cytoplasmic side of the erythrocyte membrane. Our forecasted effects are consistent with a number of documented experimental observations outlined, for example, in Background on Glc Uptake into Erythrocytes and Hypothesis and Its Basis and discussed just above. This hypothesis represents, to our knowledge, a truly novel insight into a more general mechanistic understanding of the transport of ions or small molecules, as also exemplified by our simulation of the release of Ca⁺⁺ bound via a simple electrostatic field in Example: Advection of Ions. We have suggested pathways for exploring our hypothesis, most notably biochemical pathways that would alter skeletal dynamics and affect the structure of actin, which acts as the “hub” of the skeleton JC. We note that although the imposition of shear deformation does enhance skeletal dynamics, it also causes effects that present challenges to interpreting the results. The shear deformations that enhance skeletal dynamics are an entirely natural feature of the erythrocyte skeleton and require no externally applied shear deformation.

Editor: Elsa Yan.