Protein Adsorption on Well-Characterized Polyethylene Oxide

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Protein Adsorption on Well-Characterized Polyethylene Oxide Brushes on Gold: Dependence on Molecular Weight and Grafting Density Warren Taylor*,† and Richard A. L. Jones‡ †

Materials Research Laboratory, MC 5121, University of California, Santa Barbara, California 93106-5121, United States The Department of Physics and Astronomy, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, United Kingdom

‡

ABSTRACT: The adsorption of lysozyme protein was measured ex situ on well-characterized gold surfaces coated by end-tethered polyethylene oxide brushes of various molecular weights and controlled grafting densities. The adsorbed amount of protein for different molecular weight brushes was found to collapse onto one master curve when plotted against brush coverage. We interpret this relationship in terms of a model involving site-blocking of the adsorption of proteins at the substrate and discuss the role of the physical attraction of PEO segments to gold. We account for our observation of a simple exponential relationship between protein adsorption and normalized brush coverage with a simple protein adsorption model. In contrast to other studies in similar systems, we do not observe protein adsorption on brushes at high grafting density, and we suggest that this discrepancy may be due to the solubility effects of salt upon the brushes, influencing their protein binding affinity, in the limit of high grafting density and high brush volume fraction.

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with proteins.19,20 However, light scattering experiments conducted by Bloustine et al. have found that PEO has a weak affinity for lysozyme proteins, which they suggest may be due to hydrogen bonding between the polymer and protein.21 It has also been shown in studies conducted in the Leckband group, that under compression, PEO brushes can become attractive to streptavidin protein-coated surfaces, and that the nature of this interaction is dependent upon temperature and the Mw of the PEO chains.22−24 Thus, the exact nature of PEO’s segment level interaction with proteins remains somewhat mysterious and complicated. The approach of polymer physics to the problem of the interaction between proteins and PEO brushes, in contrast, has been to stress generic, chain-level aspects of the interaction.1,10,11,17,18,25−35 Fundamental to all these theories is the important notion that polymer brushes resist protein adsorption due to energetically unfavorable restrictions in chain conformation. In this scenario, the chains act as entropic barriers preventing the insertion of proteins thermodynamically and kinetically. Satulovsky et al. and Szleifer have also investigated, through theory work, the situation where segments of the brush polymers are attracted to their substrate, which is an important but sometimes over looked nuance of protein adsorption at polymer brushes.26,30 In this scenario, the tethered chains no longer act solely as entropic barriers but also compete for the substrate, preventing protein adsorption enthalpically.

INTRODUCTION The adsorption of proteins is the first stage in the formation of biofilms on a wide range of surfaces, including algae on the hulls of ocean liners and plaque on the enamel of teeth.1,2 Biofilms once formed can be very difficult to remove and in some cases even resistant to antibiotics and bleach.3,4 Therefore, surface coatings that resist biofilm formation are highly sought after. One approach is to coat the surface with end-tethered polyethylene oxide (PEO) polymer brushes, which have been found to resist the initial adsorption of colloidal proteins that facilitate biofilm formation.5−16 Numerous experimental studies have investigated the adsorption of proteins at PEO brushes using an array of different substrates, tethering approaches, and analysis techniques. Most of these studies find that protein adsorption decreases with increasing polymer grafting density.5,6,11,14,15 However, it is still unclear how specific PEO brush−substrate systems resist protein adsorption and in more detail exactly why and how the grafting density and chain molecular weight influence the kinetics and adsorbed amount of protein at these surfaces. Some explanations emphasize the nature of the chemical interactions between PEO at segment level, water, and protein. PEO is charge neutral and hydrophilic, under most conditions, meaning that it will not interact with proteins through electrostatic or hydrophobic interactions. Jeon et al. point to the low refractive index of PEO (1.47), which should result in weak van der Waals forces with proteins, making them the most protein resistive of the water-soluble polymers.17,18 Another possible explanation for the weak interaction between proteins and PEO is strong hydrogen binding between PEO and water, which may create a hydration layer that prevents interactions © 2013 American Chemical Society

Received: September 5, 2012 Revised: April 24, 2013 Published: April 25, 2013 6116

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The results found in this study differ from two other works which found that similar mPEG brushes exhibit protein adsorption at high grafting densities. We suggest that this discrepancy is due to the very low salt concentration used in this study, preventing adverse solubility affects at high grafting density PEO brushes. A very low salt concentration was chosen to keep the system as physically simple as possible in order to observe the protein resistive behavior of PEO in its most unperturbed conditions. These simple conditions are essential when attempting to understand the protein resistive behavior of PEO, which has been shown in the literature to be very complicated and seemingly contradictory.

The rather simpler random sequential adsorption model of Katira et al. interprets much of the adsorbed amount of protein at polymer brushes as the result of bald spots caused by unavoidable lateral fluctuations in brush coverage.36 As the brush coverage increases the probability of bald spots drops, resulting in reduced protein adsorption. Gon et al. have also experimentally determined the influence intentional poly(Llysine) (PLL) defects have on fibrinogen protein adsorption at poly(L-lysine)-b-polyethylene glycol (PLL-PEG) bottle brush surfaces. It was found that as the PLL defect concentration was increased past a specific threshold, which corresponded to the dimensions of fibrinogen (4.5 × 4.5 × 45 nm), increased protein adsorption was observed.37 It can be seen, then, that despite its practical importance and the extensive degree to which it has been studied, protein adsorption at PEO brushes is still not fully understood, with a variety of possible theoretical interpretations. This lack of clarity is partly due to conceptual complications in interpreting protein adsorption data, at specific polymer brush surfaces, but is also due to inherent difficulties in making well-characterizable polymer brushes in which reliable trends can be determined. Challenges to fabricating suitable polymer brushes for protein adsorption experiments include the need to: (a) produce strongly adhered end-tethered brushes, (b) produce brushes that are adhered to a smooth and well-characterized chemically homogeneous surface, and (c) have accurate control and knowledge of the grafting density and molecular weight of grafted brushes over a wide range of parameter space. In previous work, we developed a novel polymer brush grafting-to procedure, utilizing a concentrated homopolymer solution, that allowed us to end tether high Mw PEO-SH chains onto relatively flat (2 nm rms) gold surfaces. 38 We demonstrated in this previous study that we could gain an accurate control of grafting density over the range of grafting densities at which polymer brushes become protein resistive. Importantly, we also demonstrated that PEO has a surprisingly strong physical attraction to the gold substrate. In this work, we show that the adsorbed amount of protein at these well-characterized end-tethered PEO-gold brushes, after 20 min (∼103 s), decreases directly with coverage, Ω = Apσ, where Ap is the area a polymer covers with a radius of gyration in solution unstretched. Plotting the data against coverage results in the collapse of data collected over differing molecular weights, onto one master curve, indicating a universal relationship. We suggest that the simplest interpretation of these results refers to the physical attractions between the PEO chains and the gold substrate, and results in brushes which site block the area they cover. Therefore, lateral fluctuations in surface coverage determine the adsorbed amount of protein. We further support this hypothesis by demonstrating a simple model where polymers are randomly grafted on to a surface, resulting in an exponential decrease in protein adsorption sites with increasing brush coverage, as follows,

μ ≈ μmax e

−c Ω

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EXPERIMENTAL SECTION

Polymers. 2k Mw thiol-terminated PEO methyl ether was purchased from Polymer Source. 5k and 20k Mw thiol-terminated PEO methyl ether was purchased from Jenkem Ltd. Polymer brushes were produced via concentrated homopolymer solution and are described in detail in a previous paper by the authors.38 Dialysis and Freeze-Drying of Lysozyme Protein. Chicken egg-white lysozyme CAS number 12650-88-3 was purchased from Sigma-Aldrich limited (Mw equals ∼14307 Da). Protein Solution Preparation Ex Situ. Lysozyme (200 mg) was weighed out into a 500 mL flask with 100 mL of ultra pure water with 1 mM of Tris-HCl buffer, pH ∼7.5. This produced a solution with a protein concentration of 2 mg/mL. The solution was allowed to equilibrate to a temperature of 22 °C in a water bath. Ex-Situ Protein Adsorption Experimental Procedure. In order to produce consistent results, the protein adsorption experiments were conducted in a batch method, with up to 9 polymer brush substrates being placed into the same beaker and submerged in 50 mL of ultra pure Tris-HCl buffered water. To begin the 20 min long protein adsorption assays, another 50 mL of 2 mg/mL lysozyme protein solution was added to the beaker, producing a protein solution of concentration 1 mg/mL. After 20 min, the sample solution was diluted with ultrapure water until the beaker only contained ultrapure water so as to avoid taking the sample through the air−liquid interface of a protein-containing solution.39 At this point, the samples were removed, rinsed with ultra pure water, dried under nitrogen, and measured via ellipsometry. Ellipsometry. A J. A. Woollam Company, Inc. ellipsometer was used to measure the thickness of the dry PEO brush layers adhered to the gold-coated silicon wafer. An initial ellipsometry scan was taken for each wafer to determine the reflectivity profile of the bare gold surface. This reflectivity was then saved as a B-spline material file, the Complete Ease software supplied by Woollam. The B-spline material file for each wafer was then used in a two-layer model to reveal the thickness of the end-attached PEO brushes, which were modeled using a Cauchy layer with parameters An = 1.45, Bn = 0.1, and Cn = 0. This method was found to produce the most consistent and accurate results with χ2 typically less than 1. Brush Characterization. The grafting densities of the polymer brushes were calculated using the following equation,

σ=

hρNA Mw

(2)

where h is the dry height of the brush layer, ρ is the density of PEO at 1 g/cm3, NA is Avogadro’s constant, and Mw is the molecular weight of the chains (which in our experiment takes the values 2k, 5k, or 20k), and σ is the grafting density of the polymer chains (chains/ nm2).13,40,41 The coverage of polymer at each surface was found by multiplying the grafting density by the area of a polymer chain, i.e.,

(1)

where μmax is the maximum adsorbed amount of protein on a bare surface and c is a fitted parameter. Hence, suitable theoretical explanations should be able to predict an exponential or similar fractional decrease in protein adsorption sites with increasing brush coverage for the PEO−gold brush system and other systems where the chains are attracted to their substrate.

Ω = σπR g2

(3)

where Rg is the radius of gyration of a PEO chain given by,

R g = (Nblk cos(ψ )/6)1/2 6117

(4)

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where N is the chain polymerization, b = 0.278 nm is the monomer length, lk = 1 nm is the Kuhn length of a monomer, and ψ = 37.5° is the monomer bond angle.36 Refractive Index and Density of Lysozyme. The refractive index of lysozyme is ∼1.4842, which is practically identical to that of PEO at 1.47.41 Therefore, a refractive index of 1.47 was used to model the protein−brush surface. Knowing the protein add-layer height and density (1.36 × 103 kg/m342), eq 2 can be used to calculate the protein density.

brush layers on silica, whereas negative net charge proteins such as fibrinogen, albumin, myoglobin, and alkaline phosphate would not.46 The result of Gon et al. contrasts with the work of Unsworth et al., who found that positive lysozyme and negative fibrinogen would adsorb to high grafting density brushes. As mentioned in the introduction of this paper, it is generally believed that proteins do not bind to PEO or at least there is only a very weak interaction; however, several studies, including those mentioned above, have shed doubt on those assumptions. In particular, several surface force measurement studies conducted in the Leckband group have found that PEO brushes show strong binding with streptavidin proteins, when under strong compressive forces.1,22−24 The nature of this attraction was also found to be dependent on temperature and the molecular weight of the chains.24 It has been hypothesized in theory work by Halperin et al. that PEO brushes under compression can become phaseseparated, producing potential protein adsorption sites.1,33 PEO is an unusual polymer in that its solubility in water is dependent upon temperature, salt concentration, and polymer volume fraction.1,20,47−49 This dependence manifests itself in an upper and lower solubility gap, which is speculated to be due to the breakdown of hydrogen bonding between the polymer and the surrounding water structure.20,47−49 Therefore, it may be the case that in poor solvent conditions (i.e., high brush volume fraction, high salt, or high temperature conditions), PEO is actively protein binding. Of importance to the study presented here is the possibility that the presence of salt at high volume fraction brushes (high grafting density) induces protein adsorption through solubility affects. There is some evidence to support this hypothesis from the literature concerning protein crystallization in PEO solutions, where Galkin et al. found that the liquid−liquid metastable phase boundary of protein−PEO solutions is salt concentration dependent.50 A possible interpretation of this solubility hypothesis is that the local water structure is disrupted at high volume fraction brushes in the presence of salt, resulting in PEO−lysozyme hydrogen bonding being preferred over PEO−water hydrogen bonding. This hypothesis is indirectly supported by molecular simulations, conducted by Jain et al., showing that PEO will hydrogen bond with lysine peptides.51 Bloustine et al. also suggest that hydrogen bonding between lysozyme and PEO is responsible for attractive interactions observed in their light-scattering experiments.21 Alternatively, other salt-induced solubility affects, other than protein−polymer hydrogen bonding, such as chain hydrophobicity or dehydration may account for increased protein adsorption at high grafting density brushes. The experiments conducted here are done so with no added salt. Only 1 mM of Tris-HCl buffer was added, and no high grafting density protein adsorption was observed, whereas Unsworth et al. used a PBS buffer presumably of high salt content and saw protein adsorption at high grafting densities.13,40,45 Gon et al. also used a PBS buffer with 0.008 M Na2HPO4 and 0.002 M of KH2PO4 and also saw protein adsorption. It is worth noting that the solubility of PEO in water is particularly sensitive to sodium and potassium phosphate salts.46 Therefore, it could be the case that the solubility effects of salt, tip the scales for PEO being protein binding or repellent at high volume fraction brushes. If this is true, high Mw brushes would exhibit better protein resistivity as protein adsorption can be reduced at lower grafting densities and therefore lower

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RESULTS AND DISCUSSION Ex Situ Ellipsometry Experiments. Figure 1 shows the adsorbed amount of lysozyme protein (mg/m2) plotted against

Figure 1. Protein density plotted against grafting density, σ, of the PEO brush layers. ■, ●, and ⧫ are 20, 5, and 2 kDa molecular weight chains, respectively.

the grafting density of PEO brushes of molecular weight 2, 5, and 20 kDa and clearly shows a molecular weight and grafting density dependence for the adsorption of lysozyme protein at PEO brushes. As the brush grafting density increases, the adsorbed protein density drops down to zero. However, as the molecular weight of the chains decreases, the brush grafting density at which the surfaces become protein resistant increases. This observation is in accord with a number of other studies in the literature, though there are some apparently conflicting observations, which will be discussed next.5−7,14,15 Protein Adsorption at High Grafting Density Brushes. We see no increased protein adsorption at higher grafting densities, in contrast to studies conducted by Unsworth et al. and Gon et al.13,40,43,44 Unsworth et al. observed increased adsorption at high grafting densities for lysozyme and fibrinogen adsorption at 750, 2k and 5k thiol-terminated PEO methyl ether polymers deposited on gold. They found that such adsorption was not seen when hydroxyl terminated 600 Mw PEO (−OH) was used and suggest that at high grafting densities, either (A) the methyl ethyl (−OCH3) end groups present a high density of hydrophobic groups for adsorption at the edge of the brush or (B) the hydroxyl (−OH) end groups prevent the adsorption of protein at the edge of brush, which would have otherwise occurred due to strained hydration affects at high grafting density brushes.13,40,45 It has also been observed by Gon et al. that proteins with a net positive charge such as Lysozyme would adsorb at high grafting density PLL−PEO 6118

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For instance, the four data points collected before the 20k Mw brushes become protein resistive, having approximately the same grafting density, but the adsorbed amount of protein ranges from approximately 0.7 to 2 mg/m2, with no coherent correlation, implying large systematic errors or inconsistencies at these low grafting density high molecular brushes. However, it is possible to determine that the coverage at which the 20k Mw brushes becomes protein resistive is approximately the same as the 2k and 5k series and, from the data points that are available, that the shape of the curve is not drastically different. Pasche et al. also found a similar collapse in the adsorbed amount of HSA protein at physically adsorbed poly(L-lysine)graf t-poly(ethylene glycol) brushes when plotted against the ethylene oxide surface monomer density (σN).15 The data of Pasche et al. were also compared by Dalsin et al. to HSA adsorption on titanium oxide surfaces coated with 5k Mw endtethered mPEG-DOPA polymer brushes, and a good agreement was found.6 Pasche et al. and Dalsin et al. attribute this relationship with EG monomer density to the chains being in a stretched polymer brush conformation. In recent experimental and theory work by Genzer et al., the initial adsorbed amount of protein at poly(2-hydroxyethyl methacrylate) (PHEMA) brushes is shown to scale with monomer surface density (σN).25 However, for timescales greater than 10−1 s, they assert that this relationship only applies for polymer brushes in poor solvent, which is the case for the PHEMA/water system. For polymer brushes in good solvent, the adsorbed amount no longer follows a universal relationship with σN, becoming increasingly dependent upon N or Mw. Water is a good solvent for PEO; however, a coverage relationship is still observed for the data presented here after 103 s, using high protein concentrations (1 mg/mL), implying that an alternative explanation is required for the PEO−gold system discussed here. A possible explanation is that PEO polymers are physically attracted to gold. In previous work by the authors of this paper, it was found that nonthiol-terminated PEO has a remarkably strong affinity for gold, forming physically bound layers several nanometers thick. These physically bound layers required sonication for 15 min in chloroform before they were completely removed.38 Therefore, the end-tethered PEO brushes described here may also be bound via the physical adhesion of PEO monomers with the gold substrate. Theory work by Satulovsky et al. and Szleifer have explored the affect polymer brush substrate attraction has on the kinetics and thermodynamic control of protein adsorption.26,30 For instance, brushes which are not attracted to their substrate will act as entropic barriers, primarily reducing the kinetics of adsorption over the equilibrium adsorbed amount, while polymer brushes, which are attracted to their substrate, act less like entropic barriers but rather compete for the surface, reducing the equilibrium adsorbed amount of protein, without drastically altering the kinetics of the adsorption process. The notion that PEO is acting as a surface competing barrier and not as an entropic barrier opens up the possibility that protein adsorption at such surfaces is dominated by the presence of bald spots caused by lateral fluctuations in chain density. Katira et al. have explored the affect the random distribution in chain grafting has on protein adsorption at PEO brushes using a random sequential model.6 They have compared this random sequential adsorption model to selected data available in the literature and have found some good agreement. Next, we will

volume fractions, avoiding the onset of solubility- or compression-induced protein adsorption. Coverage Relationship. The adsorbed protein density was plotted against the coverage of polymer upon the surface using eq 3, Ω = σπRg2. This coverage assumes that each polymer covers an area equivalent to its radius of gyration and that when this coverage exceeds a value of 1, the brushes are overlapping and in the brush regime. It must be noted that this term is a dimensionless quantity relating the Mw and the grafting density to the area covered by spherical polymers at the surface. The result of plotting the protein adsorption data in this way can be seen in Figure 2. It is immediately obvious from the

Figure 2. (A) Protein mass/area plotted against brush surface coverage for brushes made from 2, 5, and 20 kDa PEO-SH chains. (B) Ln(protein mass/area) plotted against brush surface coverage for brushes made from 2, 5, and 20 kDa PEO-SH chains. ⧫ 2k Mw, ● 5k Mw, and ■ 20k Mw PEO chains. Solid line is exponential fit to data, chisq = 1, coefficient values: c = 1 and μmax = 3. Dashed line is exponential fit, chisq = 0.8, coefficient values: c = 0.9 and μmax = 2.8.

disappearance (or heavy reduction) of the molecular weight dependence that there is a strong scaling relationship between this definition of brush coverage and the adsorbed amount of protein. It must be noted that for the 20k Mw brushes, it is much harder to evaluate whether the adsorption curve completely collapses upon the 2k and 5k Mw series with the same shape, as less reliable data points were obtained at grafting densities before the brushes became protein resistive. Clearly, some of the points in the 20k Mw weight series are erroneous. 6119

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particular importance to the results shown here, the adsorbed amount of protein at brushes which were attracted to their substrate was found to be dependent upon the molecular weight of the chains. Increasing the molecular weight of the chains resulted in the surface attracted polymer brushes becoming protein resistive at lower grafting densities and over a similar range of grafting density as those shown in Figure 1. The SCMF theory model of Szleifer gives chain-level mechanistic explanation for protein adsorption at PEO brushes. However, we suggest for the system in hand, where PEO has an exceptionally strong affinity for the tethering substrate, PEO effectively outcompetes the protein for the surface, and therefore lateral fluctuation in surface coverage becomes the dominate factor, determining the relationship between polymer coverage and the adsorbed amount of protein. Therefore models such as the one presented here or rational functions such as the available area function in the RSA model of Katira et al. must also be considered alongside models stressing chainlevel mechanisms for protein adsorption at PEO brushes tethered on surfaces to which they are physically attracted.6 The model presented here does not take into account intermolecular interactions such as the solubility affects of salt. As the experiments conducted here were done so with a very low salt concentration such solubility affects can be ignored. However, further theoretical work could extend this model to include the potential affects of high salt conditions, where secondary adsorption is assumed to occur at high grafting density regions of the brush, which increase in number exponentially with increasing grafting density.

demonstrate a simple model describing the adsorbed amount of protein as a fractional decrease in adsorption sites with increasing polymer coverage. Exponential Model. In order to explain the coverage relationship, which is seen in the data presented here, a simple model for the reduction in protein adsorption with increasing coverage of polymer at the surface was developed. We assume that each polymer covers an area proportional to its radius of gyration in solution and that the chains are free to overlap and are placed randomly upon the surface. We then assume that the polymers are acting like site-blocking barriers, completely outcompeting protein for attachment to the surface, preventing adsorption over the area they cover, due to the strong physical attraction between the polymer and the surface. In such a scenario, the proportion of surface not covered with polymer, p(open), is proportional to A(n)/A(0) where A(n) is the area not covered by polymer as a function of n, the number of polymers at the surface, and A(0) is area available with 0 polymers present. It is now of interest to look at how A(n) changes as more polymer is added to the surface. For the limit n → n + 1, the following differential for the area covered can be found, ⎛ A(n) ⎞ dA = −A poly ⎜ ⎟ dn ⎝ A(0) ⎠

(5)

where Apoly is the area covered by one polymer. The above differential then has the classic solution as follows, A(n) = A(0)e−A poly n / A(0)

■

(6)

If we then assume that the adsorbed amount of protein, μ, at the surface is proportional to the area not covered by polymer A(n) and also take note that Apolyn/A(0) is equal to the polymer coverage Ω as n/A(0) = σ and Ω = Apolyσ then we are left with the following expression, μ = μ(0)e−Ω

CONCLUSION The adsorbed amount of lysozyme protein on gold surfaces coated with thiol-terminated mPEO brushes of different Mw and grafting density was measured ex situ via ellipsometry. It was found that the adsorbed amount of protein at these surfaces roughly decreases exponentially with increasing brush coverage and that the following expression supplies a reasonable fit to the data μ ≈ μmaxe−cΩ . We interpret this result in terms of a model in which PEO brushes on gold act primarily as site-blocking barriers, due to the strong, attractive affinity between gold and the PEO brush chains, and in which lateral fluctuations in brush coverage determine the adsorbed amount of protein. We also suggest that the solubility effects of salt on high-volume fraction brushes account for the lack of protein adsorption at high grafting densities.

(7)

Note that the adsorbed amount of protein decreases exponentially with increasing polymer coverage. Figure 2 shows the protein adsorption data fitted with the above protein adsorption model; it can be seen by visual inspection that the model fits the data fairly well when plotted linearly against protein density (plot A) and logarithmically against ln(protein density) (plot B), thus demonstrating that the process is exponential in nature. (Note, protein density values which are 0 will not appear on the logarithmic plot as these tend toward negative affinity when plotted in such a manner). To demonstrate how well the exponential model fits the data, a fitting parameter c was added as an exponent as follows, μ = μ(0)e−cΩ. Fitting the data in igor yielded the following curve as shown in Figure 3 with the fitting parameters c = 0.9 ± 0.1 and μmax = 2.8 ± 0.2 with a chisq of 0.822. The fitting parameter c = 0.9 is ∼1 and strongly indicates that the adsorbed amount of protein at PEO brushes on gold is approximate following an exponential decay with coverage. Enforcing c = 1 and μmax = 3 and fitting in igor gave a chisq of 1, further supporting the notion that the adsorbed amount of protein is following an exponential decay with increasing polymer coverage. Szleifer calculated the adsorption isotherms for lysozyme adsorption at PEO brushes which were attracted or not attracted to their substrate using a single-chain mean-field theory, which takes into account the protein and polymer chemical potentials and intermolecular interactions.30 Of

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We would like to thank the EPRSC and White Rose DTC for funding this project. We would also like to thank Jonathan Landy of the University of California, Santa Barbara for his help formulating the exponential model presented here.

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REFERENCES

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