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* Woodruff School of Mechanical Engineering, Coulter Department of Biomedical Engineering, and Institute for Bioengineering and Bioscience, Georgia Institute of Technology, Atlanta, Georgia 30332; and Cardiovascular Biology Research Program, Oklahoma Medical Research Foundation, and Department of Biochemistry and Molecular Biology and Oklahoma Center for Medical Glycobiology, University of Oklahoma Health Sciences Center, Oklahoma City, Oklahoma 73104
Correspondence: Address reprint requests to Cheng Zhu, E-mail: cheng.zhu{at}me.gatech.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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; McEver, 2001, 2002). In this mechanically stressful environment, different forces are applied to selectin-PSGL-1 bonds at various rates. In part, the relationship between off-rate and force determines how efficiently tethers produce rolling and how rapidly and stably cells roll (Yago et al., 2004).
Bell (1978) was the first to suggest that the dissociation of adhesive receptor-ligand complexes, here termed bonds, which provide anchoring between cells or between a cell and the extracellular matrix, could be influenced by force, just as gaseous chemical reactions are affected by pressure. Although not validated, all published work to date has assumed that off-rate depends on the instantaneous force level (Bell, 1978; Dembo et al., 1988; Evans and Ritchie, 1997; Izrailev et al., 1997). This assumption has provided a theoretical framework for conceptualizing how applied force might alter the bond lifetime and how loading rate might affect the unbinding force of a receptor-ligand bond. It has also been the basis for determining the relationship between off-rate koff and force f from two types of measurements. The first type is bond lifetimes measured in a range of constant forces, which was used to obtain the first experimental estimate of koff(f) (Alon et al., 1995). All bond lifetime measurements have been made using a flow chamber (Alon et al., 1995, 1997, 1998; Chen and Springer, 2001; Dwir et al., 2001, 2002, 2003; Smith et al., 1999; Ramachandran et al., 1999, 2001; Yago et al., 2002, 2004) except for our recent work, which used atomic force microscopy (AFM) (Marshall et al., 2003; Sarangapani et al., 2004). The second type of measurement is unbinding forces measured in a range of constant loading rates, which have recently been popularized by the theory of dynamic force spectroscopy (DFS) analysis (Merkel et al., 1999; Evans et al., 2001; Evans, 2001; Evans and Williams, 2002) (see Materials and Methods below). The unbinding force measurements have been made using AFM (Fritz et al., 1998; Yuan et al., 2000; Li et al., 2003; Zhang et al., 2002, 2004; Hanley et al., 2003, 2004); a biointerface force probe (BFP) (Merkel et al., 1999; Simson et al., 1999; Strunz et al., 1999; Evans et al., 2001; Evans and Williams, 2002), or a microcantilever (Tees et al., 2001).
However, the above two measurement methods have often yielded drastically different koff versus f relationships for the same interactions, e.g., selectins dissociating from their ligands (Alon et al., 1995, 1997, 1998; Chen and Springer, 2001; Dwir et al., 2001, 2002, 2003; Smith et al., 1999; Ramachandran et al., 1999, 2001; Yago et al., 2002, 2004; Marshall et al., 2003; Sarangapani et al., 2004; Fritz et al., 1998; Evans et al., 2001; Evans and Williams, 2002; Hanley et al., 2003, 2004; Zhang et al., 2004). This is quite puzzling because it has been a generally accepted notion that off-rates govern both receptor-ligand bond lifetimes and bond strength. As such, one expects to be able to determine off-rates from either bond lifetime or unbinding force measurements. It is difficult to identify the causes for the discrepant results because of the many potential differences involved, as different laboratories obtained their results using different reagents and techniques. Using AFM to measure off-rates of P-selectin dissociating from PSGL-1 or from an antibody, we showed that the discrepancies were results of the two types of measurements—unbinding forces and bond lifetimes—themselves. Analyzing these data with a new method of time-to-dissociation analysis, we showed that the off-rate depended not only on force but also on force history. This finding reconciles previous discrepancies and provides a new paradigm for understanding how force regulates receptor-ligand dissociation.
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MATERIALS AND METHODS |
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; Sarangapani et al., 2004). Briefly, cantilevers (ThermoMicroscopes, Sunnyvale, CA) of spring constants 4–13 pN/nm, each calibrated in situ by thermal fluctuation analysis (Hutter and Bechhoefer, 1993), were functionalized by capturing a monomeric recombinant soluble PSGL-1 (sPSGL-1) (Yago et al., 2002) with a nonblocking monoclonal antibody (mAb) PL2 against PSGL-1 (Moore et al., 1995). PL2 or a blocking anti-P-selectin mAb G1 (Geng et al., 1990) were adsorbed on the cantilever tip (Fig. 1 A) followed by blocking with 1% bovine serum albumin in Hanks' balanced salt solution. Membrane P-selectin purified from human platelets (Moore et al., 1994) was reconstituted into glass-supported, polyethylenimine (PEI)-cushioned lipid bilayers (Fig. 1 A) using the method of vesicle fusion (Wong et al., 1999, McConnell et al., 1986). The P-selectin density was kept low to achieve infrequent binding, which is required to ensure binding to be mediated by a small number of (most likely just one) bond(s) (Zhu et al., 2002). Cantilever tips bearing sPSGL-1 or G1 were repeatedly brought into contact with the P-selectin-bilayer to allow bond formation and retracted to record the rare adhesion events (<20%). Three sets of data—unbinding forces, bond lifetimes, and survival times–were collected along two families of force histories: constant-rate loading, where force increased linearly with time, and constant-force loading, where force initially increased linearly with time and then became constant once a prescribed level was achieved (Fig. 1 B). The unbinding force (or holding force in the bond lifetime experiment) was measured from the force difference at the point of bond failure (Fig. 1 B). The bond lifetime was measured from the instant when the bond was fully loaded to a desired force to the instant of bond dissociation at that force. In both assays, the survival time, or the time to dissociation, was measured from the instant when the bond was first loaded to the instant of bond dissociation, regardless of the history of force application. The loading rate rf was determined as the ratio of unbinding force (or holding force) to survival time (or ramping time) because the f versus t curves during ramping were linear in the force range tested. Force averaging was used to circumvent thermally driven force fluctuations (Marshall et al., 2003; Sarangapani et al., 2004).
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 | (1a–c) |
; Dembo et al., 1988; Evans and Ritchie, 1997; Izrailev et al., 1997; Evans, 2001; Zhu, 2000; Strunz et al., 2000; Hummer and Szabo, 2003). Along a constant-rate (rf) force history f = rft, the time dependence of p and koff in Eq. 1 can be transformed to their respective force dependence, which yields the respective cumulative probability pc for a bond to unbind at force not exceeding f and the probability density pd for a bond to unbind at force f,
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; Evans et al., 2001; Evans, 2001; Evans and Williams, 2002; Hummer and Szabo, 2003). The "bond strength" fm, defined as the most probable unbinding force and measured as the peak force from the unbinding force histogram, is plotted against the logarithm of the loading rate rf (cf. Fig. 2 B). This is referred to as a "strength spectrum", which often appears as a nearly piecewise linear curve. Such strength spectra have been interpreted using an off-rate expressed as multiple Bell (1978) models in a serial arrangement (Merkel et al., 1999; Evans et al., 2001; Evans, 2001; Evans and Williams, 2002):
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; Evans et al., 2001; Evans, 2001; Evans and Williams, 2002):
 | (4) |
Fitting Eq. 4 to the strength spectra, i.e., fm versus rf data (cf. Fig. 2 B), allows evaluation of the parameters ai and ki, which are interpreted as related to the locations and differences, respectively, in heights of the ith energy barrier along the force-driven dissociation path (Merkel et al., 1999).
The bond lifetime data were analyzed by Eq. 1c after setting as time 0 the instant when the desired force level had just been achieved along a constant-force force history. It follows from the assumption of off-rate being a function of force only that koff becomes a constant for t > 0 and that Eq. 1c is reduced to p = exp(–kofft). The reciprocal off-rate 1/koff at a given force was then estimated from three lifetime measures: the mean [
t
= –
0
t(dp/dt)dt)] and standard deviation {
(t) = [–0(t – t)2(dp/dt)dt]1/2} of lifetimes and the reciprocal negative slope of the ln(No. of events with a lifetime 
t) versus t plot (Marshall et al., 2003; Sarangapani et al., 2004).
The survival time data were analyzed by a new method, which allows determination of the complete function koff(t) from a single distribution of survival times measured along an arbitrary force history. By comparison, to determine koff(f) from the above DFS or lifetime analysis requires, respectively, multiple unbinding force histograms or multiple bond lifetime histograms, which have to be measured along their respective families of force histories. The new method has removed the restrictive assumptions that koff is a single-valued function of f and that it depends on t only through f. The two remaining general assumptions are: a), the dissociation of single molecular bonds obeys Eq. 1 along any force history; and b), once the force history is specified, the off-rate becomes a single-valued but general function of the survival time t, which includes but is not limited to its dependence on t through f. To develop this method, we solved koff from Eq. 1:
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We noted that the right-hand side of Eq. 5 could be measured as 1 – p, and d(1 – p)/dt corresponded to the respective cumulative frequency and histogram of survival time data. To minimize irregular fluctuations caused by numerical differentiation of finite data points, the measured cumulative frequency data were smoothed by cubic spline fitting, which was then differentiated. The result was substituted along with the fitted cumulative frequency into Eq. 5 to estimate koff(t) without any a priori assumption of its functional form.
For unbinding forces measured at a single loading rate, i.e., along a given constant-rate force history where force and time are related by f = rft, Eq. 5 can be rearranged to obtain an explicit expression for off-rate as a function of force:
 | (6) |
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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). The unbinding force histograms exhibited typical single peaks and shifted rightward toward larger forces as the loading rate was increased (Fig. 2 A), similar to previously reported histograms (Merkel et al., 1999; Hanley et al., 2003). To apply the DFS analysis (Merkel et al., 1999; Evans et al., 2001; Evans, 2001; Evans and Williams, 2002), the "strength spectra" were generated by plotting the peak force fm against the loading rate rf in a log scale, which displayed two line segments for each of the interactions (Fig. 2 B). The force dependence of off-rate koff was determined by fitting these data to Eq. 4, which was derived based on a double Bell model in series (i.e., N = 2 in Eq. 3) assumed a priori for koff(f).
The bond lifetimes measured at the constant-force loading experiments were exponentially distributed. At each force level, the 1/koff was determined from the mean or standard deviation of lifetimes or from the –1/slope of the ln(No. of events with a lifetime t) versus t plot, which show good agreement, as predicted from the first-order dissociation kinetics. The directly measured bond lifetime versus force data (points) were compared to the 1/koff versus f relationship (black dashed curves) previously determined from the DFS analysis of the unbinding force measurements (Fig. 2 C).
Unexpectedly, the 1/koff values determined from the two assays differed drastically. For the P-selectin-sPSGL-1 interaction (Fig. 2 C, left panel), the directly measured bond lifetime data behaved as a catch-slip transitional bond, where force first prolonged and then shortened bond lifetimes (Marshall et al., 2003). In sharp contrast, the 1/koff curve obtained from the unbinding force data decreased monotonically with force, exhibiting only slip bond characteristics. For the P-selectin-G1 interaction (Fig. 2 C, right panel), the directly measured bond lifetime data followed a straight line in the semilog plot, indicating a single exponential function for koff(f) that was well described by the Bell (1978) model. By comparison, the 1/koff curve obtained from the unbinding force data included two line segments in the semilog plot (Fig. 2 B), thereby requiring two exponential functions for its description, which was 
10-fold smaller than that determined from the bond lifetime data in the force range tested (Fig. 2 C, right panel).
In addition to the DFS analysis (Fig. 2 B), the off-rate was estimated (Fig. 2 C, solid curves) directly from the unbinding force data (Fig. 2 A, color-matched histograms) using a new method without a priori assumption as to its functional form (see Materials and Methods, Eq. 6). This method utilized the entire histogram and cumulative frequency data as opposed to only the peak value of the histogram, and it enabled evaluation of koff(f) from unbinding forces measured at a single loading rate as opposed to requiring a range of loading rates.
Two observations can be made from this analysis of the unbinding force data: a), the 1/koff values (Fig. 2 C, colored solid curves) were considerably different from the directly measured bond lifetime data for both P-selectin-sPSGL-1 and P-selectin-G1 interactions; and b), the 1/koff versus f curves estimated from individual unbinding force histograms were qualitatively similar to, but quantitatively different from each other and from the 1/koff versus f curve estimated from the DFS analysis of all histograms (Fig. 2 C). These observations raised questions about the validity of Eq. 3 that was assumed a priori. Additionally, the 1/koff versus f curve was loading-rate sensitive—the larger the rf the smaller the 1/koff, which provided initial evidence for the force history dependence of the off-rate (Fig. 2 C). The loading-rate sensitivity was not a result of a compliant linkage to the adhesive interface, because the loading rates were measured experimentally, which had already accounted for the nearly linear elasticity of the interacting molecules (B. T. Marshall, K. K. Sarangapani, J. Wu, M. B. Lawrence, R. P. McEver, and C. Zhu, unpublished data).
Off-rates derived from bond lifetimes fail to predict measured unbinding forces
Alternatively, we used the force dependence of off-rate determined from the bond lifetime measurements to predict the unbinding forces according to Eq. 2b. The predicted unbinding force distributions differed drastically from those actually measured (Fig. 3 and data not shown for histograms at other loading rates). For the P-selectin-sPSGL-1 interaction, the koff(f) obtained from the bond lifetime data predicted a minimum at 15 pN followed by a maximum at 50 pN in the unbinding force histogram at a loading rate of 300 pN/s (Fig. 3 A, solid bars). By comparison, the unbinding force histogram directly measured at rf = 300 pN/s had no minimum and peaked at 15 pN (Fig. 3 A, open bars). For the P-selectin-G1 interaction, the koff(f) obtained from the bond lifetime data predicted an unbinding force distribution (Fig. 3 B, solid bars) that was shifted rightward toward larger forces relative to the measured distribution (Fig. 3 B, open bars). The peak force moved from 20 pN to 40 pN in the histogram at rf = 100 pN/s. Thus, although both can be interpreted using the same theoretical framework, which is based on the concept that force modulates the off-rate of receptor-ligand dissociation, the bond lifetime and unbinding force data result in two koff(f) functions that drastically differ not only quantitatively but also qualitatively.
Survival time analysis reveals bond stabilization once ramping stops
One possible explanation for the discrepancy might be that, in the constant-force bond lifetime assay, a weaker, shorter-lived subpopulation of bonds broke during ramping before the desired level of force was achieved. This could select a stronger, longer-lived subpopulation of bonds that survived the force ramping phase for the bond lifetime measurements, resulting in a smaller koff than that estimated from the unbinding force measurements of both subpopulations of bonds. To examine this possibility, we added to the bond lifetime data the previously omitted unbinding events that occurred during the ramping phase and replotted the extended data in survival time histograms. To allow direct comparison, the unbinding force histograms were also converted to survival time histograms.
For P-selectin-sPSGL-1 (Fig. 4 A) or P-selectin-G1 (Fig. 4 B) interactions, the cumulative frequencies from both the bond lifetime and unbinding force assays displayed identical behavior while the bonds were being loaded by a ramp force (solid symbols). Only after the force histories deviated was a difference in the cumulative frequencies evident: curves from the bond lifetime assay (open symbols) increased with survival time less rapidly than that from the unbinding force assay (solid circles). Therefore, the same population of bonds existed in both assays at the point where the force histories diverged. Interestingly, clamping the force at a constant level appeared to stabilize the bonds, resulting in their slower dissociation than that resulted from continuing to ramp up the force at a constant rate. Thus, the differences in the koff(f) obtained from the two assays were caused by the change in force histories, not by the omission of unbinding events during the ramping phase of the bond lifetime assay. Clamping the force at higher and higher levels stabilized the P-selectin-G1 bonds less and less, giving rise to their slip bond characteristic (Fig. 4 B). By contrast, increasing the holding force stabilized the P-selectin-sPSGL-1 bonds initially more and then less, resulting in their catch-slip transitional bond characteristic (Fig. 4 A).
Off-rate as a functional of loading history rather than a function of force
We have established that the history of force application affects the force dependence of off-rate. To directly compare the off-rates determined from different measurements, we calculated koff as a function of survival time along a given force history with our new method (Eq. 5) using the cumulative frequency data without a priori assumptions as to how koff depended on time and/or force. The history of force application was plotted on the f-t (x-y) plane (Fig. 5, A and B), and the reciprocal off-rate (z axis) was shown as a three-dimensional (3D) curve (Fig. 5, C and D). The projection of the koff curve on the f-t plane corresponds to the force history along which the bonds were loaded.
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In the bond lifetime experiment, force histories were represented by two-line segments on the f-t plane (Fig. 5, A and B): a straight line from the origin (the ramping portion) followed by lines parallel to the t axis (the holding portion). The 1/koff values along the ramping portions were identical to those obtained from an unbinding force experiment performed at the same loading rate. However, the bond lifetime data diverged from the unbinding force data when the force histories turned onto the holding portions. The 1/koff values jumped to higher levels and remained constant, which can be clearly seen from the projections of the 3D 1/koff curves onto the t-1/koff (y-z) plane (Fig. 5, E and F).
By analyzing both unbinding force and bond lifetime experiments in the same framework of survival time, the force dependence of off-rate obtained from the bond lifetime experiment was not biased by exclusion of any data. The different relationships between koff and f were a result of the different force histories. The jumps in the 1/koff values resulted from the corner points in the survival time cumulative frequency curves (cf. Fig. 4), which reflected the stabilization of bonds by stopping the ramping and holding the force constant.
The projection on the f-1/koff (x-z) plane of a 3D 1/koff curve along the holding portion of a constant-force force history was a single point because the off-rate remained the same value for all times when the bonds were held at a constant force. The locus of many such points (dotted curves in Fig. 5, C and D) over a range of forces produced a 1/koff versus f curve. Again, the catch-slip transitional bond was seen for the P-selectin-sPSGL-1 interaction and the slip bond was seen for the P-selectin-G1 interaction, agreeing with the bond lifetime data in Fig. 2 C.
The differences in koff values at the same f (Fig. 2 C) and at the same t (Fig. 5, E and F) indicated that the off-rate was not a function of force or of time only. Closer inspection revealed that koff was not even a single-valued function of the dual variables f and t, but it depended on the entire history of force application to the bond. This force-history dependence was most evident by comparing the 1/koff values at the same endpoint on the f-t plane evaluated along two different histories: a), a constant-rate force history with a low rate (orange lines in Fig. 5, A and B), and b), a high-rate history that ramped until the desired force was reached and then held at a constant force until the desired time was reached (two-segment green lines in Fig. 5, A and B). Even though this endpoint (intersection of the two force histories, indicated by a black open circle in Fig. 5, A and B) was arrived in the same duration of time to the same level of force, the two 1/koff values (orange and green solid circles in Fig. 5, C and D) were clearly very different. The same can be seen in Fig. 5, E and F, where the two distinct pairs of orange and green solid circles are projected on the t-1/koff plane.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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, 1997, 1998; Chen and Springer, 2001; Dwir et al., 2001, 2002, 2003; Smith et al., 1999; Ramachandran et al., 1999, 2001; Yago et al., 2002, 2004; Marshall et al., 2003; Sarangapani et al., 2004; Fritz et al., 1998; Evans et al., 2001; Evans and Williams, 2002; Hanley et al., 2003, 2004; Zhang et al., 2004). It has been difficult to identify the causes of these discrepancies because the experiments were performed in different laboratories using different reagents and techniques to measure different quantities that were analyzed differently. Unbinding forces were measured by AFM (Fritz et al., 1998; Hanley et al., 2003, 2004; Zhang et al., 2004) and BFP (Evans et al., 2001; Evans and Williams, 2002) and analyzed by DFS, and bond lifetimes were measured in flow chambers (Alon et al., 1995, 1997, 1998; Chen and Springer, 2001; Dwir et al., 2001, 2002, 2003; Smith et al., 1999; Ramachandran et al., 1999, 2001; Yago et al., 2002, 2004; Marshall et al., 2003; Sarangapani et al., 2004) and by AFM (Marshall et al., 2003; Sarangapani et al., 2004) and analyzed by the ln(No. of events with a lifetime t) versus t plot. Some authors attributed these discrepancies to the higher force and/or temporal sensitivities of the AFM and BFP techniques than the flow chamber, which might enable the former to detect interactions supported by fewer and/or shorter-lived bonds than the latter (Evans et al., 2001; Hanley et al., 2003). However, our recent bond lifetime measurements with AFM and flow chamber yielded closely matched koff versus f relationships—catch-slip transitional bonds for P- and L-selectins interacting with PSGL-1 and slip bonds for interactions with their respective mAbs (Marshall et al., 2003; Sarangapani et al., 2004)—that reaffirmed the assertion that the intrinsic off-rates for the same molecular interactions should be independent of the experimental techniques used to measure them (Zhu, 2000; Strunz et al., 2000).
In this study, we eliminated most of the potential differences by measuring the unbinding forces and bond lifetimes in the same laboratory with the same reagents and the same AFM instrument; using cantilevers of similar spring constants to contact bilayers prepared the same way with the same approach velocity, contact time, and contact force. The same numbers of bonds must have been formed during the contact phase of the two assays and been detected as such; yet the off-rates estimated from unbinding forces and bond lifetimes were still very different. Although quite surprising at first glance, this unexpected finding, obtained from carefully controlled experiments and analyses, can be rationalized by reexamining the assumptions underlying the analysis by which off-rates are evaluated from the data. It is possible that the kinetic mechanism assumed here—the first-order irreversible unbinding of individual bonds along a single thermodynamic pathway for dissociation as described by Eq. 1—may overly simplify the interactions in question. Were this the case, the koff(f) evaluated from our analysis would have been apparent rather than intrinsic off-rates. Thus, our data suggest that kinetic mechanisms more complex than Eq. 1, which has been assumed in all publications except a recent work (Evans et al., 2004), may be required to describe the dissociation of P-selectin from sPSGL-1 and from G1. Alternatively, the assumption that off-rates were single-valued functions of force only may not be always true. In fact, replacing it with a more general assumption that off-rates can depend on the entire history of force application has reconciled the discrepant koff(f) as reflecting the different types of measurements—bond lifetimes versus unbinding forces—that were made along different force histories.
Since Bell (1978) proposed that force could influence off-rate, koff has only been considered a single-valued function of f (Bell, 1978; Dembo et al., 1988; Evans and Ritchie, 1997; Izrailev et al., 1997). Our data suggest that koff may depend on the entire history of force application rather than the instantaneous value of force, thereby challenging the existing paradigm. The concept of force affecting kinetic rate is based on the idea that work done by the force can be superimposed on the binding potential of the interacting molecules, which tilts the energy landscape, thereby altering the rate of dissociation. Thermodynamically, however, work is a path function rather than a point function—its value depends on the path along which the force is applied rather than on the particular force value at any single point in the path. Although it is valid to add the path-independent conservative work onto the potential energy, this is not the case for the path-dependent dissipative work because it simply represents nonrecoverable losses that are not additive to the potential energy (Izrailev et al., 1997; Balsera et al., 1997). It also seems reasonable that the relative proportions of the conservative and dissipative work should depend on the history of force application, which would then give rise to a force history dependence of the off-rate. Thus, the concept of force-history dependence of off-rate does not contradict but rather generalizes the concept of force dependence of off-rate. Not only is the former more fundamental and realistic than the latter, but the latter can also be reduced from the former as a special case if the force histories are restricted to those of constant-forces only. This generalized concept reconciles the apparent discrepancies in the koff versus f relationships obtained from unbinding force and bond lifetime measurements (this work and Alon et al., 1995, 1997, 1998; Chen and Springer, 2001; Dwir et al., 2001, 2002, 2003; Smith et al., 1999; Ramachandran et al., 1999, 2001; Yago et al., 2002, 2004; Marshall et al., 2003; Sarangapani et al., 2004; Fritz et al., 1998; Evans et al., 2001; Evans and Williams, 2002; Hanley et al., 2003, 2004; Zhang et al., 2004). It further provides a new paradigm for understanding how force regulates biological interactions.
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ACKNOWLEDGEMENTS |
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This work was supported by National Institutes of Health grants AI44902 and HL65631. B.T.M. was a recipient of the Whitaker Foundation Graduate Fellowship.
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FOOTNOTES |
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Submitted on July 28, 2004; accepted for publication November 9, 2004.
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REFERENCES |
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