Structure

Article

Biochemical Implications of a Three-DimensionalModel of Monomeric Actin Bound toMagnesium-Chelated ATPKeiji Takamoto,1,2,* J.K. Amisha Kamal,1,2 and Mark R. Chance1,2

1 Case Center for Proteomics, Case Western Reserve University, 10090 Euclid Avenue, Cleveland, OH 44106, USA2 Lab address: http://casemed.case.edu/proteomics/

*Correspondence: keiji.takamoto@case.eduDOI 10.1016/j.str.2006.11.005

SUMMARY

Actin structure is of intense interest in biologydue to its importance in cell function andmotility mediated by the spatial and temporalregulation of actin monomer-filament intercon-versions in a wide range of developmental anddisease states. Despite this interest, the struc-ture of many functionally important actin formshas eluded high-resolution analysis. Due to thepropensity of actin monomers to assemble intofilaments structural analysis of Mg-bound actinmonomers has proven difficult, whereas high-resolution structures of actin with a diverse ar-ray of ligands that preclude polymerization havebeen quite successful. In this work, we providea high-resolution structural model of the Mg-ATP-actin monomer using a combination ofcomputational methods and experimental foot-printing data that we have previously published.The key conclusion of this study is that thestructure of the nucleotide binding cleft definedby subdomains 2 and 4 is essentially closed,with specific contacts between two subdo-mains predicted by the data.

INTRODUCTION

Actin is a ubiquitous and important protein in eukaryotes

and is extremely well conserved from yeast to man. Actin

binding of nucleotides and its interactions with other pro-

teins in the cell control the spatial and temporal assembly

and disassembly of the cytoskeletal network; the careful

regulation of this network has a profound influence on

cell motility (Paavilainen et al., 2004; Schoenenberger

et al., 2002; Wear et al., 2000; Winder, 2003). Even the

nature of the metal ion bound to the nucleotide with the

actin structure can profoundly alter the ability of actin

monomers to assemble into filaments. In spite of its bio-

logical importance, the structure of the Mg2+-ATP-bound

form of the actin monomer (called G-actin) is not known.

Structure 15,

The determination of the high-resolution structure of

Mg2+-G-actin is hampered by the propensity of actin to

polymerize. Generation of crystals suitable for X-ray dif-

fraction as well as NMR studies typically requires higher

concentrations than the critical concentration for actin

polymerization for the Mg2+-nucleotide-bound forms of

the actin monomer. The high-resolution crystal structures

of G-actin, whose richness and depth are a nearly unique

resource for this investigation, never the less have either

Ca2+-nucleotide bound to actin or, if Mg2+-bound nucleo-

tide is used, actin is cocrystallized with ligands such as

DNase I (Kabsch et al., 1985, 1990; Suck et al., 1981), gel-

solin segment 1 (Mannherz et al., 1992; Vorobiev et al.,

2003), vitamin D binding protein (Otterbein et al., 2002;

Swamy et al., 2002; Verboven et al., 2003), or macrolides

(Allingham et al., 2004; Klenchin et al., 2003; Morton et al.,

2000; Reutzel et al., 2004; Yarmola et al., 2000). These li-

gands typically prevent polymerization. Although struc-

tures of Ca2+- and Mg2+-G-actin bound to ligands are

overall similar in many respects, significant differences in

structure and function between these species in the

absence of other bound ligands in solution have been

consistently reported, including biochemical/biophysical

analyses such as limited proteolysis (Chen et al., 1995;

Strzelecka-Golaszewska et al., 1993), fluorescence stud-

ies (Frieden and Patane, 1985; Moraczewska et al., 1999;

Selden et al., 1989; Valentin-Ranc and Carlier, 1991; Zim-

merle et al., 1987), and molecular dynamics simulations

(Wriggers and Schulten, 1997).

Recently, we have investigated the solution structure of

Mg2+-ATP-actin using hydroxyl-radical footprinting and

mass spectrometry (MS); these experiments have in-

cluded a comparison of Ca2+-ATP-actin in the presence

and absence of gelsolin segment 1 (GS1) (Guan et al.,

2003). The measured side-chain solvent accessibilities

of Ca2+-ATP-actin in the absence of GS1 are very similar

to that in the presence of GS1 overall and consistent with

the accessible surface area (ASA) calculated from solved

crystal structures. In contrast, Mg2+-ATP-actin in the

absence of GS1 is quite different in its side-chain surface

accessibilities, particularly in subdomains (SD) 2 and 4.

These differences are reversed in the presence of GS1,

indicating the structure of the Mg2+-ATP-actin/GS1 com-

plex in solution is consistent with crystal structure data.

39–51, January 2007 ª2007 Elsevier Ltd All rights reserved 39

Structure

Model of Mg2+-ATP-Actin Monomer

Hydroxyl-radical footprinting and MS have been proven

to be powerful tools for probing protein structure (Guan

et al., 2003; Rashidzadeh et al., 2003) and conformational

change (Kiselar et al., 2003a, 2003b), as well as for probing

protein-ligand (Gupta et al., 2004) and protein-protein

interactions (Guan et al., 2002, 2004; Liu et al., 2003). As

the experimental details of the technique have matured

and become reliable (Guan and Chance, 2005; Takamoto

and Chance, 2006), we and others have attempted to use

the data in conjunction with comparative modeling tech-

niques to provide unique structural models (Gupta et al.,

2004; Sharp et al., 2005). In this study, we extend our

previous work (Guan et al., 2003) with visualization of

our descriptive prediction by generating an atomic model

of the Mg2+-ATP-actin monomer structure using footprint-

ing and crystallographic data. The computational strategy

uses rigid-body rotations and translations of actin subdo-

mains, primarily guided by known subdomain rearrange-

ments from the range of actin crystallographic structures,

in order to generate a structure consistent with surface

accessibility data predicted by footprinting. In addition

to an atomic model of the Mg2+-ATP-actin monomer, we

also propose a specific mechanism for cleft closure

mediated by changes in metal-ion coordination.

RESULTS

Rigid-Body Rearrangements of Actin

Domain/Subdomains Derived from

an Actin Crystallographic ‘‘Database’’

The wealth of actin structures available in the literature

provides a nearly unique opportunity for examining the

range of conformations accessible to actin in its various

ligand-bound states. Actin is composed of two domains

that are termed the large and small domains, and each do-

main is composed of two subdomains (Figures 1A and

1B). Table 1 provides a ‘‘database’’ of structures that

encompass a number of relative arrangements of the var-

ious actin subdomains. These are the major structures

that are used in this paper to provide an analysis of the

range of motions of the actin subdomains. Our approach

in this section is to survey these structures and determine

patterns of subdomain motions.

Our previous footprinting data (Guan et al., 2003)

indicated that the large cleft (nucleotide binding cleft) be-

tween SD2 and SD4 is in a more closed configuration for

monomeric actin bound to Mg2+-ATP compared to the

Ca2+-ATP form. We surveyed the available crystal struc-

tures to ascertain the range of relative motions of the large

cleft. Most actin crystal structures (Table 1) are very

similar, with an overall backbone rmsd of 0.8 A for 13

structures (including Protein Data Bank ID codes 1RFQ-A

and 1RFQ-B, 1IJJ-A and 1IJJ-B, 1NM1, 1D4X, 1EQY-A,

1NWK, 1QZ6, 1YAG, 1MA9, 1AQK, and 1NLV). However,

there are some crystal structures that show significant de-

viations from the majority of solved structures. One of the

most prominent examples is crystal structure 1HLU (Chik

et al., 1996), which has profilin bound to the small cleft

40 Structure 15, 39–51, January 2007 ª2007 Elsevier Ltd All rig

between SD1 and SD3; this structure exhibits a ‘‘super-

open’’ nucleotide cleft, with a large movement of the

ATP/cation complex within the cleft. This structure can be

related to other more ‘‘typical’’ structures through domain

movements by rotational/sheer transformation (Chik et al.,

1996; Page et al., 1998). A second example is 1RFQ

Figure 1. Domain Rigid-Body Movements Observed in Actin

Crystal Structures

(A) The movement of the large domain relative to SD1 observed in crys-

tal structure 1HLU. The models shown in tin and silver are 1YAG and

1HLU, respectively. The movements of Ca atoms are indicated by

arrows. The colors of arrows indicate the size of movements ranging

from blue (small) to orange (large). The center of rotational movement

and normal vector are shown by the yellow sphere and long arrow.

(B) Schematic representation of large-domain movement.

(C) The movements of SD2 relative to subdomain 1. The white arrows

indicate deviations of the G63 Ca atom position from 1YAG.

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Structure

Model of Mg2+-ATP-Actin Monomer

Table 1. List of Crystal Structures Used for Analyses

Protein Data Bank ID Code Species Cocrystal Cofactors Resolution Comments

Crystal Structures of Interest

1YAG Yeast GS1b Mg2+/ATP 1.90 ‘‘Template’’ molecule (with

complete D loop)

1RFQ-Ba Rabbit LARc Mg2+/ATP 3.00 Closed large cleft with alteredside-chain geometries

1QZ5 Rabbit KABd Ca2+/ATP 1.45 Closed large cleft

1HLU Bovine Profilin Ca2+/ATP 2.65 Large cleft in ‘‘super-open’’state, b-actin

Crystal Structures with Complete D Loop

1J6Z Rabbit TMR Ca2+/ATP 1.54 TMR affects C-terminal structure?

1YVN Yeast GS1 Mg2+/ATP 2.10 V159N mutant of 1YAG

2BTF Bovine Profilin Sr2+/ATP 2.55 With Sr2+. b-actin

1ATN Rabbit DNase I Ca2+/ATP 2.80 D loop interacts with DNase I

1IJJ Rabbit LAR Mg2+/ATP 2.85 D loop that interacts with another

chain’s small cleft

1LCU Rabbit LAR Ca2+/ATP 3.50

Crystal Structures with Partial D Loop

1D4X C. elegans GS1b Ca2+/ATP 1.75

1C0G Chimeric GS1b Ca2+/ATP 2.00 Chimeric, Q228K/T229A/A230Y/E360H

1MDU Chicken GS1b Ca2+/ATP 2.20 Actin-trimer

1C0F Chimeric GS1b Ca2+/ATP 2.40 Chimeric Dictyostelium/Tetrahymena

1DEJ Chimeric GSe Mg2+/ATP 2.40 Chimeric, Q228K/T229A/A230Y/

A231K/S232E/E360H

1H1V Human GSe Ca2+/ATP 3.00

Important crystal structures of monomeric actin. The first set is used for modeling process or strategy; the second set is actin struc-

tures with complete D-loop backbone coordinates; and the last set is a partial but relatively better coverage of D-loop backbonecoordinates. Except for 1RFQ, the interasymmetric unit interactions between their D loop and the other molecule are unknown.

1RFQ does not have contact with other molecules.a 1RFQ-B, chain B of asymmetric unit in 1RFQ.b Gelsolin segment 1.c Latrunculin A.d Kabiramide C.e Gelsolin.

(Reutzel et al., 2004), which has two chains in the asym-

metric unit whose structures are quite different from

each other. Chain A exhibits a ‘‘standard’’ cleft structure

similar to those of the vast majority of solved structures.

However, chain B (1RFQ-B) is interesting as it shows

a ‘‘closed’’ form of the cleft, with very different geometries

for the residues inside the cleft. The closed form in chain B

of the crystal structure exemplifies a database entry that

qualitatively satisfies an important aspect of our footprint-

ing data, that is, it provides an example of how relative

subdomain motions can result in a closed nucleotide cleft.

Crystal structure 1QZ5 (Klenchin et al., 2003) shows a

closed large cleft as well. In this structure, the geometry in-

side the large cleft is almost identical to 1YAG (although it

has a Ca2+ ion instead of an Mg2+ ion). To achieve cleft

closure compared to canonical actin forms, SD2 moves

Structure 15,

toward SD4 without tilting (unlike 1RFQ-B), minimizing ste-

ric conflicts between the turn in SD2 (residues 62–65) and

the top of the helix in SD4 (residues 200–205). Although our

hydroxyl-radical footprinting results indicate more signifi-

cant closure of the cleft than observed in these structures,

these structures can serve as ‘‘qualitative’’ templates for

understanding the Mg2+ form of G-actin in solution. The

B factors of the atoms in SD2 of 1QZ5 are relatively high,

indicating the relative mobility of SD2. This suggests the

likelihood of high relative mobility of SD2 (not only the D

loop but also the SD2 core) in solution (see the Supple-

mental Data available with this article online).

Vector Analysis of Subdomain Relative Positions

The structure 1YAG is used as our standard structural

template having high-resolution (1.90 A) and well-defined

39–51, January 2007 ª2007 Elsevier Ltd All rights reserved 41

Structure

Model of Mg2+-ATP-Actin Monomer

waters within the nucleotide binding pocket. The struc-

tures of Protein Data Bank ID codes 1HLU (Chik et al.,

1996), 1QZ5 (Klenchin et al., 2003), 1RFQ-B (Reutzel

et al., 2004), 1J6Z (Otterbein et al., 2001), 1ATN (Kabsch

et al., 1990), 1S22 (Allingham et al., 2004), 1EQY-A

(Mannherz et al., 1992), 1IJJ-A (Bubb et al., 2002), 1MA9

(Verboven et al., 2003), 1KXP (Otterbein et al., 2002),

and 2BTF (Schutt et al., 1993) were compared with 1YAG,

and differences between Ca/backbone coordinates were

visualized using Tcl scripts for visual molecular dynamics

(VMD). The script also calculates the average center

position and rotational normal vector of movement of

Ca/backbone coordinates.

Figure 1A shows the relative movements (vector) of the

large domain with respect to SD1 in the comparison of

1YAG (the standard) and 1HLU (most open nucleotide

cleft structure). The arrows indicate the Ca movements

(arrows from coordinates in 1YAG to 1HLU). As the move-

ments are relative, we chose to show the large-domain

movement relative to SD1 in this representation because

SD2 also shows significant movement against SD1. As

seen in Figure 1B, this movement is 8.9� rotational (hinge)

movement that has a center position in the junction be-

tween SD1 and SD3 (around Q137-A138). The axis of ro-

tation goes from the front to the back side of the molecule

(where the front side is defined as the surface of the actin

molecule that exposes ATP). Figure 1B illustrates this

movement (the rotational angle is exaggerated for clarity).

A similar movement is reported in previous analyses and

refinement of F-actin fiber X-ray data including alterations

of the nucleotide binding pocket (Lorenz et al., 1993; Tirion

et al., 1995).

The SD2 subdomain is extremely mobile even in crystal

structures. Figure 1C compares 11 SD2 structures with

that of 1YAG; SD2 is observed in many orientations rela-

tive to SD1. The directions of rotation of these crystal

structures (relative to 1YAG) are shown in Figure 1C by

arrows. All of these differences can be mediated by

hinge movements having pivot points near P32-S33 and

Y69-P70.

Intracleft Interactions

There are a number of interactions between the ATP/

cation complex and the protein within this cleft that are

important for understanding large-cleft structure and the

possible mechanisms of the structural changes. The sub-

domains are not strongly connected by hydrogen-bond

networks or other forms of interactions; the majority of

the interactions between subdomains are mediated

through a hydrogen-bond network with the ATP molecule

and with waters in the pocket of the cleft. SD1 has interac-

tions with phosphate oxygen atoms of ATP through resi-

dues S14, G15, L/M16, and K18 (Figure 2A). Connections

between SD1 and SD2 are mediated only by the hydrogen

bonds between S14 and G74 (Figure 2B) (Chen et al.,

1995; Kabsch et al., 1990; Schuler, 2001). This link is

severed in 1RFQ-B due to subdomain rearrangements.

Although the metal ion is located near the interface of sub-

domains 1 and 2, it only interacts with SD1 by hydrogen

42 Structure 15, 39–51, January 2007 ª2007 Elsevier Ltd All rig

bonds through waters (no inner-sphere coordination)

and no interaction is formed with SD2 residues.

The interactions between SD3 and SD4 are also mainly

mediated through ATP. In addition, there is a poorly

packed hydrophobic cluster at the interface of the two

Figure 2. Important Interactions Involving the ATP Molecule

Brown and silver colors are 1YAG and 1FRQ, respectively. Hydrogen

bonds and coordination bonds are shown by broken lines.

(A) N-terminal b strand and ATP-metal interactions.

(B) Hydrogen bonds that connect subdomains 1 and 2 and ATP.

(C) Interactions with the Mg2+ ion. Water molecules observed in 1YAG

are shown as blue balls.

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Structure

Model of Mg2+-ATP-Actin Monomer

Table 2. Probe Residue Solvent Accessibilities from Crystal Structures and Experimentally DeterminedHydroxyl-Radical Reactivities

Crystal Structures

1ATN 1YAG 1RFQB 1RDW Footprinting Data

Ca Mg Mg Mg Solution G-Actin

DNase I GS1 LAR LAR Ca Mg

Residue

Number Residue

Side-chain

ASA (A2)

Side-chain

ASA (A2)

Side-chain

ASA (A2)

Side-chain

ASA (A2)

Rate

constant

(s�1)

Rate

constant

(s�1)

Peptide

(Residue

Numbers)

21 Phe 34.32 23.87 37.13 34.39 0.65/0.70 0.32/0.43 19–28 11–23

44 Met 79.35 (16.31)a 137.63 (35.96)a N.D.b N.D.b 33 20 40–50

47 Met 104.58 (30.94)a 113.37 (24.81)a N.D.b N.D.b 33 20

53 Tyr 14.26 6.60 64.67c 55.88c 0.69 0.32 51–61

67 Leu 28.70 22.16 25.46 40.25 0.40 0.08 63–68

69 Tyr 80.46 98.57 68.54 66.81 1.1 0.18 58–72

200 Phe 0.36 3.61 10.79 5.50 1.10/0.83 0.26/0.24 197–206196–207

201 Val 82.01 65.45 108.42 101.42 1.10/0.83 0.26/0.24

202 Thr 81.61 72.45 43.79 73.26 1.10/0.83 0.26/0.24

243 Pro 43.26 63.97 72.23 68.46 0.69/0.75 0.26/0.30 239–254

242–253

362 Tyr 12.59 9.05 15.18 11.75 0.72 0.38 362–372

367 Pro 45.75 40.74 39.49 41.88 0.72 0.38

371 His 27.79 26.53 34.71 25.89 0.72 0.38

374 Cys N.D. 3.53 49.40 24.29 8.5 3.6 363–375

375 Phe N.D. 157.72 77.56 44.38 N.D.d N.D.d

a Values in parentheses are ASAs for the sulfur atom in the side chain.b These crystals lacking observed D-loop structures.c These values are affected by the lack of a D loop (larger than expected).d The rate constant in the peptide is dominated by C374 and thus could not be determined.

subdomains. The adenosine moiety of ATP is located in

the pocket between the two subdomains, with hydrogen

bond and van der Waals contacts with surrounding resi-

dues. Thus, SD3, SD4, and ATP form an interdependent

set of interactions.

Metal-Ion Coordination

In most nonactin protein structures with ATP bound, mag-

nesium ion coordinates to at least one residue from the

protein (14 structures out of 16 nonredundant structures

examined; data not shown; Dudev et al., 1999). In the

case of actin, the magnesium ion (or calcium ion) forms in-

ner-sphere coordination bonds with phosphate oxygens

and water molecules, but not side-chain oxygen atoms,

with a coordination number of 6 (or 7 for Ca2+). A notice-

able difference between 1YAG (or other structures) and

1RFQ-B is that residue atoms Q137:OE1 and D11:OD1,

OD2 are closer to the Mg2+ ion in 1RFQ-B (3.0, 3.5, and

4.0 A) compared to 1YAG (4.4, 4.3, and 4.2 A, respec-

tively). Figure 2C shows the difference in the geometries

surrounding the magnesium ion. Unfortunately for 1RFQ,

at a resolution of 3.0 A, ordered water molecules are not

observed within the nucleotide binding cleft. However,

Structure 15,

some water molecules observed in 1YAG must be radi-

cally reorganized in 1RFQ-B, as these water molecules

would clash with side-chain oxygen atoms (Figure 2C,

pink side-chain oxygen atoms overlapping with blue water

oxygen atoms). Based on these data, we used the geom-

etry inside the large cleft from 1RFQ-B as a template for

our structural modeling of the nucleotide binding cleft.

Comparison of Crystallographic Data and

Hydroxyl-Radical Footprinting Data

Table 2 summarizes the calculated ASAs and rate con-

stants of modification of actin peptides analyzed by

hydroxyl-radical footprinting (Guan et al., 2003).

The hydroxyl-radical footprinting data for Ca2+-ATP-G-

actin are generally consistent with the calculated ASA,

that is, solvent-accessible, reactive residues show oxida-

tion and inaccessible residues are not appreciably oxi-

dized. However, a number of probe residues in SD2 and

SD4 and at the C terminus (SD1) show decreased rates

of modification for Mg2+-ATP-G-actin where the solvent

accessibilities for the structure are similar to those for

Ca2+-ATP-G-actin. These sites that show protections

from oxidation for Mg- versus Ca-actin are illustrated in

39–51, January 2007 ª2007 Elsevier Ltd All rights reserved 43

Structure

Model of Mg2+-ATP-Actin Monomer

Figure 3. For example, probe residues in the D loop are

modestly protected from oxidative modifications (�40%

reduction in rate constant), whereas residues 200–202

and 243 also experience strong reductions in modification

rate (�75% and �60%, respectively). L67 and Y69 lo-

cated inside the large cleft (Figure 3) experience an 80%

reduction in modification rates. Consistent with these find-

ings are limited proteolysis data, where cleavage between

K68 and Y69 is almost completely suppressed in Mg2+-

G-actin compared to Ca2+-G-actin (Chen et al., 1995;

Strzelecka-Golaszewska et al., 1993). A closure of the

large cleft is consistent with these data.

Modeling Strategy: Overall Structural

Considerations

In the previous section, we analyzed possible relative

movements of domains/subdomains. In light of these

analyses, we can understand the cleft structures in each

case. DNase I binding prevents large-cleft closure by

prohibiting the movements of SD2 and the large domain

by interacting with both of them (Kabsch et al., 1990).

Latrunculin A binds between SD2 and SD4 (Bubb et al.,

2002; Reutzel et al., 2004); this prevents movements of

those subdomains. Gelsolin segment 1 (Mannherz et al.,

1992; McLaughlin et al., 1993) and vitamin D binding pro-

tein (Otterbein et al., 2002) prevent the domain movement

between SD1 and SD3 blocking corresponding large-

domain movements. Some macrolides (kabiramide C,

jaspisamide A, and ulapualide A; Allingham et al., 2004;

Klenchin et al., 2003) bind to the small cleft and block its

movement as well. This is summarized in Figure 4A. It is

important to note that macrolides are small enough to fit

into small or large clefts and do not seem to prevent poly-

merization by steric hindrance as opposed to actin binding

Figure 3. Analyses of Hydroxyl-Radical Reactivity and Struc-

tures

The protection sites in Mg2+-G-actin (compared with Ca2+-G-actin) are

displayed with residues. Colors are coded as blue (strong protection)

to red (enhancement) by reactivity changes.

44 Structure 15, 39–51, January 2007 ª2007 Elsevier Ltd All rig

proteins that result in steric blockage. It appears that

these macrolides interfere with polymerization by prevent-

ing domain/subdomain motions.

The analysis of the various crystal structures and the

footprinting data (Guan et al., 2003) provides significant

clues about how the structure of Mg2+-G-actin differs

from that of Ca2+-G-actin. First, the footprinting data indi-

cate that the large cleft between SD2 and SD4 is almost

completely closed. Second, the analyses of the various

crystal structures indicate that movement of the SD2 core

is a rigid-body movement. The movement of SD2 toward

the large domain alone cannot explain the protections of

Figure 4. Schematic Representation of Domain/Subdomain

Movements

(A) The effects on subdomain movements by ligand binding.

Actin binding proteins (top) bind to actin and prevent polymerization,

possibly by steric hindrance or interference of domain movements.

Macrolides are small enough to fit into small or large clefts and interfere

with the domain movements. They may prevent polymerization by this

mechanism.

(B) The proposed mechanism for large-cleft closure. SD2 movement

alone cannot explain the complete closure of the large cleft (top right).

The partial closure by large-domain movement brings SD4 and SD2

close enough to allow interactions between the two subdomains, re-

sulting in complete cleft closure.

hts reserved

Structure

Model of Mg2+-ATP-Actin Monomer

both L67 and Y69. Y69 is located deep within the large cleft

and cannot be buried by SD2 rigid-body movement. In ad-

dition, neither M44 nor M47 can reach SD4 to form pair-

wise interactions to protect P243 and M44/47. This sug-

gests that both the large domain and SD2 need to move

as rigid bodies relative to SD1 in order to explain protec-

tions observed in SD2 and SD4 (Figure 4B).

The hydrogen-bond network among phosphate, S14,

and G74 is the only one actually connecting SD2 to SD1

and is observed in almost all structures (Schuler, 2001).

This linkage is severed in the 1RFQ-B structure. We spec-

ulate that latrunculin A prevents the movement of SD2 fur-

ther from the observed structure although the geometry

inside the large cleft is changed. As a result, the hydrogen

bonds are broken. This hydrogen-bond linkage was main-

tained in the model structure. We constructed a series of

structures with combinations of different rotational angles

and normal vectors for SD2 in order to determine the best-

fit model for the hydroxyl-radical data.

Regions Excluded from Modeling

Although there is evidence for structural variation in the

C-terminal region (Y362 to F375 changes solvent accessi-

bility in hydroxyl-radical footprinting; Guan et al., 2003)

and other data (Frieden and Patane, 1985; Strzelecka-

Golaszewska et al., 1993; Valentin-Ranc and Carlier, 1991;

Zimmerle et al., 1987), we do not have clear evidence to

support the specific structural differences. Thus, we have

not modeled the structural differences in this region. F21

experiences an �50% reduction in modification rate in

Mg2+-G-actin compared to Ca2+-G-actin, but this change

cannot be explained with our modeling strategy. Our

speculation is that the C-terminal (residues 336–375)

and N-terminal (residues 1–33) regions interacting with

SD1 are also affected by the movements of SD2 (con-

nected to the N-terminal region) and the large domain

(connected to the C-terminal region) and mediate these

additional structural differences between the two forms.

Model of the Mg2+-ATP-Actin Monomer

In general, the hydroxyl-radical footprinting data are

very consistent with known crystal structures. Solvent-

accessible surface areas and rate constants of oxidation

on side chains in known structures are in good correlation

(Guan et al., 2004; Guan and Chance, 2005; Takamoto

and Chance, 2006; Xu and Chance, 2005). This is not sur-

prising, as the size of the hydroxyl radical is very close to

that of a water molecule. Although it is not easy to com-

pare rate constants among different side chains (such as

Leu and Phe), it is very quantitative and reliable for the

same residue in different conformational states. The site

of oxidation is determined by MS/MS analyses that are

well established and routine. With the use of different

proteases, protein sequence coverage by MS analysis is

usually 80%–90% and most of the probe residues can

be detected. In our previous study, we covered �90% of

the actin sequence with trypsin and Asp-N proteases.

Thus, it is important to note that we have used high-quality

data for our modeling.

Structure 15, 3

Figure 5A shows the final model and the template struc-

ture for Mg2+-G-actin. The residues indicated to change

conformation in the footprinting experiments are shown

in stick-and-bubble format for both structures. The struc-

tures are aligned at SD1 in order to show both large-

domain and SD2 movements. The large domain rotates

toward the small domain and SD2 rotates to the front

(as defined previously) and toward the large domain.

Figure 5B show the backbone movement analyzed by

VMD/Tcl scripts. Figures 6A and 6B show a magnified view

of the SD2/SD4 interface in the model. In order to make

contact between D-loop residue M44 and the hydropho-

bic pocket in SD4 around protected residue P243, SD2

was moved toward the front side along with its movement

toward the large domain. As shown in Figure 6C, the M44

side chain is inserted into the hydrophobic pocket and

forms a contact with the P243 side chain. M47 cannot

form this interaction with the hydrophobic pocket without

Figure 5. The Model

(A) Overlaid structure of the model (silver) and template structure 1YAG

(brown). The residues that experience protections are displayed as

sticks and bubbles. Structures are aligned by subdomain 1.

(B) The difference between the model and template structure 1YAG.

The arrows indicate the differences of backbone atom positions. The

sphere and long arrow indicate the center of rotation and its normal

vectors for rigid-body movements (green for large domain and yellow

for subdomain 2).

9–51, January 2007 ª2007 Elsevier Ltd All rights reserved 45

Structure

Model of Mg2+-ATP-Actin Monomer

Figure 6. Magnified Views of Interface

between SD2 and SD4

(A) Template structure 1YAG.

(B) The model. The residues that experience

protection are colored blue (in SD2) or red

(SD4).

(C) The interaction between the SD4 hydropho-

bic pocket and D-loop M44 side chain. The

side chain of P243 is colored green while the

side chain of M44 is displayed as bubbles (sul-

fur is colored yellow).

(D) Newly formed hydrogen-bond network be-

tween SD2 and SD4 in the model. Residues are

silver in SD2 and brown in SD4. The possible

p-stacking is indicated by surface models on

residues Y69 and R183.

forming improper bond angles in the context of these

rigid-body movements. The two residues (M44 and

P243) are simultaneously protected from solvent expo-

sure through the ‘‘pairwise’’ interactions. On the other

hand, M47 is exposed to solvent almost completely.

This makes predictions that could be tested in future foot-

printing experiments using high-resolution tandem MS.

In systematic mutational analysis experiments on yeast

actin, the double-mutation A204E/P243K abolishes poly-

merization (Joel et al., 2004). These mutations introduce

bulky and charged residues at the place where contacts

are formed in our model, likely blocking this conforma-

tional change. However, because these residues are also

involved in the intermolecular contacts of the F-actin

model, it is also possible that the mutations may hinder

F-actin filament assembly.

The closure of the large cleft in our model allows the

formation of a hydrogen-bond network between SD2

and SD4 (Figure 6D). In our model, seven new hydrogen

46 Structure 15, 39–51, January 2007 ª2007 Elsevier Ltd All rig

bonds can be formed between SD2 and SD4, as indicated

in the figure. Also, Y69 and R183 are in good locations for

p-stacking interactions (this interaction may not be critical

to form the structure, as the R183A/D184A mutant has no

significant change in phenotype). In a comparison of ASA

between 1YAG (rabbit sequence) and the model (Figure 7),

M47’s side-chain ASA increased by almost 50%. On the

other hand, M44 experiences almost complete burial of its

side chain. If we take into account the extremely dynamic

nature of the D loop, it is reasonable to assume that

both M44 and M47 are much more accessible than the

structure observed in 1YAG. Overall, the accessibility of

M44/47 seen in footprinting is in good agreement with

the model. The ASA changes in Y53 can be explained in

the same way. The D-loop structure is highly mobile, and

in the crystal structure the side chain of Y53 in 1YAG is

covered by the D loop and appears buried, but this does

not reflect the ensemble average experienced by the fluc-

tuating structure. In the model, although the ASA of the

Figure 7. Changes in Accessible Surface

Areas of Probe Residues

The red bars are ASAs of 1YAG (rabbit) and the

blue bars are differences between 1YAG and

the model. Negative values indicate less ex-

posed in the model.

hts reserved

Structure

Model of Mg2+-ATP-Actin Monomer

Y53 side chain increases about 25 A2, the D-loop structure

should be stiff through its formation of an interaction with

SD4. Thus, if the dynamic nature of structural changes in

SD2 is considered, this residue can experience a lower

reactivity consistent with the structural changes seen in

the model.

DISCUSSION

Possible Mechanism of Cleft Closure

Our model is based on rigid-body movements of domains

and subdomains observed in solved crystal structures.

We applied changes strictly to the backbone angles that

are involved in domain movements because we do not

have any data that indicate there is large-scale reorgani-

zation of structure other than domain movements. The

large-domain movement relative to SD1 narrowed the

large cleft including the nucleotide binding pocket as pro-

posed in previous reports of F-actin structure (Belmont

et al., 1999; Lorenz et al., 1993; Tirion et al., 1995). In the

crystal structures except 1RFQ-B, the metal ion is coordi-

nated only by water molecules and phosphate oxygen

atoms. The narrowed cleft results in a closer proximity of

side-chain oxygen atoms (D11:OD1, OD2, and Q137:OE1)

to metal ions that coordinate phosphate oxygen atoms. In

our model, the side chains of these residues form inner-

sphere coordination to Mg2+. As Mg2+ ions strongly prefer

to coordinate to oxygen atoms of side chains instead of

those of water molecules with a likely gain of free energy

(Dudev et al., 1999), it is reasonable to assume that ex-

change of inner-sphere coordinations between water mol-

ecules and oxygen atoms in D11 and Q137 occurs spon-

taneously, as thermal fluctuations bring side-chain oxygen

atoms close enough to the metal ions mediating the ex-

change reaction. This can then be a driving force for the

movement of the large domain toward SD1. Interestingly,

there is no strong interaction between SD1 and SD3—no

hydrogen-bond network nor hydrophobic interactions.

Thus, there is no discernable preference (or penalty) for

these subdomain motions compared to the structures ob-

served in the crystals. Once the large domain is locked in

the closed form by formation of inner-sphere coordination

between side-chain oxygen atoms and the metal ion, SD2

and SD4 are closer to each other and in the range where

a detailed hydrogen-bond network (and possibly p-stack-

ing) can form between them. This hydrogen-bond network

and p-stacking stabilize and lock a ‘‘super-closed’’ form

of the large cleft. The invariant residue Y69 appears to

be a key residue for the interaction between SD2 and

SD4. It possibly forms hydrogen bonds with the residues

in SD4 at the bottom of the large cleft and acts as a latch.

Mutations of residues forming interactions between SD2

and SD4 (Figure 6D) are reportedly lethal in yeast (Sheter-

line and Sparrow, 1994; Wertman et al., 1992) (R62, R206,

E207), although these residues located within the cleft are

unlikely to be involved in protein/protein interactions.

The contribution of the D loop to cleft closure is

documented in the literature (Khaitlina and Strzelecka-

Golaszewska, 2002). The cleavages in the D loop by

Structure 15,

ECP32 protease (G42-V43) or subtilisin (M47-G48) greatly

stimulate the tryptic susceptibility of residues within SD2

(R62 and K68). The cleavage of the D loop apparently

makes those accessibilities of tryptic cleavage sites in-

crease. Thus, the D loop contributes to the stability of the

closed structure although it is not clear how much contri-

bution to the stability comes from the D loop. Our model

predicts the interaction between M44 and the SD4 hydro-

phobic pocket contributes to stabilization of the closed

structure of the large cleft. The observations above sup-

port our model, as the D-loop/SD4 interaction is a strong

candidate for the change in dynamics of SD2.

Catalytic Residue

As a consequence of the large-cleft closure by large-

domain movements relative to SD1, the geometry inside

the nucleotide binding pocket is altered from those seen in

typical crystal structures. The narrowed cleft brings D11,

Q137, and D154 side chains closer to the metal ion and

the b and g phosphates while maintaining the hydrogen-

bond network already in place.

D11 and Q137 are likely to be involved in the inner-

sphere metal-ion coordination. On the other hand, D154

seems to be too far from the metal ion to coordinate but

is located in a good position to form an inline attack of the

g phosphate. The double-mutant D154A/D157A is lethal in

yeast (Wertman et al., 1992); however, the S14C/D157A

double mutant does not show significant change in

ATPase activity (Schuler et al., 1999). Thus, it is very likely

that the D154A mutation is directly involved in lethality. As

suggested in the literature (Schuler, 2001), D154 appears

to be a prime candidate for the catalytic residue from both

previous reports and our model.

Actin Model Consistency with Nucleotide

and Metal-Ion Exchange Rates

The stability constants of the Mg2+- and Ca2+-ATP com-

plexes are similar (4.2 and 4.0, respectively) (Williams,

1970), and the exchange rate constants of the metal

ions and ATP in water are quite rapid (3.3 3 105 and 2.0 3

105 s�1) (De La Cruz and Pollard, 1995; Pecoraro et al.,

1984). On the other hand, the exchange rates of Mg2+

and Ca2+ bound to ATP-G-actin are quite slow compared

with the rates observed in water and are different from

each other (association rates: 2.3 3 105 and 2 3 107

M�1 s�1; dissociation rates: 0.0015 and 0.014 s�1 for

Mg2+- and Ca2+-ATP-G-actin, respectively, as summa-

rized in Table 3) (Gershman et al., 1991; Selden et al.,

1989; Sheterline and Sparrow, 1994). These data indicate

that the metal ions are significantly stabilized within the

cleft by limited diffusion or coordination bond formation

with protein side-chain oxygen atoms. The difference in

association/dissociation with ATP between Mg2+ and

Ca2+ is not significant in water but is an order of magnitude

different in the case of actin-nucleotide-bound forms. Our

model predicts that one factor explaining the difference in

stability is coordination of Mg2+ with oxygen atoms of the

protein. The other factor is the difference in accessibility of

metal ions (steric effect). The exchange rates of ATP for

39–51, January 2007 ª2007 Elsevier Ltd All rights reserved 47

Structure

Model of Mg2+-ATP-Actin Monomer

Table 3. Exchange Rate Constants of Nucleotides/Metal Ions for Different Species of Actin

Rate Constant Mg2+ Ca2+ Species

Metal ion Exchange rate 3.3 3 105 s�1 2.0 3 105 s�1 ATP/metal ion

Metal ion Association rate 2.3 3 105 M�1 s�1 2.0 3 107 M�1 s�1 Metal ion/ATP + actin

Metal ion Dissociation rate 0.0015 s�1 0.014 s�1 Metal ion/ATP/actin

ATP Exchange ratea 5.0 3 10�4 s�1b 1.5 3 10�2 s�1c Metal ion/ATP/actin

Metal ion Affinity (Kd) 10 nM 2 nM ATP-G-actin

a The exchange rate is highly dependent on metal-ion concentrations.b Mg2+ concentration of 1 mM.c Ca2+ concentration of 0.1 mM.

both Mg2+- and Ca2+-G-actin are even more different (5 3

10�4 and 1.5 3 10�2, respectively) (Sheterline and Spar-

row, 1994). As ATP exchange rates are tightly connected

to free metal concentrations, the metal-ion exchange

rate would be a limiting factor for nucleotide exchange

(Kinosian et al., 1993). Even if we attribute all of the metal

exchange rate difference to coordination states, still there

is a difference in ATP exchange rates. Thus, this ATP ex-

change rate difference may be related to the steric effect

of the cleft closure. These observed exchange rate differ-

ences between Mg2+- and Ca2+-G-actin are quite consis-

tent with our model (closed cleft and direct coordination of

Mg2+ to the side-chain oxygen atoms).

Implications for F-Actin Models

The Holmes model of F-actin (Holmes et al., 1990; Lorenz

et al., 1993; Tirion et al., 1995) has been widely accepted

and is largely consistent with X-ray fiber diffraction (Bel-

mont et al., 1999; Holmes et al., 1990; Lorenz et al.,

1993; Oda et al., 2001; Tirion et al., 1995; Wu and Ma,

2004), cryoelectron microscopy (Schmid et al., 2004),

and atomic force microscopy data (Shao et al., 2000; Shi

et al., 2001). We have examined the F-actin structure by

side-chain solvent accessibility utilizing hydroxyl-radical

footprinting (Guan et al., 2005). Although our data were

mainly consistent with the F-actin model, there are some

disagreements with the Holmes model. The most appar-

ent difference is the status of the large cleft. The protection

sites in SD4 (residues 200–202 and 243) and the D loop

(residues 40, 44, and 47) form contacts in the intermolec-

ular interfaces; there is no disagreement with footprinting

data for these residues. However, the hydroxyl-radical re-

activity within the large cleft is significantly lower in F-actin

compared with Ca2+-G-actin. A closed cleft is also ob-

served in a reconstructed model from cryo-EM data

(Schmid et al., 2004), although the degree of closure

seems not to be enough to explain the almost 90% reduc-

tion in hydroxyl-radical reactivity compared to the open

form of Ca2+-G-actin.

The ADP and ADP-BeF3� forms of fiber diffraction data

show differences in the status of the large cleft (Belmont

et al., 1999); the latter form indicates a more closed cleft.

The replacement of ADP with ADP-BeF3� also diminishes

susceptibility of F-actin to tryptic cleavage within SD2

(Muhlrad et al., 1994). As ADP-BeF3� mimics ATP or

48 Structure 15, 39–51, January 2007 ª2007 Elsevier Ltd All righ

ADP-Pi (Muhlrad et al., 1994), this form represents ATP

or ADP-Pi-F-actin before it releases Pi. These observations

indicate that in the F-actin filament, hydrolysis of ATP to

ADP and subsequent release of Pi correlate with large-cleft

opening. On the other hand, when Mg-ATP-G-actin is in-

corporated into the filament, we suggest it is in the highly

closed form seen here; this Mg-ATP form of actin has a

lower critical concentration than Ca2+-ATP-G-actin. Catal-

ysis mediated by the Mg2+ ion may drive conformational

changes among the coordination bonds between side

chains and the metal ion, leading to rearrangements of

subdomains 2 and 4 and the generation of a more open

cleft within the ADP-F-actin structure. This would provide

a more flexible and dynamic SD2, which is thought to be

a key factor in attracting binding proteins and driving fila-

ment dynamics (Schmid et al., 2004).

Conclusion

Understanding monomeric G-actin structure is important,

as it is relevant to many protein/protein interactions in

which actin participates. Hydroxyl-radical footprinting is

a powerful technique to probe surface accessibility of

macromolecules. The technique is a local measure as it

reports side-chain accessibility. Such information pro-

vides strict restraints for estimating changes in structure

and in defining domain interactions. In conjunction with

a computational approach, a structural model of the Mg-

ATP-actin monomer was built that is consistent with ex-

perimental data. This model provides novel insights into

the conformational rearrangements driving actin filament

dynamics and provides a unique structural insight into ac-

tin monomer structure. This combined computational and

experimental approach is particularly useful for modeling

the structural changes of proteins when we do not have

access to crystal or solution structures for conformations

of functional interest, but where we may have suitable

starting templates for modeling the structure. Further

developments in computational modeling will ease the

integration of the experimental into the computational

approach as a filtering/validation tool.

EXPERIMENTAL PROCEDURES

Rigid-Body Movement Analyses

Tcl scripts for molecular viewer VMD (Humphrey et al., 1996) were de-

veloped for the analysis and visualization of rigid-body rotational

ts reserved

Structure

Model of Mg2+-ATP-Actin Monomer

movements of domains/subdomains. The scripts accept the range of

the target residues in two structures for rigid-body movement and then

calculate vectors of atom movements and averaged normal for these

vectors; they then search for the atoms within the reference structure

that are nearest to the center of motion. The algorithm is based on the

assumption that there are pivot points in bonds within the backbone

allowing rigid-body rotational movements (the movements should be

explained by rotations of some bond angles). The algorithm is based

on the following step-by-step calculations.

Define structures A and B, where A is the reference structure and B

is the target structure for movement calculation. The total number of

atoms (Ca orbackbone atoms) isnwithin the target range. The total num-

ber of the atoms in reference structure A (Ca or backbone atoms) is N.

Atoms in structure A: ai (i = 1 to n), coordinates Pai.

Atoms in structure B: bi (i = 1 to n), coordinates Pbi.

Atoms for the center C: cj (j = 1 to N), coordinates Pcj.

The midpoint of ai and bi: mi (i = 1 to n), coordinates Pmi.

The vector between points ai and bi: v!i = Pbi � Pai.

The vector between points mi and cj: u!ij = Pmi � Pcj.

The angle formed by the two vectors v!i and u!ij :

qij =

����acos

�ð v!i � u!jiÞj v!ij

�� u!ji

�������:

The averaged angle for the point Pcj:

�qj =1

n

Xn

i = 1

�qij �

p

2

�ð j = 1to NÞ:

The SD of the angles for atom Pcj:

SDj =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni = 1ððqij � p

2

�� �qj

�2

n� 1

sð j = 1to NÞ

The averaged angle between the real center of movement cr and the

two atoms in structures A and B should be p2; thus, �qj is 0. We need to

find the atom that gives the averaged angle closest to 0. In order to

avoid the cases where �qj happens to be close to 0 but angles vary

as qij can be positive or negative, the score is calculated by the follow-

ing formula:

Sj = �qj 3 SDj :

Then, the script calculates the average positions of the top scored

five atoms. This point is defined as gc.

Now, we need to calculate the normal vector for the rotational move-

ments. The normal vector for the rotational movements should be ap-

proximately perpendicular to all vectors from the center of movement

to the midpoint of atoms in the two structures A and B. The normal vec-

tor is also perpendicular to the vector vi between ai and bi. In order to

minimize the contribution from small vectors that tend to have larger

deviation from ideal ones (the vectors are extended by unitization

along with ‘‘error’’), ‘‘normalized’’ normal vectors are weighed by their

norms. After the average normal vector is calculated, it is renormalized

to the unit vector.

The vector between points mi and gc: w!i = Pmi � gc.

The normal vector of rotation for points ai and bi:

n!

i =v!i 3 w!i

j v!i jjw!i j

(crossproduct).

Averaged (weighed) vector:

n!

av =1

n

Xn

i = 1

ðj v!i jjw!i j, n!

iÞ:

Normal vector (unit vector):

n!

unit =n!

av

j n!av j:

Structure 15

The resultant averaged normal vector and center of rotation is

visualized along with vectors vi within VMD by Tcl commands in the

script.

Accessible Surface Area Calculations

The accessible surface areas of side chains are calculated using the

GETAREA 1.1 (Fraczkiewicz and Braun, 1998) web server with 1.4 A

probe radius (http://www.scsb.utmb.edu/cgi-bin/get_a_form.tcl) with

additional atom type library and residue type entries. In our previous

report, we used rabbit skeletal muscle actin. As ASA greatly depends

on the side chain, the sequence of rabbit skeletal muscle actin was

mapped on 1YAG using MODELLER 7v7(Sali and Blundell, 1993).

This structure is designated as 1YAG (rabbit sequence) in the following

sections.

Computational Modeling

Model construction started from a ‘‘template’’ structure of 1YAG (rab-

bit sequence). The rotational movement toward the small domain was

applied to the large domain (residues 138–332) at the pivot residue 138

with the normal vector calculated in 1HLU (Chik et al., 1996). The rota-

tional movement toward SD2 was applied as a combination of several

movements calculated based on the analyses of several crystal struc-

tures. The rotations were applied over four residues from specifically

selected pivot points (residues 34–37 and 66–69) in order to prevent

too much rotation to single bonds that can lead to improper bond an-

gles. All rotational transformations of coordinates were performed us-

ing Tcl scripts within VMD. Modeled structures were generated with

combinations of different angles and normal vectors. ASAs of side

chains were calculated for these structures in order to assess the pre-

liminary models. The best model in the series of models by rigid-body

movements was chosen according to the consistency with expected

changes in ASAs of hydroxyl-radical probes.

The model was then subjected to manual inspections/manipulations

of side chains in the Swiss-PdbViewer (Guex and Peitsch, 1997) in or-

der to better satisfy the hydroxyl-radical data. The rotamer of side

chains was selected to maximize hydrogen-bond formations within

the large cleft. As the large cleft between SD2 and SD4 was closed

by moving the large domain relative to SD1, the nucleotide binding

pocket was also affected and became significantly narrower than in

the template structure 1YAG. The coordinates of the phosphates of

ATP were manually adjusted within VMD using Tcl scripts by rotating

the phosphate-oxygen bond and fit into the narrower cleft interactively

in order to avoid steric clashes. The D loop was constructed manually

in the Swiss-PdbViewer by rotating the phi and psi angles of the tem-

plate structure. This process was guided mainly to allow M44 to reach

the hydrophobic pocket in SD4 while at the same time avoiding im-

proper angles. At this stage, the model structure was examined by

PROCHECK (Laskowski et al., 1993) and Verify3D (Bowie et al., 1991)

for model validity (Figure S1). The model coordinates were then passed

to MODELLER 7v7 (Sali and Blundell, 1993) as a template for adjusting

and correcting the stereochemical parameters. MODELLER was also

used to adjust the geometry of the residues and metal ion in the nucle-

otide binding pocket with manually assigned restraints according to the

1RFQ-B (chain B) structure (Reutzel et al., 2004). After further manual

inspections/adjustments in VMD and energy minimization within the

Swiss-PdbViewer for certain side chains, the final structural model

was subjected to PROCHECK and Verify3D for the assessment of

structure validity.

Visualization of Structures

The structures were visualized using POV-Ray ray-tracer (Version 3.6)

with output from VMD (Version 1.8.3) and manual editing of scene files.

Some presentations were directly defined by POV-Ray scene. All op-

erations were done on a Power Macintosh G5 with Mac OS X except

for energy minimization within the Swiss-PdbViewer that was per-

formed on a Windows 2000 workstation (OS X version is an a version

and energy minimization is only partially implemented).

, 39–51, January 2007 ª2007 Elsevier Ltd All rights reserved 49

Structure

Model of Mg2+-ATP-Actin Monomer

Supplemental Data

Supplemental Data include two figures and Supplemental Results

and are available at http://www.structure.org/cgi/content/full/15/1/

39/DC1/.

ACKNOWLEDGMENTS

This work is supported in part by the Biomedical Technology Centers

Program of the National Institutes for Biomedical Imaging and Bioen-

gineering (P41-EB-01979).

Received: July 26, 2006

Revised: November 6, 2006

Accepted: November 18, 2006

Published: January 16, 2007

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