A triangle lattice model that predicts transmembrane helix configuration using a polar jigsaw puzzle

Takatsugu Hirokawa1,2, Junichi Uechi1, Hiroyuki Sasamoto1, Makiko Suwa3 and Shigeki Mitaku1,4

1 Department of Biotechnology, Tokyo University of Agriculture and Technology, Nakacho, Koganei, Tokyo 184-8588, 2 Ryoka System Inc., Computational Science and Technology Division, Irifune, Urayasu,Chiba Prefecture 279-0012 and 3 Helix Research Institute, Inc.,1532–3 Yana, Kisarazu-shi, Chiba 292, Japan


    Abstract
 Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 References
 
We developed a method of predicting the tertiary structures of seven transmembrane helical proteins in triangle lattice models, assuming that the configuration of helices is stabilized by polar interactions. Triangle lattice models having 12 or 11 nearest neighbor pairs were used as general templates of a seven-helix system, then the orientation angles of all helices were varied at intervals of 15°. The polar interaction energy for all possible positions of each helix was estimated using the calculated polar indices of transmembrane helices. An automated system was constructed and applied to bacteriorhodopsin, a typical membrane protein with seven transmembrane helices. The predicted optimal and actual structures were similar. The top 100 predicted helical configurations indicated that the helix-triangle, CFG, occurred at the highest frequency. In fact, this helix-triangle of bacteriorhodopsin forms an active proton-pumping site, suggesting that the present method can identify functionally important helices in membrane proteins. The possibility of studying the structure change of bacteriorhodopsin during the functional process by this method is discussed, and may serve to explain the experimental structures of photointermediate states.

Keywords: bacteriorhodopsin/lattice model/membrane protein/polar interactions/structure prediction/transmembrane helix


    Introduction
 Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 References
 
Many of the targets of pharmaceuticals are receptor proteins located in the cytoplasmic membrane (Strader et al., 1995Go). One of the most important receptor families is that of the G-protein-coupled receptor, which has seven hydrophobic transmembrane helices (Watson and Arkinstall, 1996Go). Therefore, considerable information has been gathered regarding the structure of G-protein-coupled receptors (Schertler et al., 1993Go; Davies et al., 1996Go; Unger et al., 1997Go; Herzyk and Hubbard, 1998Go), ion pumps with seven transmembrane helices, bacteriorhodopsin (Henderson and Unwin, 1975Go; Henderson et al., 1990Go; Grigorieff et al., 1996Go; Kimura et al., 1997Go; Pebay-Peyroula et al., 1997Go; Essen et al., 1998Go; Luecke et al., 1998Go, 1999aGo; Belrhali et al., 1999Go; Sato et al., 1999Go) and halorhodopsin (Havelka et al., 1995Go; Kolbe et al., 2000Go), using X-ray and electron diffraction. The results indicated that the seven transmembrane helices are almost all arranged in parallel and form a bundle of antiparallel helices. The helices are amphiphilic, with their polar surfaces oriented towards the inside of the bundle. Furthermore, structural studies of bacteriorhodopsin during the photocycle have revealed structural changes in the helical configuration (Subramaniam et al., 1993Go; Kamikubo et al., 1996Go, 1997Go; Oka et al., 1999Go) and also in the hydrogen bonding networks of side chains and water molecules in membrane (Belrhali et al., 1999Go; Edman et al., 1999Go; Luecke et al, 1999bGo; Subramanian et al., 1999).

Several theoretical methods using various algorithms have been proposed with which to analyze and predict the structures of membrane proteins with seven transmembrane helices (Jähnig, 1992Go; Baldwin, 1993Go, 1997; Cronet et al., 1993Go; Donnelly et al., 1993Go; Du and Alkorta, 1994Go; Taylor et al., 1994Go; Tuffery et al., 1994Go; Herzyk and Hubbard, 1995Go; Suwa et al., 1995Go; Bowie, 1999Go; Koshi and Bruno, 1999Go; Pilpel et al, 1999Go). Among these, the method proposed by Taylor et al. (1994) is very convenient because a single amino acid sequence is a sufficient basis upon which to reconstruct the three-dimensional (3D) structure of a membrane protein. They adopted a lattice model as the structural template of the arrangement of transmembrane helices. The configuration of seven transmembrane helices is determined according to the basic premise that the sides of helices with more hydrophobic and variable residues face the lipid bilayer phase. Bowie (1999) estimated the number of folds of membrane proteins, including seven-helix-type membrane proteins. The folds may serve for extending the lattice models to more general templates. A novel index of amino acids was developed by Pilpel et al. (1999) for predicting the orientation of transmembrane helices, which may improve the performance of 3D structure prediction of membrane proteins.

However, their procedure may not address structural change during the functioning process, for example the photocycle of bacteriorhodopsin, because a unique index value is allocated to each amino acid irrespective of its protonation state. Studies of bacteriorhodopsin photointermediates have revealed that the structure change of bacteriorhodopsin is coupled with the protonation and deprotonation of essential residues in transmembrane helices. The Schiff base of Lys216 with retinal is protonated in the ground state and deprotonated in the photointermediate states of M and N. Comparisons between the various states of bacteriorhodopsin show that the later photointermediates have a different 3D structure to the ground state (Subramaniam et al., 1993Go, 1999Go; Kamikubo et al., 1996Go; Spudich and Lanyi, 1996Go; Kamikubo et al., 1997Go; Sass et al., 1997Go, 1998Go; Oka et al., 1999Go). Sass et al. (1997, 1998) explicitly stated that these structural changes are caused by the altered charge distribution in the protein.

Suwa et al. (1995) proposed a method with which to predict the structure of membrane proteins based on the assumption that polar interactions are essential for stability of the tertiary structure. They developed a probe helix method with which to characterize the polar interaction field around a transmembrane helix. The polar interaction energy between a transmembrane helix and a probe helix consisting of serine–alanine block copolymer was calculated, in which the copolymer spans the alternative halves of a transmembrane helix (Suwa et al., 1992Go). The energy value was used as the index of helix surface polarity, which corresponds to the serine stretch of the probe helix. Suwa et al. (1995) evaluated the polar interaction energy of the seven-helix system of bacteriorhodopsin, assuming the experimental position of transmembrane helices. The best arrangement of seven helices obtained from the calculated polar interaction energy was the same as that of the experimental structure and the orientation of helices was also very similar to the experimental helix configuration. This method may address the structural change of a membrane protein such as bacteriorhodopsin in the functioning processes, because the algorithm is based on the charge distribution in the protein, which changes owing to the protonation states of polar residues.

This study extends the method of Suwa et al. (1995) by adopting the lattice models of seven helices to survey a wider structural space. The predicted and experimental structures were very similar, strongly suggesting that the algorithm adopted in this work reflects the essential part of the interactions by which the 3D structure of bacteriorhodopsin is stabilized. Furthermore, the 100 optimal structures of bacteriorhodopsin in the triangle lattice indicated a preference for a triangle consisting of helices C, F and G.


    Methods
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 Abstract
 Introduction
 Methods
 Results
 Discussion
 References
 
Principle of specific binding between helices

We assumed that the tertiary structure of a membrane protein is stabilized mainly by polar interactions between transmembrane helices. Binding specificity arises from the difference in the polar area on the sides of helices. Figure 1aGo shows schematically the mechanism of specific binding. For example, when helices A and C have polar areas only in their upper and lower halves, respectively, they do not effectively bind each other. However, when helix B has two polar areas on opposite sides in the upper and lower halves, then helices A and C will be connected through helix B. In this way, the configuration of helices may be determined as a jigsaw puzzle, in which the indentation of pieces consists of polar sides of helices. We therefore call this method the polar jigsaw puzzle (PJP) (Suwa et al., 1995Go).



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Fig. 1. Schematic diagram describing the specific binding between transmembrane helices (a) and probe helix method for simplifying polar interaction field of helices (b). Polar interaction energy between a transmembrane helix and a probe helix of serine–alanine copolymer is calculated and then used as the polar index of a helix. Using two probe helices and rotating a transmembrane helix, 48 polar index values were obtained for simple estimation of helix–helix binding energy (c).

 
Interaction energy between transmembrane helices

The PJP assumes that the total interaction energy E between a pair of transmembrane helices has two terms, the polar interactions of the Ehelix at the hydrophobic region and the rubber elasticity Eloop at the region linking two neighboring helices:

(1)
The importance of polar interactions in the binding between transmembrane helices was proven experimentally by penetration experiments on intrinsic membrane protein, e.g. bacteriorhodopsin (Mitaku et al., 1988Go, 1995Go; Kukita and Mitaku, 1993Go; Mukai et al., 1999Go). Although another mechanism of interhelix binding, namely the packing effect, has been established for the dimerization of a single-helical membrane protein, glycophorin (Fleming et al., 1997Go; MacKenzie et al., 1997Go), multispanning membrane proteins such as bacteriorhodopsin seem to be stabilized mainly by polar interactions that are enhanced in the non-polar environment of membranes (Steitz et al., 1982Go; Mitaku et al., 1988Go, 1995Go; Kukita and Mitaku, 1993Go). The energy term of a loop segment is zero for two helices that are separated by other transmembrane helices.

Here, the pair polar interaction between two transmembrane helices Ehelix is approximately expressed by the following equation:

(2)
where the assumption is that the angular dependence and the distance dependence of the polar interaction energy can be separated. The angular function B({theta}ij, {theta}ji) represents the binding energy between helices i and j at a distance r0, which we set to 1.2 nm. The dipole–dipole interaction energy generally has an inverse cubic distance dependence. Since the polar interaction energy between helices will be given by integrating many dipole–dipole interactions, we assumed an inverse square distance dependence of the interhelix binding energy.

The latter term of total energy is the effective interaction energy, arising from the structure of loop segments. We assumed a rubber elastic potential function for this term:

(3)
where Lii + 1 is the length of a loop segment between the ith and the next helix. An elastic constant C of 0.8 kcal/mol.nm was used in this calculation. The most important effect of rubber elastic potential here is that two transmembrane helices linked by a short loop segment cannot split apart.

Simplified expression of polar interactions between helices

In the PJP framework, the polar interaction energy Ehelix between transmembrane helices is simplified using the polar index of a helix (Suwa et al., 1995Go), which expresses the strength of polarity of a helix, by dividing its surface into vertical and angular directions, as shown in Figure 1bGo. The values of the polar index were estimated by calculating the interaction energy of a transmembrane helix using N-probe (N–Ala2–Ser11–Ala10–Ala2–C) and C-probe helices (N–Ala2–Ala10–Ser11–Ala2–C). These probes characterized the distribution of polar residues in each half of the transmembrane helices, as shown in Figure 1cGo. The value of the polar index varies according to the number and the polarity of residues that face the serine block of the probe helix. Rotating a transmembrane helix by 15°, this method generates 24 values for the polar index per probe helix. Therefore, 48 values of the polar index were obtained for each transmembrane helix, which expressed the vertical and angular distribution of the polar residues.

The angular function of the interaction energy B({theta}ij, {theta}ji) can then be described in terms of the polar index P({theta}).

(4)
where subscripts i and j represent the i- and j-helices, respectively. The superscripts u and l denote the halves of a transmembrane helix: u represents the N-terminal halves of even-numbered helices and the C-terminal halves of odd-numbered helices; l indicates the N-terminal halves of odd-numbered helices and the C-terminal halves of even-numbered helices. Therefore, u and l in bacteriorhodopsin denote the cytoplasmic and external halves of transmembrane helices, respectively. Equation 4Go means that the interaction energy between i- and j-helices increases when the polar surfaces of these helices face each other.

Although this expression for the polar interaction energy is very simple, it agrees well with the pair energy calculation of various model helices (Suwa et al., 1995Go). The coefficient A is the factor used to adjust the simplified energy function to the direct calculation. When the interacting helices have the same type of charge, the factor A is –0.02 (kcal/mol)–1 and otherwise the value is –0.14 (kcal/mol)–1.

We applied the probe helix method in the present study using CHARMm-QUANTA (Molecular Simulation) on an Indy (Silicon Graphics) (Brooks et al.,1983Go). The dielectric constant of the system used for the probe helix method was set at 2.0. The configuration of the two helices was antiparallel and the standard distance between helices was 1.2 nm. The cut-off parameters were 0.8 nm for non-bonded interaction and 120° for hydrogen bonding.

Lattice models of seven transmembrane helices

The total energy in the PJP framework decreases upon contact of the polar surface of helices. Therefore, the lower energy of a multi-helix system should be realized for the structures with more nearest neighbor pairs of helices. Figure 2Go shows six structures of a seven-helix system in the triangle lattice, in which the number of the nearest neighbor pairs is 12 (L12) and 11 (L11A, L11B, L11Bm, L11C and L11Cm). The structures L12 and L11A are symmetric, whereas the structures L11Bm and L11Cm are the mirror images of L11B and L11C, respectively. The position of transmembrane helices in the real structure of bacteriorhodopsin, for example, is not identical with any of the six models. However, the lattice model may be used as the initial basis for a more realistic structure prediction of membrane protein structure. This will be reported elsewhere.



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Fig. 2. Six kinds of helix configurations in a triangular lattice, having 12 (L12) and 11 nearest neighbor pairs (L11A, L11B, L11Bm, L11C, L11Cm).

 
The simplification of energy calculation by Equation 4Go has considerable advantages. A very large structural space can be surveyed in a reasonable time and the energy landscape of a multi-helix system becomes very smooth with only a few energy minima. Therefore, it is easy to identify the grand energy minimum from the energy landscape, which is calculated within the framework of the PJP. The present study shows that all possible configurations of the helices of a seven-helix system in seven lattice points could be examined for six models in Figure 2Go. The rotational angles of all helices were also varied by 15°. Although the total number of structures, 1.387x1014 (= 6x7!x247), was sizable, a very large structural space was surveyed and the grand minimum of a seven-helix system was obtained within the framework of the lattice model.

Bacteriorhodopsin

We applied the present method to the seven transmembrane helices of bacteriorhodopsin, because this protein has many polar residues in the hydrophobic regions and is therefore suitable for analysis using PJP. Seven transmembrane regions of bacteriorhodopsin were taken from the structure defined by Grigorieff et al. (1996): A, 9–31; B, 38–62; C, 77–100; D, 105–127; E, 134–157; F, 166–191; G, 202–226. Table IGo shows the amino acid sequences of the seven transmembrane regions together with the ratio of polar residues in those regions. The average ratio of polar residues was about 30% when all residues with polar groups such as Tyr and Trp were classified as polar residues. This value is large enough to determine structures by their polar interactions.


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Table I. Transmembrane helix regions of bacteriorhodopsin and the ratio of polar residues in the helix regions
 

    Results
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 Abstract
 Introduction
 Methods
 Results
 Discussion
 References
 
The prediction was obtained from the following procedures: (1) characterization of polar interaction field around TM helices using probe helices, (2) predicting energetically preferable structures in the lattice model and (3) extracting characteristic triangles of transmembrane helices that may be functionally important.

Figure 3Go shows the polar interaction energy profile of seven TM helices in bacteriorhodopsin as a function of their orientational angles. The open and filled circles indicate the polar interaction energies of upper (external) and lower (cytoplasmic) halves of TM helices, respectively, that are obtained from the calculated energy between a TM helix and one of N- and C-probe helices. The values of the polar interaction energy are used as the polar indices in Equation 4Go. Namely, the polar interaction energy with a probe helix was assumed to represent the strength of polarity of the helix surface.



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Fig. 3. Angular dependence of polar interaction energy of seven transmembrane helices in bacteriorhodopsin. Solid and open circles represent cytoplasmic and external halves of transmembrane helices, respectively. Using these values as polar indices, the helix–helix binding energy was calculated by Equation 4Go.

 
The characteristics of the polar interaction field differed widely among helices. For example, the interaction energy among helices A, B, D and E showed only a small angular dependence of the interaction energy, whereas helices C, F and G exhibited large dips of energy. The polar indices of the TM helices C, F and G were large owing to the following charged residues: Arg82, Asp85 and Asp96 for helix C; Lys172 and Arg175 for helix F; Asp212 and Lys216 for helix G.

The interhelix binding energy Ehelix of pairs of helices was calculated using Equation (4)Go. Each diagram in Figure 4Go shows the energy landscape in orientational angle space of two helices. The vertical and horizontal axes represent rotational angles of a pair of TM helices. A gray scale represents the intensity of binding energy with darker spots indicating higher binding energy. The numbers above the interaction map represent the length of linking segments between TM helices. Polar interactions of pairs containing helices C, F and G were significantly stronger than those of other helix pairs. Therefore, helices C, F and G will be closely packed in energetically preferable structures.



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Fig. 4. Energy landscapes of interhelix binding for all pairs of transmembrane helices. Horizontal and vertical axes of each diagram are orientational angles of two helices. The numbers above the triangle map represent the length of loop segments between helices. Pixel density in diagrams shows interhelix binding strength, i.e. binding energy.

 
We calculated the structural energy of the seven TM helices from bacteriorhodopsin using Equations 1–4GoGoGoGo. All combinations of helices and lattice points in the structural space and all orientational angles at 15° intervals for all six models were examined as shown in Figure 2Go. We determined a minimum energy of 247 structures of different orientational angles for each combination of helices and lattice points. Figure 5Go shows that the histogram of the minimum energies is almost symmetrical.



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Fig. 5. Energy histogram of the seven transmembrane helices in bacteriorhodopsin. Comparing 247 energy values for each positioning of all helices, the lowest energy was selected and plotted in the histogram.

 
The most preferable configuration of the seven transmembrane helices of bacteriorhodopsin is shown in Figure 6Go, in which the experimental structure is also included for comparison (Grigorieff et al., 1996Go). The positioning of the seven helices is similar to that of the experimental structure, in that the helices are in a clockwise arrangement from the N-terminal end and helix C is located at the center of the protein. The orientation of helices is also very similar to the real structure, because polar residues are oriented to the inside of the protein. The functionally essential residues, Asp85, Asp96 and Lys216, face each other. Consequently, the configuration of the seven helices in bacteriorhodopsin was accurately predicted by the polar jigsaw puzzle method.



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Fig. 6. Comparison of the most preferable structure from calculated (a) and experimental (b) structures of bacteriorhodopsin. The orientation and arrangement of seven helices in the predicted structure are very similar to those in the experimental structure. Ribbon models were of zbrd (PDB) drawn by INSIGHT II of Molecular Simulation Inc. (Kraulis, 1991Go).

 
Figure 7Go shows schematically the best 20 structures, highlighting helix triangles made of helices C, F and G. Although the top 20 structures appear highly diverse, the arrangements of helices C, D, E, F and G are not so diverse. The arrangement of the five helices is the same for the top four structures. To analyze systematically the characteristics of the structures, we focused on the triangles formed by transmembrane helices. Figure 7Go shows the CFG triangle, in which helices C, F and G are arranged clockwise and its mirror image in the top 20 structures. Among these, 17 structures contained the CFG triangle and two had its mirror image. Only one structure among the 20 did not contain a CFG or CGF triangle. The calculation shows that the CFG triangle is predominant in the structure of bacteriorhodopsin.



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Fig. 7. The best 20 arrangements of transmembrane helices, having the 20 lowest energies in Figure 5Go. Helix triangles of CFG clockwise and anticlockwise are highlighted in the helix arrangements. Seventeen arrangements contained the triangle of CFG helices.

 
Figure 8Go shows systematic analyses of the top 100 structures based on the occurrence of various helix triangles. The proportion of triangles was calculated as a function of the number of structures from the top. A CFG triangle occurred most frequently among all possible triangles, while the proportion of triangles ABC and ACG rapidly decayed. Triangles CDE and CEF are persistent among the preferred structures, but the decay of the proportion was more rapid than that of the CFG triangle. The persistence of a triangle among the energetically preferable structures seems to correlate well with strong interactions among the three helices, which contain functionally important residues.



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Fig. 8. Proportion of various helix triangles in the best 100-helix arrangements. The proportion is plotted as a function of the number of arrangements from the top: (a) helices ABC, ACB and ABG; (b) helices ACG, AGC and ACF; (c) helices CFG, CGF, BFG and DFG; (d) helices CEF, CFE, DEF and EFG; (e) helices CDE, CED and CDF.

 

    Discussion
 Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 References
 
The goal of membrane protein structure prediction is to develop a means of predicting 3D structure from a single amino acid sequence. However, the problem has to be simplified into several steps, because the process to achieve this goal is very complex and difficult. The purpose of the present study was to predict the approximate positioning and orientation of transmembrane helices based on the interactions in transmembrane regions. The key results may be summarized as follows. We developed a new method with which to predict the configuration and orientation of transmembrane helices within the framework of the PJP method using a triangle lattice model. The configuration of transmembrane helices in bacteriorhodopsin was accurately predicted by applying this method to the amino acid sequences of transmembrane helices. The best structure predicted by this method was very similar to the experimental structure. The most persistent triangle of transmembrane helices in the top 100 predicted structures of bacteriorhodopsin was that composed of helices C, F and G, which also corresponded to the functionally most important helices.

This method has the following advantages. First, the helix configuration is determined based upon polar interactions between helices in the present study. In the framework of this method, the predicted structure depends not only on the amino acid sequences but also on the protonation states of charged residues. This algorithm has the potential advantage of predicting the structural change of a membrane protein during the functioning process, because function is usually coupled with the protonation and deprotonation of polar residues at an active site (Oka et al., 1999Go; Subramaniam et al., 1993Go, 1999Go; Kamikubo et al., 1996Go, 1997Go; Spudich and Lanyi, 1996Go; Sass et al., 1997Go, 1998Go). Second, the optimal structure and important triangles of transmembrane helices can be revealed by this method. Information about an active site arising from analyses of an important helix triangle will be useful for molecular biologists, who investigate functionally important amino acids. Third, the method is not associated with structural frustration, which is a serious problem in the molecular simulation of proteins. A helix is assumed to be a rigid rod, having a polar interaction field of smooth angular dependence. Therefore, the energy landscape of the total system becomes smooth and the determination of the grand energy minimum is guaranteed in the framework of this method. Fourth, the structure of a membrane protein with seven transmembrane helices may be automatically predicted. Therefore, predictive ambiguity, arising from manual handling of structures during the processes of molecular modeling, is avoided.

However, this method also has disadvantages and limitations. The physical interaction dominant for the structure formation is assumed to be the polar interaction in the central region of transmembrane helices. In general, polar interaction is significant in a non-polar environment because of a low dielectric constant. Therefore, polar interaction in the hydrophobic region of a membrane should be an important factor for binding between transmembrane helices. In fact, the seven helices of bacteriorhodopsin have a considerably large proportion of polar residues, as seen from Table 1Go, which is a prerequisite of the PJP. The importance of polar interactions for interhelix binding in bacteriorhodopsin was previously proven by alcohol denaturation experiments (Mitaku et al., 1988Go, 1995Go; Kukita and Mitaku, 1993Go; Mukai et al., 1999Go). However, the present method is not applicable to membrane proteins containing transmembrane helices without polar residues because such helices cannot bind with other helices in the framework of the PJP. The packing (Fleming et al., 1997Go; MacKenzie et al., 1997Go) of transmembrane helices may be the binding mechanism of such helices. A statistical index of amino acids such as kPROT (Pilpel et al., 1999Go) may be useful for solving the problem of 3D structure predictions of membrane proteins with only a few polar residues.

Another limitation is that of the size of the protein being analyzed. The present prediction system focuses on membrane proteins with seven transmembrane helices. This type of membrane proteins is also very important, because many are G-protein coupled receptors, which constitute the major factor by which a cell communicates with the environment and with other cells. However, the types of membrane proteins are highly diverse. For example, many channels and pumps have more than 10 transmembrane helices. When the number of transmembrane helices increases, the number of templates of lattice points also increases, but very rapidly. The combination of helices and lattice points also increases greatly, rendering prediction very difficult. The method of sampling of structural templates proposed by Bowie (1999) will serve for surveying the structure of membrane proteins with more transmembrane helices.

Despite these limitations, this automated method may be used for several purposes. The structure predicted by the present method may be used as the initial basis upon which to determine a more detailed structure. The next steps of structure prediction will be to refine the positioning of helices, to remove the restriction of lattice points and to determine the tilting of helices with respect to the normal membrane. Such refinements are promising when applied to bacteriorhodopsin not only in the native state but also in the functional intermediate states and will be reported elsewhere. Making this prediction system internationally available through the Internet will help molecular biologists. Important triangles of transmembrane helices are provided by this system, even if many energetically preferable structures are not similar to the true structure. Therefore, the number of possible sites of mutation that can be used for studying functionally important residues will be greatly reduced. The mode of structural change of a membrane protein such as bacteriorhodopsin during its functioning process can be discussed according to the prediction, since stabilization of the helix configuration by polar interaction is assumed.


    Notes
 
4 To whom correspondence should be addressed. E-mail: mitaku{at}cc.tuat.ac.jp Back


    Acknowledgments
 
This work was supported by a Grant-in-Aid for scientific research on priority areas from Mombusho (Ministry of Education, Science, Sports and Culture of Japan).


    References
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 Introduction
 Methods
 Results
 Discussion
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Received February 23, 2000; revised July 7, 2000; accepted August 23, 2000.