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Chromatin organization studied by computer simulation of models of growing polymer chains
J. theor. Biol. (1977) 64, 221-235
Chromatin Organization Studiedby Computer Simulation of Models of Growing Polymer Chains GEORGE RUSSEV AND KALIN
DUDOV
Institute of Biochemistry, Bulgarian Academy of Sciences, 13 Sofia, Bulgaria (Received 19 January 1976) An attempt was made to discriminate between the contribution of the unspecific electrostatic and weak interactions on one hand, and specific pattern recognitions, on the other, to chromatin structure. Several models imitating the organization of growing polymer chains were computer simulated and their degree of adequacy to chromatin was analysed. In this way four levels of organization of chromatin were distinguished. (i) The binding of histones to DNA, which is due to electrostatic interactions and gives rise to a nucleohistone thread; (ii) selforganization of the nucleohistone thread into condensed structure due to weak interactions; (iii) generation of mixed structures containing both condensed and stretched regions due to the simultaneous and competitive progression of two reactions-the self-organization of the nucleohistone thread and its complexing with non-histone proteins; (iv) discrimination between regions which should be condensed and regions which should remain stretched. It was shown that under certain conditions the first three levels of organization occur spontaneously, being accounted for by non-specific interactions only. For the fourth level of organization to be achieved, additional information is necessary.
1. Introduction Chromatin, the interphase form of the genetic material in eukaryotic cells, mainly consists of DNA, histones and non-histone acidic proteins. In spite of the fact that during recent years a massive body of experimental and theoretical work has been done, the biological role of the two main protein components of chromatin is still not fully understood. It is generally believed that histones serve as unspecific blockage of genetic material and that block can be presumably taken off by specific non-hi&one proteins (see McClure
& Hnilica, 1972; Stein, Spelsberg & Kleinsmith, 1974). Even less is known about the events taking place in the process of chromatin organization and the role of the two protein classes in the generation and preservation of this 221
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organization. Recently it has been shown that chromatin is composed of repeated structural units (Olins & Olins, 1974; Clark & Felsenfeld, 1974; Oosterhof, Hozier & Rill, 1975) and several excellent works have described their properties (Clark & Felsenfeld, 1974; Kornberg & Thomas, 1974; Sahasrabuddhe & van Holde, 1974; Nell, 1974). A few theoretical works have also been published in which the organization of these structural units was explained on the basis of specific histone-histone interactions (Kornberg, 1974; van Holde, Sahasrabuddhe & Shaw, 19743; Baldwin, Boseley & Bradbury, 1975; Hyde & Walker, 1975; Thomas & Komberg, 1975). Nevertheless, there is still a long way to be gone towards the complete understanding of chromatin structure and no simple and quantitative description of its organization is yet available. In the present paper, using a computer simulation of imitation models as a probe, we tried to establish the levels of organization and the structural hierarchy in chromatin; to analyse the nature of the forces responsible for the generation and maintenance of these levels; and to look for the material basis of the interactions in terms of molecules with necessary properties. 2. The Model We shall analyse systems containing a growing DNA chain, histones and non-histone proteins. DNA should be thought of as a flexible polymer built up by successive attachment of arbitrary, negatively-charged structural elements to the growing end of the chain. The histones and the non-histone proteins will be represented by oligomers of positively or negatively-charged arbitrary structural elements, respectively. The number of positive elements in this system is supposed to be equal to that of the negative elements composing the DNA chain. The number of negatively-charged elements representing acidic proteins is not defined and can be varied. Here the number of the free positive and negative protein structural elements, not engaged in mutual complexes, is considered. No concrete meaning of the structural elements is given for the time being in terms of dimensions and/or number of charges. By “minus” a certain portion of DNA or acidic protein is denoted bearing arbitrary, but constant, negative charge and by “plus” -a certain portion of basic protein having positive charge capable of neutralizing that of the “minus”. The combination of “plus” and “minus” gives a neutral structural element. Combination of such neutral elements with “plus” or “minus” gives positively- or negatively-charged elements again. Combinations among neutral elements are additive (Fig. 1). All structural elements are supposed to possess spherical symmetry and equal size.
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(y-J+@=@
o+o=;
FIG. 1. A scheme showing the possible interactions among the arbitrary elements consideredin the model.
(A)
CASE I
Let us consider a system containing proteins and DNA described as above. Suppose the only interactions taking place in this system are due to the electrostatic forces of repulsion among the likely charged and of attraction among the unlikely charged elements. It is obvious that the only reaction taking place in this system will be the combination of “minuses” and “pluses” and after a given period of time, provided finite length of DNA, all negative charges along the DNA chain will be neutralized. As a result, a uniform, randomly-coiled chain, built up of neutral elements (Structure I) will be spontaneously generated. (B)
CASE II
Let us consider a system described under Case I, but here, in addition to the electrostatic interactions, weak interactions of attraction among the neutral elements are also admitted. It is evident that in this system two processes will simultaneously take place-the process of neutralization of the negative charges along the growing DNA chain and the process of further organization of the formed neutral elements due to the weak interactions. Because of the uniformity of the nucleohistone thread, accepted so far, the velocity of this second process is supposed to be a constant. (1) Suppose this velocity is greater than the velocity of the first process. The problem to be solved here is the successive arrangement of mutually attracting elements.
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This process of second level organization was computer simulated (see Appendix) and the result was a uniform, tightly-packed, spherical (provided finite length of DNA) structure (Structure II), with individual structural elements arranged in a crystal-like fashion. (2) Suppose the velocity of formation of successive neutral elements is greater than the velocity of their organization. Here the problem is the spatial arrangement of a chain built up of mutually attracting elements. It can be shown that this case can formally be brought to the previous one, i.e. the case of successive arrangement, assuming the organization of the chain begins at certain, spatially separated points and that the velocity of germ formation is negligible compared with the velocity of self-organization. (C)
CASE
III
Let us consider the system described under Case II, but here, in addition, weak interactions of attraction between charged and neutral elements are also permitted. Here a possibility arises newly-formed neutral elements to complex with charged elements rather than to stick to the already existing neutral elements. The portions of DNA where this has happened will be considered as “defective” because they will not organize according to the rules determining the organization of the neutral elements and will, therefore, bring about defects in the uniformity of Structure II. As far as the velocity of organization of the successive neutral elements is constant, the number and the lengths of the defective regions will depend on the concentration and the lengths of the protein molecules accounting for the formation of defective regions. Several such structures were computer simulated (see Appendix), varying the percentage of the defective elements and the lengths of the defective regions. It was shown that on increasing the percentage of defective elements the compactness of the structure decreased and finally it was converted into Structure I, i.e. a random coil with very low packing ratio. However, the mode of this transition depends on the length of the defective regions. Thus an increase of the percentage of defective elements organized in sequences with length shorter than certain critical length, even to high values, do not lead to loss of the compactness of the structure, and on further increase a steady transition to random coil takes place (Fig. 2, curves 1,2). On the contrary, even very low percentages of defective elements, when organized in sequences longer than this critical value, lead to sharp decrease in the compactness of the structure, further increase of the percentage having small effect on it (Fig. 2, curves 3, 4, 5). In these cases the transition from the compact Structure II to Structure I processes through intermediate structures characterized by regular repetition of organized and nonorganized regions with dimensions depending on the particular parameters
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FIG. 2. Dependenceof packing ratio (fl) on percentageof defectiveelements(7. 100)for structures containing defective regions of 1, 2, 3, 4, and 5 successivedefective elements. In the figure idealized,mean curvesderived from severalcomputer simulations are shown.
of the model (Structure III, Fig. 3). This “mixed” structure preserves its main characteristic, i.e. alteration of the type of organization of the individual
elements on increasing the percentage of defective elements to certain value, afterwards being rapidly converted into a random coil. It can be shown that the condition for loss of the compactness of Structure II is the length of the defective regions 2r1, where I is the number of defective elements per region, and r is the radius of the individual structural elements, to become great enough that the distance between the formed condensed regions becomes greater than the admitted radius of action of the weak interactions, thus preventing these regions from being complexed together. For our particular model this requirement is given by the equation: 21.1-r 2 R-i-C
(1) where R is the mean radius of the condensed regions (Fig. 3), and C = 3r
(see Appendix) is the radius of action of the weak forces. The packing ratio j? of the condensed regions, expressed as a ratio of the volume occupied by structural elements to the total volume of the region, is given by the equation: (2)
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FIG. 3. A schematic representation type of structuring.
AND
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of Structure III, characterized by alternation
of the
where Nis the total number of elements, n is the number of defective elements, and (N-n)/(n/l) is the mean number of elements per condensed region. Substituting R of equation (2) into equation (l), and setting q = n/N, after a transformation equation (1) becomes : (3) Equation (3) gives for each particular q a value of 1 determining the transition of Structure II into Structure III [Fig. 4 (A)]. An examination of this equation indicates that on increasing q, 1 decreases approaching 2. On the other hand it can be shown that for each 1, on reaching certain q, formation of successive defective regions, without any condensed regions between them, takes place, making the existence of Structure III impossible. Assuming 2 as the minimal number of elements forming a condensed region, the condition for transition of Structure III into Structure I is given by the equation: N-n 2<---. n/l Setting again rj = n/N and transforming, I<-
we obtain : 2rl
1-q’
(5)
Equations (3) and (5) determine the area of stability of Structure III in terms both of percentage of defective elements and length of defective regions
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80
60
0
2
4
6
8
I
FIG. 4. A diagram showing both percentageof defective elements (7. 100) and length of the defectiveregions (0, determining the existenceof Structure III. The two bordering curves(A) and (B) were built up exploiting equations (3) and (5) respectively.
(Fig. 4). It is seen from Fig. 4 that for 1 greater than the critical value determined by equation (3), which for our particular case is 2, the transition of compact structure of type II to a random coil necessarily processes via mixed structures of type III. 3. Application to Chromatin The assumptions made in the model mirror the main properties of DNA, histones and non-histone proteins and although formal, the performed analysis can be used for description of the organization of the genetic material. (A)
RELATION
OF THE
SIMULATED
STRUCTURES
TO CHROMATIN
The highly packed structure II resembles the organization of the genetic material in its most condensed and inactive state, i.e. nucleoprotamine or nucleohistone in spermatozoa. There is some experimental evidence that there it forms tightly-packed, non&stone protein-lacking, crystal-like
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structures (Zimmermann, 1974; Marushige & Marushige, 1974). It is possible by simulating several successive DNA replications in presence or absence, respectively, of non-histone proteins to transform Structure II into Structure III and vice versa, but it will be shown elsewhere. The mixed structures analysed under Case III are in agreement with our present knowledge of chromatin organization in the interphase nuclei. To estimate the degree of adequacy of these structures to real chromatin structure we have to substitute a continuous flexible uniform cylinder with radius equal to that of the arbitrary structural elements for the discontinuous chain of spheres we have previously worked with. The radius of this cylinder should correspond to the radius of the nucleoprotein fibres, e.g. about 15 A (Bram & Ris, 1971; Haapala & Sauer, 1973). On the basis of this assumption the length of the DNA segments corresponding to our arbitrary elements will be about 30 A, i.e. about ten base pairs, which is in agreement with the data showing a ten base pairs DNA segment as a repeated structural element of the nucleosomes (Noll, 1974). The mean length of DNA covered by a single histone is somewhere about 45 A (see van Holde et al., 19743) and, therefore, in our model the histone population will be represented by a mixture of positively charged monomers and dimers. The length of DNA covered by a single non-histone protein molecule depends on both-its molecular weight and conformation. It hardly could be quantitatively estimated, but as far as the mean molecular weight of the non-histone proteins is about 50 000 (Garrard et al., 1974), values of about 100 A are quite plausible and it means that in our model the non-histone proteins should be represented as three tetramers of negatively-charged structural elements. Spontaneous assembly in a system containing DNA, histones and nonhistone proteins defined as above will take place and the mixed Structure III will necessarily be generated with non-histone proteins accounting for the stretched regions, the histones being preferentially involved in complexing with DNA. It should be noted that the same structure will spontaneously arise when a limited amount of histones is permitted in the system, free DNA portions accounting for the stretched regions in this case. This structure (Fig. 5) is composed of statistically regular repetition of packed spherical regions with mean diameter of 90 A, containing about 360A DNA and eight histone molecules each and stretched regions with mean length of about 30-4OA containing DNA, histones and non-histone proteins. It should be noted here that the length of the stretched regions is less than the length of the defective regions because in our model the formation of the condensed regions begins at the ends of the stretched regions and, therefore, some parts of the defective regions are buried in the condensed bodies (Fig. 3). We can avoid this, excluding the possibility of neutral elements to
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360
240
I 20
A FIG. 5. An x, y projection of a computer simulated arrangement of 120 structural elements, 32 of them (25 %) being defective and distributed in eight defective regions containing four successive elements each. The structure is outlined with a smooth curve and the dimensions in A are calculated assuming the diameter of the nucleoprotein thread being 30 A. The mean diameter of the condensed bodies is 90 A and the mean length of DNA portions packed in the condensed bodies is 380 A (120 base pairs).
occur in some volume around the stretched regions, thus simulating repulsion between nucleohistone segments and nucleohistone segments complexed with non-histone proteins, but we do not believe that there are enough grounds to do so. It is easy to show that the characteristic of the structure represented in Fig. 5, i.e. composition (DNA: histone: non-histone proteins = 1: 1:0*25), melting and sedimentation behaviour, X-ray diffraction, mode of enzyme degradation, etc., would be in good agreement with the experimental data for chromatin. Two points should be mentioned which made the structure described above different from the other chromatin models. They are the presence of non-histone proteins and the structure of the condensed bodies. (1) Our model, for the first time, takes into account the non-histone proteins and charges them with active structural role, namely to prevent some portions of chromatin from being condensed. (2) The structure of the condensed bodies, built up on the basis of the requirement for minimum free energy (Plate l), differs from the structures previously suggested. Model building experiments show that it is more probable than supercoiled or palindromelike organized structures with the histones in the core (van Holde ef al., 1974b; Baldwin et al., 1975; Hyde & Walker, 1975) because in these cases a considerable tension arises due to the necessity to unfold DNA double helix.
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FIG. 6. Flow chart of the computer program hydrophobic and mixed polymers.
K. DUDOV
I
for simulation
of the organization
of
(B) CHROMATIN FOURTH LEVEL OF ORGANIZATION
The three structures we have dealt with here are spontaneously generated by statistical assembly of elements on the basis of the electrostatic and weak interactions only, without any specific or even preferential recognition of the elements to take place. They illustrate the three levels of chromatin organization-randomly-coiled nucleoprotein fibre; nucleoprotein fibre, uniformly packed; and nucleoprotein fibre packed in such a way that different structural elements (condensed and stretched regions) are formed. Here it should be emphasized that all these three types of structures are by no means a unique privilege of chromatin and, as it was shown, under certain conditions are necessarily formed regardless of the chemical nature of the material.
PLATE 1. 36 cm long rod with an average diameter of 2 cm, made out of plastic material s folded to illustrate the organization of the nucleohistone thread in the condensed bodies n Fig. 5.
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The first two structures are completely determined, because, provided infinite length of DNA, the individual structural elements are homogeneously distributed. The third structure, however, is not completely determined because the elements are not structurally equal (they may occur in the stretched or in the condensed regions), and under the conditions accepted so far, their positions are not determined. Two identical computer simulations gave structures identical in respect to their statistical characteristics--composition, packing ratio, etc., but not identical in respect to the spatial environment of the elements: some of the elements found in the first case in the condensed regions were found the second time in the stretched regions and vice versa. Thus it turned out that Structure III possessed a fourth level of organization-the mode of distribution of the material between the condensed and stretched regions which can not be achieved on the basis of probability interactions only. The question whether this fourth level of organization is of any significance for the specificity of chromatin structure appears to be a very important one. If, for instance, it turns out that the belonging of a DNA portion to one of the two forms of organization (condensed and stretched regions) or to a given combination of them is important for its transcription, some other requirements have to be met in order for this fourth level to be explained and described. 4. Discussion
Recently several efforts have been made to deduce the chromatin structure from its composition, bearing more or less in mind the particular properties of chromatin components (Kornberg, 1974; van Holde et al., 1974b; Baldwin et al., 1975; Hyde & Walker, 1975). Some of the authors have taken a symmetry-consideration approach (Hyde & Walker, 1975) and others pattern recognition approach (Kornberg, 1974; van Holde et al., 19743). One feature all these models have in common is a description of the information transfer from composition to structure in terms of reality and specificity of the histone molecules. Such an approach, taking into account the recent data about the individuality of the histone molecules and the specificity of their interactions is, obviously, the most natural and exact one. Unfortunately, it can hardly be quantitated. Instead of using a finite number of individual histone molecules, we took advantage of some of their most general properties and also of the most general properties of the non&stone proteins and succeeded in developing the statistical model of chromatin organization. This model, at its present state, is obviously not suitable for fine description of chromatin structure. Instead, however, it can be used in studying the structural responses to changes of some parameters as protein T.B. 16
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concentrations and velocity rate constants. Furthermore, it enabled us to understand clearly the principal border to which the statistical considerations can describe the chromatin structure, and beyond which more subtle and specific interactions, have to be involved. For instance it seems quite possible to reconstruct chromatin identical to the native chromatin in respect to all properties having statistical background--electron microscopy appearance, composition, sedimentation behaviour, etc., and this assumption is in agreement with some experimental data (Paul & More, 1972; Chi-Born-Chae, 1975; Oudet, Gross-Bellard & Chambon, 1975). On the other hand it is rather doubtful whether the reconstitution of tissue specific chromatin, provided the mode of distribution of the condensed and stretched regions along DNA is important for the tissue specificity of chromatin, is even possible. Finally it should be noted that some of the main characteristics of chromatin structure-the dimensions of the repeated units, the length of DNA per such a unit, the state of DNA in the stretched regions (naked or complexed and if complexed, with what), etc., are still known with some uncertainty (compare Kornberg, 1974 and van Holde et al., 19743). For this reason the structure in Fig. 5 should be considered only as an illustration of the ability of the statistical approach to simulate the generations of systems possessing properties compatible to those of chromatin. Nevertheless the simulation and analysis of such structures might prove useful in understanding the chromatin organization.
REFERENCES K. (1975).
Nature, Land. 253,
S. & RIS, H. (1971). J. mokc. Biol. 56, 325. . (1975). Biochemistry 14,900. CLARK, R. J. & FELSENFELD, G. (1974). Biochemistry 13, 3622. GARRARD, W. T., PEARSON, W. R., WAKE, S. K. & BONNER, J. (1974).
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BALDWIN,
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Res. Comm. 58, 50. HAAPALA, 0. K. & SAUER, M.-O. (1973). Nature, New Biol. 244, 195. VAN HOLDE, K. E., SAHURABUDDHE, CH. G., SHAW, B. R., VAN BRUGGEN, ARNBERG, A. C. (1974a). Biochem. Biophys. Res. Comm. 60, 1365. VAN HOLDE, K. E., SAHASRABUDDHE, CH. G. & SHAW, B. R. (19746). Nucl. 1579. HYDE, J. E. & WALKER, I. 0. (1975). Nucl. Acid Res. 12, 405. KORNBERG, R. D. (1974). Science, N. Y. 184,868. KORNBERG, R. D. & THOMAS, J. 0. (1974). Science, N. Y. 184,865. MARUSHIGE, Y. & MARUSHIGE, K. (1974). Biochim. Biophys. Actu 340,498. MCCLURE, E. M. & HNILICA, L. S. (1972). Sub.-Cell. Biochem. 1, 311. NOLL, M. (1974). Nucl. Acid. Res. 11, 1573. OLINS, A. L. & Ohms, D. E. (1974). S&me, N. Y. 183, 330.
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D. K., HOZIER, J. C. & RILL, R. L. (1975). Proc. natn. Ad. Sci. U.S.A. 633. OUD~ZT, P., GROSS-BELLARD, M. & CHAMBON, P. (1975). Cell 4,281. PAUL, J. & MORE, J. R. (1972). Nature, New Biol. 239, 135. SAHASRABUDDHE, CH. G. & VAN HOLDE, K. E. (1974). J. Biol. Chem. 249, 152. STEIN, G. S., SPELSBERO, TH. C. & KLEINSMITH, L. J. (1974). Science, N. Y. 183,817. THOMAS, G. 0. & KORNBERG, R. D. (1975). Proc. natn. Acad. Sci. U.S.A. 2626. ZIMMERMANN, H.-P. (1974). Cytobiol. 9, 144. OOSTERHOF,
233 72,
APPENDIX
Computer Simulation of the Organization of Growing Hydrophobic and Mixed Polymers Let us define the process of growing a hydrophobic polymer chain as a stepwise elongation due to successive attachment of arbitrary, mutually attracting structural elements with spherical symmetry to one of its ends. Attention must be directed to the fact that the definition of these elements is based exclusively upon considerations of identity and do not refer to their chemical nature or size. No restrictions on the angles between successive elements are imposed. The elongation rate is determined by the concentration of the monomer only and under indefinite dilution tends to zero. Here it will be considered much lower than the rate of the chain self-organization based on the requirement to minimize the free energy of the system. Therefore, each successive element bound to the growing end of the chain will tend to orientate in such a way that to contact the greatest possible number of other elements : jiI Dij = min
(i = 1,2,. . .) N; i #j).
This tendency will be limited by two restrictions: (1) the reality of the chain and (2) its continuity. The first restriction required the distance D between the centers of any two elements to be equal to, or greater than 2r: D,j 2 2r (i = 1,2,. . ., N;j = 1,2,. . ., N; i #j) (7) Where r is the radius of the arbitrary structural elements. The second requirement is the distance between the centers of two successive elements to be equal to 2r: D i,i+l = 2r (i=1,2 ,..., N-l). 63) The equation (6) can be written in a slightly different way: (i=1,2 ,..., AJ) (9) jFniDij=min
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where Szi is the multitude of elements j in a region with a centre the ith element and radius C: Ri = &(j: Dij < C; i # j]. Several computer simulations have been performed, varying c and it was shown that for C > 3r the final structure was a tightly-folded chain with individual elements arranged in a crystal-like fashion, while shorter C did not utilize all possible contacts among the structural elements. For this reason and to save machine time, which is moderately expensive, C = 3r was finally adopted, moreover, this value being compatible with the radius of action of the weak forces. The foregoing considerations about the organization of hydrophobic polymers are valid for mixed polymers as well, with the only difference that the organization of the chain will be interrupted at the points where hydrophylic elements occur. Here as “mixed”, polymers containing both hydrophobic and hydrophylic elements are considered. The two types of elements can be randomly distributed, the distribution mirroring their concentration ratio, or otherwise. The hydrophylic elements are supposed to be mutually repulsing but no further quantitation of this repulsion is made. For the sake of simplicity, regions of successive hydrophobic elements, whenever formed, are thought as straight lines. On the basis of all these considerations a program for MINSK-22M (USSR) was written, which can be briefly outlined as follows (Fig. 6): Block 1. Introduces the input parameters N, n, I, r and c, where N = total number of elements per chain, II = number of hydrophylic elements, 1 = length of the hydrophylic regions, r = radius of the individual structural elements, c = distance within which the elements are considered as neighbours. Block 2. Determines the distribution of the hydrophobic and hydrophylic elements along the chain using a random number generator (subroutine S). Control whether the ith element is or is not hydrophobic. Block 3. Generates the organization of the regions consisting of successive hydrophobic elements : (1) determines all triplets of elements, the ith element being necessarily one of the triplet, for which the position of the (i+ 1)th element can be chosen in such a way that the distance between it and the centres of the elements of the triplet is 2r; (2) determine the co-ordinates of all (i+ 1) positions, remembering only those satisfying requirement (7) and having maximal number of neighbours belonging to Qi; (3) if more than one position with equal number of neighbours within and satisfying (7) are possible, a random number is generated to determine the position of the (,i+ I)th element. Block 4. Determines the co-ordinates of the hydrophylic elements by generating random numbers and choosing those satisfying requirements (7) and (8).
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The computer program produces for any given set of input parameters a table displaying: (1) the sequential number of the element (i); (2) the co-ordinates of the element (xi, yi, z,); the sequential numbers of the elements forming the triplet from which the position of the ith element has been derived; (3) the number of neighbours within c. The hydrophobic and hydrophylic e!ements in the table are separated by horizontal lines.
Chromatin Organization Studiedby Computer Simulation of Models of Growing Polymer Chains GEORGE RUSSEV AND KALIN
DUDOV
Institute of Biochemistry, Bulgarian Academy of Sciences, 13 Sofia, Bulgaria (Received 19 January 1976) An attempt was made to discriminate between the contribution of the unspecific electrostatic and weak interactions on one hand, and specific pattern recognitions, on the other, to chromatin structure. Several models imitating the organization of growing polymer chains were computer simulated and their degree of adequacy to chromatin was analysed. In this way four levels of organization of chromatin were distinguished. (i) The binding of histones to DNA, which is due to electrostatic interactions and gives rise to a nucleohistone thread; (ii) selforganization of the nucleohistone thread into condensed structure due to weak interactions; (iii) generation of mixed structures containing both condensed and stretched regions due to the simultaneous and competitive progression of two reactions-the self-organization of the nucleohistone thread and its complexing with non-histone proteins; (iv) discrimination between regions which should be condensed and regions which should remain stretched. It was shown that under certain conditions the first three levels of organization occur spontaneously, being accounted for by non-specific interactions only. For the fourth level of organization to be achieved, additional information is necessary.
1. Introduction Chromatin, the interphase form of the genetic material in eukaryotic cells, mainly consists of DNA, histones and non-histone acidic proteins. In spite of the fact that during recent years a massive body of experimental and theoretical work has been done, the biological role of the two main protein components of chromatin is still not fully understood. It is generally believed that histones serve as unspecific blockage of genetic material and that block can be presumably taken off by specific non-hi&one proteins (see McClure
& Hnilica, 1972; Stein, Spelsberg & Kleinsmith, 1974). Even less is known about the events taking place in the process of chromatin organization and the role of the two protein classes in the generation and preservation of this 221
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organization. Recently it has been shown that chromatin is composed of repeated structural units (Olins & Olins, 1974; Clark & Felsenfeld, 1974; Oosterhof, Hozier & Rill, 1975) and several excellent works have described their properties (Clark & Felsenfeld, 1974; Kornberg & Thomas, 1974; Sahasrabuddhe & van Holde, 1974; Nell, 1974). A few theoretical works have also been published in which the organization of these structural units was explained on the basis of specific histone-histone interactions (Kornberg, 1974; van Holde, Sahasrabuddhe & Shaw, 19743; Baldwin, Boseley & Bradbury, 1975; Hyde & Walker, 1975; Thomas & Komberg, 1975). Nevertheless, there is still a long way to be gone towards the complete understanding of chromatin structure and no simple and quantitative description of its organization is yet available. In the present paper, using a computer simulation of imitation models as a probe, we tried to establish the levels of organization and the structural hierarchy in chromatin; to analyse the nature of the forces responsible for the generation and maintenance of these levels; and to look for the material basis of the interactions in terms of molecules with necessary properties. 2. The Model We shall analyse systems containing a growing DNA chain, histones and non-histone proteins. DNA should be thought of as a flexible polymer built up by successive attachment of arbitrary, negatively-charged structural elements to the growing end of the chain. The histones and the non-histone proteins will be represented by oligomers of positively or negatively-charged arbitrary structural elements, respectively. The number of positive elements in this system is supposed to be equal to that of the negative elements composing the DNA chain. The number of negatively-charged elements representing acidic proteins is not defined and can be varied. Here the number of the free positive and negative protein structural elements, not engaged in mutual complexes, is considered. No concrete meaning of the structural elements is given for the time being in terms of dimensions and/or number of charges. By “minus” a certain portion of DNA or acidic protein is denoted bearing arbitrary, but constant, negative charge and by “plus” -a certain portion of basic protein having positive charge capable of neutralizing that of the “minus”. The combination of “plus” and “minus” gives a neutral structural element. Combination of such neutral elements with “plus” or “minus” gives positively- or negatively-charged elements again. Combinations among neutral elements are additive (Fig. 1). All structural elements are supposed to possess spherical symmetry and equal size.
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(y-J+@=@
o+o=;
FIG. 1. A scheme showing the possible interactions among the arbitrary elements consideredin the model.
(A)
CASE I
Let us consider a system containing proteins and DNA described as above. Suppose the only interactions taking place in this system are due to the electrostatic forces of repulsion among the likely charged and of attraction among the unlikely charged elements. It is obvious that the only reaction taking place in this system will be the combination of “minuses” and “pluses” and after a given period of time, provided finite length of DNA, all negative charges along the DNA chain will be neutralized. As a result, a uniform, randomly-coiled chain, built up of neutral elements (Structure I) will be spontaneously generated. (B)
CASE II
Let us consider a system described under Case I, but here, in addition to the electrostatic interactions, weak interactions of attraction among the neutral elements are also admitted. It is evident that in this system two processes will simultaneously take place-the process of neutralization of the negative charges along the growing DNA chain and the process of further organization of the formed neutral elements due to the weak interactions. Because of the uniformity of the nucleohistone thread, accepted so far, the velocity of this second process is supposed to be a constant. (1) Suppose this velocity is greater than the velocity of the first process. The problem to be solved here is the successive arrangement of mutually attracting elements.
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This process of second level organization was computer simulated (see Appendix) and the result was a uniform, tightly-packed, spherical (provided finite length of DNA) structure (Structure II), with individual structural elements arranged in a crystal-like fashion. (2) Suppose the velocity of formation of successive neutral elements is greater than the velocity of their organization. Here the problem is the spatial arrangement of a chain built up of mutually attracting elements. It can be shown that this case can formally be brought to the previous one, i.e. the case of successive arrangement, assuming the organization of the chain begins at certain, spatially separated points and that the velocity of germ formation is negligible compared with the velocity of self-organization. (C)
CASE
III
Let us consider the system described under Case II, but here, in addition, weak interactions of attraction between charged and neutral elements are also permitted. Here a possibility arises newly-formed neutral elements to complex with charged elements rather than to stick to the already existing neutral elements. The portions of DNA where this has happened will be considered as “defective” because they will not organize according to the rules determining the organization of the neutral elements and will, therefore, bring about defects in the uniformity of Structure II. As far as the velocity of organization of the successive neutral elements is constant, the number and the lengths of the defective regions will depend on the concentration and the lengths of the protein molecules accounting for the formation of defective regions. Several such structures were computer simulated (see Appendix), varying the percentage of the defective elements and the lengths of the defective regions. It was shown that on increasing the percentage of defective elements the compactness of the structure decreased and finally it was converted into Structure I, i.e. a random coil with very low packing ratio. However, the mode of this transition depends on the length of the defective regions. Thus an increase of the percentage of defective elements organized in sequences with length shorter than certain critical length, even to high values, do not lead to loss of the compactness of the structure, and on further increase a steady transition to random coil takes place (Fig. 2, curves 1,2). On the contrary, even very low percentages of defective elements, when organized in sequences longer than this critical value, lead to sharp decrease in the compactness of the structure, further increase of the percentage having small effect on it (Fig. 2, curves 3, 4, 5). In these cases the transition from the compact Structure II to Structure I processes through intermediate structures characterized by regular repetition of organized and nonorganized regions with dimensions depending on the particular parameters
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FIG. 2. Dependenceof packing ratio (fl) on percentageof defectiveelements(7. 100)for structures containing defective regions of 1, 2, 3, 4, and 5 successivedefective elements. In the figure idealized,mean curvesderived from severalcomputer simulations are shown.
of the model (Structure III, Fig. 3). This “mixed” structure preserves its main characteristic, i.e. alteration of the type of organization of the individual
elements on increasing the percentage of defective elements to certain value, afterwards being rapidly converted into a random coil. It can be shown that the condition for loss of the compactness of Structure II is the length of the defective regions 2r1, where I is the number of defective elements per region, and r is the radius of the individual structural elements, to become great enough that the distance between the formed condensed regions becomes greater than the admitted radius of action of the weak interactions, thus preventing these regions from being complexed together. For our particular model this requirement is given by the equation: 21.1-r 2 R-i-C
(1) where R is the mean radius of the condensed regions (Fig. 3), and C = 3r
(see Appendix) is the radius of action of the weak forces. The packing ratio j? of the condensed regions, expressed as a ratio of the volume occupied by structural elements to the total volume of the region, is given by the equation: (2)
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FIG. 3. A schematic representation type of structuring.
AND
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of Structure III, characterized by alternation
of the
where Nis the total number of elements, n is the number of defective elements, and (N-n)/(n/l) is the mean number of elements per condensed region. Substituting R of equation (2) into equation (l), and setting q = n/N, after a transformation equation (1) becomes : (3) Equation (3) gives for each particular q a value of 1 determining the transition of Structure II into Structure III [Fig. 4 (A)]. An examination of this equation indicates that on increasing q, 1 decreases approaching 2. On the other hand it can be shown that for each 1, on reaching certain q, formation of successive defective regions, without any condensed regions between them, takes place, making the existence of Structure III impossible. Assuming 2 as the minimal number of elements forming a condensed region, the condition for transition of Structure III into Structure I is given by the equation: N-n 2<---. n/l Setting again rj = n/N and transforming, I<-
we obtain : 2rl
1-q’
(5)
Equations (3) and (5) determine the area of stability of Structure III in terms both of percentage of defective elements and length of defective regions
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80
60
0
2
4
6
8
I
FIG. 4. A diagram showing both percentageof defective elements (7. 100) and length of the defectiveregions (0, determining the existenceof Structure III. The two bordering curves(A) and (B) were built up exploiting equations (3) and (5) respectively.
(Fig. 4). It is seen from Fig. 4 that for 1 greater than the critical value determined by equation (3), which for our particular case is 2, the transition of compact structure of type II to a random coil necessarily processes via mixed structures of type III. 3. Application to Chromatin The assumptions made in the model mirror the main properties of DNA, histones and non-histone proteins and although formal, the performed analysis can be used for description of the organization of the genetic material. (A)
RELATION
OF THE
SIMULATED
STRUCTURES
TO CHROMATIN
The highly packed structure II resembles the organization of the genetic material in its most condensed and inactive state, i.e. nucleoprotamine or nucleohistone in spermatozoa. There is some experimental evidence that there it forms tightly-packed, non&stone protein-lacking, crystal-like
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structures (Zimmermann, 1974; Marushige & Marushige, 1974). It is possible by simulating several successive DNA replications in presence or absence, respectively, of non-histone proteins to transform Structure II into Structure III and vice versa, but it will be shown elsewhere. The mixed structures analysed under Case III are in agreement with our present knowledge of chromatin organization in the interphase nuclei. To estimate the degree of adequacy of these structures to real chromatin structure we have to substitute a continuous flexible uniform cylinder with radius equal to that of the arbitrary structural elements for the discontinuous chain of spheres we have previously worked with. The radius of this cylinder should correspond to the radius of the nucleoprotein fibres, e.g. about 15 A (Bram & Ris, 1971; Haapala & Sauer, 1973). On the basis of this assumption the length of the DNA segments corresponding to our arbitrary elements will be about 30 A, i.e. about ten base pairs, which is in agreement with the data showing a ten base pairs DNA segment as a repeated structural element of the nucleosomes (Noll, 1974). The mean length of DNA covered by a single histone is somewhere about 45 A (see van Holde et al., 19743) and, therefore, in our model the histone population will be represented by a mixture of positively charged monomers and dimers. The length of DNA covered by a single non-histone protein molecule depends on both-its molecular weight and conformation. It hardly could be quantitatively estimated, but as far as the mean molecular weight of the non-histone proteins is about 50 000 (Garrard et al., 1974), values of about 100 A are quite plausible and it means that in our model the non-histone proteins should be represented as three tetramers of negatively-charged structural elements. Spontaneous assembly in a system containing DNA, histones and nonhistone proteins defined as above will take place and the mixed Structure III will necessarily be generated with non-histone proteins accounting for the stretched regions, the histones being preferentially involved in complexing with DNA. It should be noted that the same structure will spontaneously arise when a limited amount of histones is permitted in the system, free DNA portions accounting for the stretched regions in this case. This structure (Fig. 5) is composed of statistically regular repetition of packed spherical regions with mean diameter of 90 A, containing about 360A DNA and eight histone molecules each and stretched regions with mean length of about 30-4OA containing DNA, histones and non-histone proteins. It should be noted here that the length of the stretched regions is less than the length of the defective regions because in our model the formation of the condensed regions begins at the ends of the stretched regions and, therefore, some parts of the defective regions are buried in the condensed bodies (Fig. 3). We can avoid this, excluding the possibility of neutral elements to
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360
240
I 20
A FIG. 5. An x, y projection of a computer simulated arrangement of 120 structural elements, 32 of them (25 %) being defective and distributed in eight defective regions containing four successive elements each. The structure is outlined with a smooth curve and the dimensions in A are calculated assuming the diameter of the nucleoprotein thread being 30 A. The mean diameter of the condensed bodies is 90 A and the mean length of DNA portions packed in the condensed bodies is 380 A (120 base pairs).
occur in some volume around the stretched regions, thus simulating repulsion between nucleohistone segments and nucleohistone segments complexed with non-histone proteins, but we do not believe that there are enough grounds to do so. It is easy to show that the characteristic of the structure represented in Fig. 5, i.e. composition (DNA: histone: non-histone proteins = 1: 1:0*25), melting and sedimentation behaviour, X-ray diffraction, mode of enzyme degradation, etc., would be in good agreement with the experimental data for chromatin. Two points should be mentioned which made the structure described above different from the other chromatin models. They are the presence of non-histone proteins and the structure of the condensed bodies. (1) Our model, for the first time, takes into account the non-histone proteins and charges them with active structural role, namely to prevent some portions of chromatin from being condensed. (2) The structure of the condensed bodies, built up on the basis of the requirement for minimum free energy (Plate l), differs from the structures previously suggested. Model building experiments show that it is more probable than supercoiled or palindromelike organized structures with the histones in the core (van Holde ef al., 1974b; Baldwin et al., 1975; Hyde & Walker, 1975) because in these cases a considerable tension arises due to the necessity to unfold DNA double helix.
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FIG. 6. Flow chart of the computer program hydrophobic and mixed polymers.
K. DUDOV
I
for simulation
of the organization
of
(B) CHROMATIN FOURTH LEVEL OF ORGANIZATION
The three structures we have dealt with here are spontaneously generated by statistical assembly of elements on the basis of the electrostatic and weak interactions only, without any specific or even preferential recognition of the elements to take place. They illustrate the three levels of chromatin organization-randomly-coiled nucleoprotein fibre; nucleoprotein fibre, uniformly packed; and nucleoprotein fibre packed in such a way that different structural elements (condensed and stretched regions) are formed. Here it should be emphasized that all these three types of structures are by no means a unique privilege of chromatin and, as it was shown, under certain conditions are necessarily formed regardless of the chemical nature of the material.
PLATE 1. 36 cm long rod with an average diameter of 2 cm, made out of plastic material s folded to illustrate the organization of the nucleohistone thread in the condensed bodies n Fig. 5.
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The first two structures are completely determined, because, provided infinite length of DNA, the individual structural elements are homogeneously distributed. The third structure, however, is not completely determined because the elements are not structurally equal (they may occur in the stretched or in the condensed regions), and under the conditions accepted so far, their positions are not determined. Two identical computer simulations gave structures identical in respect to their statistical characteristics--composition, packing ratio, etc., but not identical in respect to the spatial environment of the elements: some of the elements found in the first case in the condensed regions were found the second time in the stretched regions and vice versa. Thus it turned out that Structure III possessed a fourth level of organization-the mode of distribution of the material between the condensed and stretched regions which can not be achieved on the basis of probability interactions only. The question whether this fourth level of organization is of any significance for the specificity of chromatin structure appears to be a very important one. If, for instance, it turns out that the belonging of a DNA portion to one of the two forms of organization (condensed and stretched regions) or to a given combination of them is important for its transcription, some other requirements have to be met in order for this fourth level to be explained and described. 4. Discussion
Recently several efforts have been made to deduce the chromatin structure from its composition, bearing more or less in mind the particular properties of chromatin components (Kornberg, 1974; van Holde et al., 1974b; Baldwin et al., 1975; Hyde & Walker, 1975). Some of the authors have taken a symmetry-consideration approach (Hyde & Walker, 1975) and others pattern recognition approach (Kornberg, 1974; van Holde et al., 19743). One feature all these models have in common is a description of the information transfer from composition to structure in terms of reality and specificity of the histone molecules. Such an approach, taking into account the recent data about the individuality of the histone molecules and the specificity of their interactions is, obviously, the most natural and exact one. Unfortunately, it can hardly be quantitated. Instead of using a finite number of individual histone molecules, we took advantage of some of their most general properties and also of the most general properties of the non&stone proteins and succeeded in developing the statistical model of chromatin organization. This model, at its present state, is obviously not suitable for fine description of chromatin structure. Instead, however, it can be used in studying the structural responses to changes of some parameters as protein T.B. 16
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concentrations and velocity rate constants. Furthermore, it enabled us to understand clearly the principal border to which the statistical considerations can describe the chromatin structure, and beyond which more subtle and specific interactions, have to be involved. For instance it seems quite possible to reconstruct chromatin identical to the native chromatin in respect to all properties having statistical background--electron microscopy appearance, composition, sedimentation behaviour, etc., and this assumption is in agreement with some experimental data (Paul & More, 1972; Chi-Born-Chae, 1975; Oudet, Gross-Bellard & Chambon, 1975). On the other hand it is rather doubtful whether the reconstitution of tissue specific chromatin, provided the mode of distribution of the condensed and stretched regions along DNA is important for the tissue specificity of chromatin, is even possible. Finally it should be noted that some of the main characteristics of chromatin structure-the dimensions of the repeated units, the length of DNA per such a unit, the state of DNA in the stretched regions (naked or complexed and if complexed, with what), etc., are still known with some uncertainty (compare Kornberg, 1974 and van Holde et al., 19743). For this reason the structure in Fig. 5 should be considered only as an illustration of the ability of the statistical approach to simulate the generations of systems possessing properties compatible to those of chromatin. Nevertheless the simulation and analysis of such structures might prove useful in understanding the chromatin organization.
REFERENCES K. (1975).
Nature, Land. 253,
S. & RIS, H. (1971). J. mokc. Biol. 56, 325. . (1975). Biochemistry 14,900. CLARK, R. J. & FELSENFELD, G. (1974). Biochemistry 13, 3622. GARRARD, W. T., PEARSON, W. R., WAKE, S. K. & BONNER, J. (1974).
Biochem. Biophys.
BALDWIN,
J. P., B~SELEY,
P. O., BRADBURY,
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245. BRAM,
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Res. Comm. 58, 50. HAAPALA, 0. K. & SAUER, M.-O. (1973). Nature, New Biol. 244, 195. VAN HOLDE, K. E., SAHURABUDDHE, CH. G., SHAW, B. R., VAN BRUGGEN, ARNBERG, A. C. (1974a). Biochem. Biophys. Res. Comm. 60, 1365. VAN HOLDE, K. E., SAHASRABUDDHE, CH. G. & SHAW, B. R. (19746). Nucl. 1579. HYDE, J. E. & WALKER, I. 0. (1975). Nucl. Acid Res. 12, 405. KORNBERG, R. D. (1974). Science, N. Y. 184,868. KORNBERG, R. D. & THOMAS, J. 0. (1974). Science, N. Y. 184,865. MARUSHIGE, Y. & MARUSHIGE, K. (1974). Biochim. Biophys. Actu 340,498. MCCLURE, E. M. & HNILICA, L. S. (1972). Sub.-Cell. Biochem. 1, 311. NOLL, M. (1974). Nucl. Acid. Res. 11, 1573. OLINS, A. L. & Ohms, D. E. (1974). S&me, N. Y. 183, 330.
E. F. J. &
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D. K., HOZIER, J. C. & RILL, R. L. (1975). Proc. natn. Ad. Sci. U.S.A. 633. OUD~ZT, P., GROSS-BELLARD, M. & CHAMBON, P. (1975). Cell 4,281. PAUL, J. & MORE, J. R. (1972). Nature, New Biol. 239, 135. SAHASRABUDDHE, CH. G. & VAN HOLDE, K. E. (1974). J. Biol. Chem. 249, 152. STEIN, G. S., SPELSBERO, TH. C. & KLEINSMITH, L. J. (1974). Science, N. Y. 183,817. THOMAS, G. 0. & KORNBERG, R. D. (1975). Proc. natn. Acad. Sci. U.S.A. 2626. ZIMMERMANN, H.-P. (1974). Cytobiol. 9, 144. OOSTERHOF,
233 72,
APPENDIX
Computer Simulation of the Organization of Growing Hydrophobic and Mixed Polymers Let us define the process of growing a hydrophobic polymer chain as a stepwise elongation due to successive attachment of arbitrary, mutually attracting structural elements with spherical symmetry to one of its ends. Attention must be directed to the fact that the definition of these elements is based exclusively upon considerations of identity and do not refer to their chemical nature or size. No restrictions on the angles between successive elements are imposed. The elongation rate is determined by the concentration of the monomer only and under indefinite dilution tends to zero. Here it will be considered much lower than the rate of the chain self-organization based on the requirement to minimize the free energy of the system. Therefore, each successive element bound to the growing end of the chain will tend to orientate in such a way that to contact the greatest possible number of other elements : jiI Dij = min
(i = 1,2,. . .) N; i #j).
This tendency will be limited by two restrictions: (1) the reality of the chain and (2) its continuity. The first restriction required the distance D between the centers of any two elements to be equal to, or greater than 2r: D,j 2 2r (i = 1,2,. . ., N;j = 1,2,. . ., N; i #j) (7) Where r is the radius of the arbitrary structural elements. The second requirement is the distance between the centers of two successive elements to be equal to 2r: D i,i+l = 2r (i=1,2 ,..., N-l). 63) The equation (6) can be written in a slightly different way: (i=1,2 ,..., AJ) (9) jFniDij=min
234
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where Szi is the multitude of elements j in a region with a centre the ith element and radius C: Ri = &(j: Dij < C; i # j]. Several computer simulations have been performed, varying c and it was shown that for C > 3r the final structure was a tightly-folded chain with individual elements arranged in a crystal-like fashion, while shorter C did not utilize all possible contacts among the structural elements. For this reason and to save machine time, which is moderately expensive, C = 3r was finally adopted, moreover, this value being compatible with the radius of action of the weak forces. The foregoing considerations about the organization of hydrophobic polymers are valid for mixed polymers as well, with the only difference that the organization of the chain will be interrupted at the points where hydrophylic elements occur. Here as “mixed”, polymers containing both hydrophobic and hydrophylic elements are considered. The two types of elements can be randomly distributed, the distribution mirroring their concentration ratio, or otherwise. The hydrophylic elements are supposed to be mutually repulsing but no further quantitation of this repulsion is made. For the sake of simplicity, regions of successive hydrophobic elements, whenever formed, are thought as straight lines. On the basis of all these considerations a program for MINSK-22M (USSR) was written, which can be briefly outlined as follows (Fig. 6): Block 1. Introduces the input parameters N, n, I, r and c, where N = total number of elements per chain, II = number of hydrophylic elements, 1 = length of the hydrophylic regions, r = radius of the individual structural elements, c = distance within which the elements are considered as neighbours. Block 2. Determines the distribution of the hydrophobic and hydrophylic elements along the chain using a random number generator (subroutine S). Control whether the ith element is or is not hydrophobic. Block 3. Generates the organization of the regions consisting of successive hydrophobic elements : (1) determines all triplets of elements, the ith element being necessarily one of the triplet, for which the position of the (i+ 1)th element can be chosen in such a way that the distance between it and the centres of the elements of the triplet is 2r; (2) determine the co-ordinates of all (i+ 1) positions, remembering only those satisfying requirement (7) and having maximal number of neighbours belonging to Qi; (3) if more than one position with equal number of neighbours within and satisfying (7) are possible, a random number is generated to determine the position of the (,i+ I)th element. Block 4. Determines the co-ordinates of the hydrophylic elements by generating random numbers and choosing those satisfying requirements (7) and (8).
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ORGANIZATION
235
The computer program produces for any given set of input parameters a table displaying: (1) the sequential number of the element (i); (2) the co-ordinates of the element (xi, yi, z,); the sequential numbers of the elements forming the triplet from which the position of the ith element has been derived; (3) the number of neighbours within c. The hydrophobic and hydrophylic e!ements in the table are separated by horizontal lines.