─юъєьхэЄ тч Є шч ъ¤°р яюшёъютющ ьр°шэ√. └фЁхё юЁшушэры№эюую фюъєьхэЄр : http://www.mce.biophys.msu.ru/archive/doc15823/doc.pdf
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.. () , . , , , . , . . ON THE PARAMETERS OF THE DYNAMICAL MODELS OF DNA Yakushevich L.V. (Pushchino) Mathematical models of the DNA dynamics contain parameters characterizing the internal motions and interactions. The most important of them are mass, the moments of inertia, distances between structural elements, coefficients of the rigidity. The choice of the parameters and the relations between them are very important because they determine the type of the solutions of corresponding dynamical equations. In this article we consider the problem of the choice of the parameters and obtain an optimal set of them. Introduction Dynamics of DNA is widely considered as an important factor that should be taken into account when studying the biological functioning of the molecule. Many different models of the internal DNA dynamics are known [1], and they all contain parameters characterizing the internal motions and interactions. The most important of
196

.. -- -10, 2002, .196-197

them are mass of moving structural elements, distances between them, the moments of inertia, coefficients of the rigidity. The values of the parameters and the relations between them are very important because they determine the type of the solutions of corresponding dynamical equations. In this article we consider the problem of the choice and estimation of the parameters and obtain an optimal set of them. To be concrete, we consider the parameters of the mathematical model proposed in [2] to describe rotational motions of bases around sugar-phosphate chains. The dynamical model Let us consider the homogeneous double chain consisting of only AT base pairs 1-st chain: 2-nd chain: ...AAAAAAAAAAAAAAAAAAAAAAA... ...TTTTTTTTTTTTTTTTTTTTTTT...

and write the model hamiltonian [2] H = T + V|| + V; (1) with the kinetic energy (T), the energy of interactions along the chains (V||) and the energy of interactions between bases in pairs (V) being determined by formulas T = n{(m1r12/2) (dn,1/dt)2 + (m2r22/2) (dn,2/dt)2}; V|| = n{(K1r12) [1-cos(n,1 нn-1,1)] + (K2r22) (2) [1- cos(n,2 н n-1,2)]};(3)

V = n(k1-2) {r1(r1 + r2)(1 н cosn,1) + r2 (r1 + r2)(1 н cosn,2) н (4) r1r2 [1 н cos(n,1 н n,2)]} were n,i is the angular displacement of the n-th base of the i-th chain from its equilibrium position; ri, is the distance between the center of mass of the i-th base and the nearest sugar-phosphate chain; a is the distance between neighboring bases along the chains; mi is mass of bases of the i-th chain; Ki is the coupling constant along the sugarphosphate chain; k1-2 is the force constant that characterizes interactions between bases in pairs; n = 1, 2, ... N; i = 1, 2. We suggest that N is rather large integer, so the end effects can be neglected. Dynamical equations corresponding to hamiltonian (1), can be obtained from the equations of Hamilton
197

2. ,

dn,i/dt = H/pn,i; dpn,i/dt = н H/n,i; where impulses, pn,i are determined by formula pn,i= L/n,; and Lagrangian, L, by formula L = T н V|| н V; Inserting (1) into (5) н (6) we obtain the dynamical equations m1 r12 (d2n,1/dt2) = = K1 r12[sin(n-1,1 -n,1) н sin(n,1-n+1,1)] н k r2r1sin(n,1-n,2)];
1-2

(5) (6) (7) (8)

[r1 (r1+r2)sinn,1 н (9)

m2 r22 (d2n,2/dt2) = = K2 r22[sin(n-1,2 -n,2) н sin(n,2-n+1,2)] н k1-2 [r2(r1+r2)sinn,2 н (10) r2r1sin(n,2-n,1)]; which describe rotational motions of bases around sugar-phosphate chains. Parameters Mathematical model (9) н (10) contains the following parameters: * masses of bases (m1, m2); * distances between centers of mass of the bases and the nearest sugar-phosphate chain (r1, r2); * force constant that characterizes interaction between bases in pairs (k1-2); * coupling constant along the sugar-phosphate chains (K1, K2). Let us estimate the parameters listed above and other physical values related with them. masses of bases (m1, m2) Masses of bases (mi, i = A,T,G,C) can be taken from any reference book on chemistry, where they are usually presented in terms of mass of protons (mp). mA=135,13 mp; mT=126,11 mp; mG=151,14 mp; mC=111.10 mp; (11) Inserting mass of proton
198

.. -- -10, 2002, .196-199

mp = 1,67343 т 10-27 kg; into (11) we obtain the following values for bases in AT chain m1 = mA = 226,13 т 10-27 kg; m2 = mT = 211,04 т 10-27 kg; And for bases in GC chain we have m1 = mG = 252,92 т 10-27 kg; m2 = mC = 185,92 т 10
-27

(12) (13) (14)

kg.

b) distances between centers of mass of the bases and the nearest sugar-phosphate chain (r1, r2); moments of inertia (I1, I2) To estimate the distances, we used the data on the geometry of BDNA taken from the Data Bank of National Center for Biotechnology (http://ncbi.nlm.nih.gov/Entrez) and adopted them to simplified model (1). As a result, for AT chain we obtained the following values r1 = rA = 5,8 х; r2 = rT = 4,8 х. (15) And for GC chain we obtained r1 = rG = 5,7 х; r2 = rC = 4,7 х. (16) Then the moments of inertia for bases in AT chain are I1=IA=mArA2=7607,03 m2 kg; I2=IT=mTrT2=4862,28 m2 kg. (17) And for bases in GC chain they are I1=IG=mGrG2=8217,44 m2 kg; I1=IC=mCrC2=4106,93 m2 kg. (18) the force constant (k1-2) that characterizes interaction between bases in pairs To estimate the value k1-2, let us take into account that the interactions between the bases in pairs are formed by two hydrogen bonds in the case of AT chain and by three hydrogen bonds in the case of GC chain. The energy required to broke one hydrogen bond is equal to H = 3┬7 kcal/mol [3]. For calculations let us take the value equal to H = 5 kcal/mol (20,934 kJ/mol). So, the average value of energy required to break and open one AT base pair is equal to AT = 2H = 10 kcal/mol 41,868 kJ/mol and to open one GC base pair we need the energy equal to GC = 3H = 15 kcal/mol 62,802 kJ/mol.
199

(19) (20)

2. ,

Let us take one base pair and suggest that the turn of A and T bases equal to /2 is equivalent to the breaking of hydrogen bonds between the bases. Then from formula (4) we obtain AT = (kAT) {rA(rA + rT)(1 н cos/2) + rT (rA + rT)(1 н cos/2) н rArT [1 н cos(/2 н /2)]} = = (rA + rT)2kAT. (21) And for GC base pair we have GC = (rG + rC)2kGC. (22) Inserting (19) and (20) into (21) and (22) we find the value k1-2 for AT and GC chains. Namely for AT chain we obtain kA-T = AT/(rA + rT)2 0,062 N/m; and for GC chain we have k
G-C

(23) (24)

= GC/(rG + rC)2 0,096 N/m.

d)low frequency spectrum To find the values of the low frequencies of the DNA spectrum, let us consider linear approximation of the model equations (9) н (10) m1 r12 (d2n,1/dt2) = K1 r12(n н r2r1(n,1-n,2)];
-1,1

-2

n,1

+

n+1,1)

нk

1-2

[r1 (r1+r2)n,1 (25)

m2 r22 (d2n,2/dt2) = K2 r22(n-1,2 -2n,2 + n+1,2) н k1-2 [r2(r1+r2)n,2 (26) н r2r1(n,2-n,1)]. Let us suggest that the solutions of the linear equations have the form of plane waves
n,1

=

0,1

exp[i(qz-wt)];

n,2

=

0,2

exp[i(qz-wt)];

(27)

where 0,1 and 0,2 are the amplitudes, w is the frequency and q is the wave vector. Then insert (21) into (19) н (20). As a result we obtain algebraic equations {-w2m1 r12 + 2K1 r12[1 н cos(qa)] + k1-2r1(r1+r2) н k1-2r2r1}0,1+ (28) {k1-2r2r1}0,2 = 0; {-w2m2 r22 + 2K2 r22[1 н cos(qa)] + k1-2r2(r1+r2) н k1-2r2r1}0,2+ (29) {k1-2 r2r1}0,1 = 0;
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.. -- -10, 2002, .196-201

which have nontrivial solution if {-m1 r12w2+1(q) r12} {-m2 r22 w2+2(q) r22}н{k1-2r2r1}2=0. (30) where 1(q) = 2K1[1- cos(qa)] + k1-2; 2(q) = 2K2[1- cos(qa)] + k1-2. Equation (30) determines the frequency of the plane waves w. The equation can be rewritten in a simple form a0x2 -bx + c = 0; (31) and its solution is determined by formula x1,2 = [b ▒ (b2 н 4a0c)1/2]/2a0. (32) Here we used the following notations x = w2; a0 = m1 m2; b = [m1 2(q) + m2 1(q)] = b0 + b1[1- cos(qa)]; b0 = k1-2 (m1 + m2); b1 = 2 (K1m2 + K2 m1); c = [1(q)2(q) н (k1-2)2] = c1 [1- cos(qa)] + c2 [1- cos(qa)]2; c1 = 2k1-2(K1 + K2); c2 = 4K1K2. If the wave vector q is small x1,2{[b0+(b1/2) (qa)2]▒[b0+(1/2b0)(b0b1н2 a0c1) (qa)2]}/2a0. (33) For sign "-" we have x1 (qa)2(c1/2b0); and for sign "+" we have x2 {2b0 + [b1 -(a0/b0)c1)] (qa)2}/2a0. Then for frequencies w1 and w2 we obtain w1(small q)=(qa)(c1/2b0)1/2 = (qa)[(K1 + K2)/(m1 + m2)]1/2;
1/2 2 2

(34) (35) (36)

w2(small q)(b0/a0) {1+(1/2) [(b1/2b0)н(a0/2b0 )c1)] (qa) ]}. (37) Formula (36) describes the acoustic branch in the spectrum of DNA molecule, and formula (37) н the optical branch. If q=0, we obtain that acoustic frequency is equal to zero w1AT(q=0) = w1GC(q=0) = 0. (38) Formula (37) gives a possibility to estimate the value of the low frequency in the spectrum of DNA. Indeed, for AT chain we obtain w2AT(q=0) = [kAT (mA + mT)/mAmT] and for GC chain we have w2GC(q=0) = [k
GC 1/2

0,75 т 10

+12 -1

s; s.

(39) (40)

(mG + mC)/mGmC]
201

1/2

0,94 т 10

+12 -1

2. ,

e) coupling constant along the first and second sugarphosphate chains (K1, K2) Let us estimate now the sum (K1 + K2). For the purpose, let us take into account that the sound velocity in the double polynucleotide chain is determined by v0=dw1(small q)/dq=(a)(c1/2b0)1/2=(a)[(K1+K2)/(m1+m2)]1/2; (41) Then from (41) we obtain (K1 + K2) = (v02/a2) (m1 + m2). (42) If we take v0 = 1890 m/s [4], then for coupling constants of AT chain we obtain (KA + KT) = (v02/a2) (mA + mT) 13,50 N/m; and for that of GC chain we have a very close value (KG + KC) = (v02/a2) (mG + mC) 13,56 N/m. f) the torsion energy (per base pair)(12) The torsion energy per base pair can be obtained from (3) 12 = (1/2){(K1r12) [2- cos(n,1 н n-1,1) н cos(n+1,1 н n,1)] + + (K2r22) [2- cos(n,2 н n-1,2) н cos(n+1,2 н n,2)]}; (45) (43) (44)

and from suggestion that angular displacements of bases equal to /2 are equivalent to the breaking of hydrogen bonds between the bases in pairs. This gives us to obtain the following formula for 12 12= (1/2){(K1r12) [2- cos(/2 н 0) н cos(0- /2)] + +(K2r22) [2- cos(/2 н 0) н cos(0 н /2)]} = {K1r12 + K2r22}. (46) Then for AT chain we obtain AT= {KArA2 + KTrT2} = 550,30 kcal/mol 2304,01 kJ/mol; (47) and for GC chain we have
GC=

{KGr

2 G

+ KCrC2} = 532,27 kcal/mol 2228,51 kJ/mol. (48)

g) the values of the frequencies when q = ▒/a It is interesting to estimate the values of the frequencies when q = ▒/a. For the purpose we can use formula (49) w1,2(q=▒/a) = {[b ▒ (b2 н 4a0c)1/2]/2a0}1/2, 2 where x = w ; a0 = m1 m2; b = k1-2 (m1 + m2) + 4 (K1m2 + K2 m1);
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.. -- -10, 2002, .196-203

1(q=▒/a) = 4K2 + k1-2; c = 16 K1K2 + 4 k1-2(K1 + K2). For sign "-" we have w1,AT (q=▒/a) = 10,93910988739 т 10
+12 -1

s; s. s; s.

(50) (51) (52) (53)

w1,GC (q=▒/a) = 1,037316314872 т 10 For sign "+" we obtain w2,AT (q=▒/a) = 11,32441846835 т 10 w2,GC (q=▒/a) = 1,209934083308 т 10

+12 -1

+12 -1

+12 -1

Conclusions In this article we considered the parameters of the mathematical model imitating rotational motions of bases around sugar-phosphate chains. We obtained an optimal set of the parameters and presented it in the table.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 mA mT mG mC rA rT rG rC IA IT IG IC H AT = 2H GC = 3H kA-T = AT/(rA + rT)2 kG-C = GC/(rG + rC)2 w1AT(q=0) (acoustic branch) w1GC(q=0) (acoustic branch) w2AT(q=0) = [kAT (mA + mT)/mAmT]1/2 (optic branch) w2GC(q=0) = [kGC (mG + mC)/mGmC]1/2 (optic branch) w1,AT (q=▒/a) w2,AT (q=▒/a) w1,GC (q=▒/a) 203 226,13 т 10 211,04 т 10 252,92 т 10 185,92 т 10 5,8 х 4,8 х 5,7 х 4,7 х 7607,03 m2 4862,28 m2 8217,44 m2 4106,93 m2 5 kcal/mol 10 kcal/mol 15 kcal/mol 0,062 N/m 0,096 N/m 0 0 0,75 т 10 0,94 т 10
-27 -27 -27 -27

kg kg kg kg

kg kg kg kg 41,868 kJ/mol 62,802 kJ/mol

+12 -1

s s

+12 -1

10,93910988739 т 10+12 s11,32441846835 т 10+12 s1,037316314872 т 10+12 s-

2. , 25 26 27 28 29 30 w2,GC (q=▒/a) v0 (Sound velocity) (KA + KT) = (v02/a2) (mA + mT) (KG + KC) = (v02/a2) (mG + mC) AT= {KArA2 + KTrT2 GC= {KGrG2 + KCrC2} 1,209934083308 т 10+12 s1890 m/s 13,50 N/m 13,56 N/m 550,30 kcal/mol 2304,01 kJ/mol 532,27 kcal/mol 2228,51 kJ/mol

References. 1. Yakushevich L.V., A hierarchy of dynamic models of DNA. Russian J.Phys.Chem. 69, N1, 167-171. (Translated from Zhurnal Fizicheskoi Khimii 69, N1, 180-185). 2. Yakushevich L.V. Savin A.V., Manevitch L.I., On the Internal dynamics of topological solitons in DNA. Phys. Rev. E-66, 016614, 2002. 3. Gali Prag, Computational Analysis of Macromolecules in Biotechnology. Course 2000-2001 Department of Molecular Genetics and Biotechnology, The Hebrew University н Hadassah Medical School Jerusalem, Israel (http://www.md.huj.ac.il/companal/intro5.html). 4. Hakim M.B., Lindsay S.M., Powell J., The speed of sound of DNA. Biopolymers 23, N7, 1185-1192,1984.

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