Thermal relaxation in DNA

Transcription

1 Physica A 310 (2002) Thermal relaxation in DNA O. Resendis Antonio, L.S. Garcia-Colin Departamento de Fisica, Universidad Autonoma, Metropolitana-Iztapalapa, Apdo. postal , Mexico, D.F , Mexico Received 28 January 2002 Abstract In this article we suggest a model to describe the thermal relaxation in denaturation process in a particular sequence of a long DNA, a polyc:polyg chain. We present the behavior predicted by the stochastic matrix method and compare it with the one obtained by a stochastic method based on a master equation. In order to study the relaxation process of the denaturation we analyze the fraction of bonded and unbonded base pairs according to the temperature increase and as well as their time evolution. c 2002 Elsevier Science B.V. All rights reserved. PACS: x; Dy Keywords: DNA denaturation; Melting temperature; Thermal relaxation; Master equation and stochastic matrix method 1. Introduction Thermal denaturation or melting double-stranded DNA is the process by which the two strands unbind upon heating [1 3]. From the physical point of view, this process is a helix coil thermodynamic phase transition [3 9]. Experimentally, a sample of DNA with a specic length and sequence is immersed in an aqueous solution. When the temperature in the solution increases, the fraction of bonded base pairs as a function of the temperature, referred to as the melting curve, is measured through light absorption at 260 nm [1]. In order to identify the helix coil transition, a temperature T m is commonly accepted, which is called the melting temperature, and it is dened as the temperature where half of the total base pairs is bonded [3,4]. The understanding of the DNA thermal denaturation has been an interesting challenge for the physical and chemical sciences [3 21]. The crucial point here is the prediction of Corresponding author. address: rean@xanum.uam.mx (O.R. Antonio) /02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved. PII: S (02)

2 O.R. Antonio, L.S. Garcia-Colin / Physica A 310 (2002) the melting curve from a theoretical model for a particular DNA sequence immersed in an aqueous solution with well-dened external conditions (ph and concentration of salts). Otherwise, the study of this process has produced an important preview in the understanding of some fundamental molecular mechanisms that take place in life, namely replication and transcription of DNAs [1,22,23], and in the development of new and useful technologies in the medical [24], crystallographic [25], and biological elds [26]. In the literature we can nd several theoretical treatments, all of them based on the principles of equilibrium statistical mechanics, to predict the melting temperature of any DNA sequence. These treatments can be classied essentially into three categories. The rst one contains all of those based on the one-dimensional Ising model for the ferromagnetic transition [1,3 5,14,15]. The second one is that proposed by Peyrad Bishop based upon the ideas of a Hamiltonian system [8]. The last one contains the recent models which are based on computer simulation techniques of melting dynamics [7, 21]. For a variety of DNA sequences it is possible to establish a reasonable agreement between the experimental measurements and the theoretical results predicted by all these models [3,9,13,16,17]. However, such agreements have been established under thermodynamics equilibrium conditions. The main purpose in this work is two fold. On one hand, we suggest a new method to obtain the melting curve for any DNA sequences, and on the other hand, we suggest a formalism to study the thermal relaxation in DNA. By thermal relaxation we refer to the process by which the DNA chain relaxes from one thermal equilibrium state to an other equilibrium state when we change the temperature in the solution. We expect that the study of this topic can be useful to understand the inuence that the rate heating has on the DNA denaturation process. This work is organized in this way. In the second part, we suggest a new theoretical model to obtain the melting curve of a DNA chain and in that section we apply it to a particular simple chain, a polyc:polyg. The third section is devoted to analyze the thermal relaxation model for a polyc:polyg chain which is based on the master equation. Finally, in the last section we analyze the advantages of the new methodology and outline the conclusions as well as the perspectives of this work. 2. Theoretical model In this section we present a new stochastic model to describe the DNA thermal denaturation process when it is dissolved in an aqueous solution with xed physiological conditions (ph and salt concentrations). In order to give a description of this phenomenon we appeal to the stochastic matrix method (SMM) [27,28], which has been proposed as a simple statistical tool useful to describe a process characterized by undergoing a phase transition near a critical temperature. The SMM essentially consists in suggesting a set of molecular structural arrangements, called congurations, which constitute the structural molecular basic units of the system under study. Taking into account only the nearest-neighbor interactions between the nucleotides,

3 214 O.R. Antonio, L.S. Garcia-Colin / Physica A 310 (2002) which is an approach frequently accepted in the standard literature [3,4,24], the DNA denaturation process is completely dened by all the possible combinations by pairs between the bonded adenine thymine (A T) and cytocine guanine (C G) and unbonded adenine thymine (A T) and cytocine guanine (C G) base pairs. Thus, in the nearest-neighbor approach, we can nd a nite number of possible congurations, namely ( A T ) ( A T ; TA ) ( A T ; AT ) CG :::etc : Here, the arrows designate the direction of sugar phosphate chain and the point from the carbon c 3, of one sugar unit to the carbon c 5, of the next, both carbons attached to the same phosphodiester links. In principle, all these congurations of couples of base pairs are possible and each one has a nite probability to occur at any temperature. The main hypothesis in SMM is to state that the probability of getting a conguration made of the units i and j, written as p ij, is represented by a Boltzmann function, namely ( p ij = exp G ij RT ) ; (1) where R is the universal gas constant, T the temperature in degrees Kelvin and G ij is the Gibbs free energy of the conguration ij. In this way, we construct a square matrix, M, called the stochastic matrix, whose entries represent the probability of nding each conguration at any temperature. In order to describe the denaturation process we impose that the sum of all elements on each column is equal to one [27]: [M] ij = p ij ; i;j =1;:::;k ; (2) k p ij =1: j=1 This condition is surely the physical interpretation on p ij, it means p ij 0. The matrix written in Eq. (2) is the fundamental expression in the description of the system. The statistical behavior of the system is obtained when we apply the stochastic matrix M onto an initial vector p 0 whose components represent the initial concentration of each conguration [27,28], so that in the kth step, we have p k = M k p 0 (4) and the equilibrium condition is described by a vector p 0 which has the property that it remains invariant under further action of the matrix M. According to the SMM [27,28], this stable point can be written when we calculate the eigenvector with eigenvalue = 1 for the stochastic matrix M. Condition (3) assures the existence of a eigenvector with eigenvalue = 1, which determines the statistical properties when k is large, it means that the chain has many base pairs. In order to test the present methodology we will apply the formalism to study the denaturation behavior of a simple DNA chain, polyc:polyg. In this case, all the possible congurations in the nearest-neighbor (3)

4 O.R. Antonio, L.S. Garcia-Colin / Physica A 310 (2002) approximation are four, which may be represented by the following matrix scheme: [ ] [ ] C G CG C G C G [ ] [ ] CG CG : C G CG Here C G and C G, represent bonded and unbonded base pairs, respectively. Taking into account that each conguration has a statistical weight given by the Boltzmann factor, Eq. (1), we now construct the stochastic matrix of this particular chain as M = ( a a+b b a+b b b+c c b+c ) ; where we have used the notation ( a = exp G ) 11 ; RT G ij ( b = exp G ) ( 21 = exp G ) 12 ; RT RT ( c = exp G ) 22 : RT is the Gibbs free energy corresponding to the conguration represented by the element ij in the stochastic matrix M. For the sake of simplicity, we have assumed that the symmetry property for the Gibbs free energy G ij =G ji is valid at any temperature. The successive application of M onto p 0, the initial vector which represents the initial concentration of each conguration, allows us to obtain the equilibrium state corresponding to the denaturation process at any temperature T. It is possible to show that the stable point is independent of p 0 [27,28]. Thus, the stable point is the eigenvector with eigenvalue =1 of M. Making use of Eq. (5) we nd that the components in the equilibrium condition are given by p 1 = p 2 = a + b a +2b + c ; b + c a +2b + c ; where p 1 and p 2 represent the fraction of bonded and unbonded C G base pairs, respectively. These probabilities have a strong dependence with the temperature through the variables a; b; c given in Eqs. (6), and as a consequence Eq. (7) provides the behavior of the fractions of bonded and unbonded base pairs as functions of the temperature. One stricking and new advantage in this model is that it explicitly avoids adjustable parameters within the model, for instance, the nucleation parameters used in the Poland Scheraga model, [3], the only necessary information is provided by the thermodynamics (5) (6) (7)

5 216 O.R. Antonio, L.S. Garcia-Colin / Physica A 310 (2002) Fig. 1. Melting curve for the polyc:polyg chain immersed in an aqueous solution with 1 M of NaCl and ph 7. In this graph we observe the typical behavior of the fraction of bonded base pairs as the temperature increases. measurements of the stability of the [ ] C G C G base pairs, which for simplicity we denote by CC=GG, and the Gibbs free energy that identify the hydrogen bond between the Cytocine and Guanine base pairs. Indeed, these measurements have been carried out by Santalucia et al. [24] when the aqueous solution has a concentration 1 M of NaCl. Thus, taking into account the stability values for CC=GG base pairs reported by these authors and the thermodynamical parameters reported by Saenger [22] for the reaction between the Cytocine and Guanine it is possible to analyze the results predicted by this model. The curves in Fig. 1 reveal a common feature in DNA denaturation, namely, the fraction of bonded base pairs, called helicity, decreases as the temperature increases. The melting temperature T m is the temperature at which = 1 2, this model predicts a T m for a polyc:polyg chain of 362:5 K, whereas the experimental measurements report 364:2 K when the chain is immersed in an aqueous solution with a concentration 1 M of NaCl. The agreement in spite of the simplied nature of the model is rather satisfactory. This is the main contribution in this work. 3. Thermal relaxation In this section we are interested in describing the thermal relaxation for a polyc:poly G chain. Most of the theories describing the denaturation in the literature assume that the chain is in thermal equilibrium with the aqueous solution at any temperature, and as a consequence the description is carried out from statistical mechanics in an equilibrium framework. However, we expect that there is a scale of time which characterizes

6 O.R. Antonio, L.S. Garcia-Colin / Physica A 310 (2002) the transition from one thermal equilibrium state to other states when we change the temperature of the aqueous solution from T 1 to T 2, keeping xed the ph and the concentration of salts in the aqueous solution. By thermal relaxation, we refer to the study as to how the polymeric chain in a thermal equilibrium reaches the new thermal equilibrium state. In order to describe the mechanism of the thermal relaxation of polyc:polyg we suggest a model inspired by the one-dimensional random walk. For this purpose, we assume that the polyc:polyg is constituted by N complementary base pairs, where each complementary base pair may be in one of the two possible states. The rst one, [C G], characterized a bonded state between the Cytocine and Guanine nitrogenous bases, and the second [C G], corresponds to the unbonded state between the Cytocine and Guanine nitrogenous bases. These states dene the situation of each of the complementary base pairs, although, each complementary base pair can change their state according to the temperature changes in the aqueous solution. In order to describe the thermal relaxation we imagine a random walker limited to move between these two states. The rst state is identied when all the elements of the chain are bonded, the second state represents the situation of full denaturation. In this model, the walker has a nite probability to stay in each site and its movement depends on the temperature of the solution. The melting temperature T m is the temperature where the probabilities of nding the random walker in both the states are equal. From the physical point of view, we expect that our system behaves as follows: if the system polyc:polyg in aqueous solution is kept at a temperature less than the T m then the walker will have a probability one to stay in state 1. When the temperature in the solution is close to T m then the probability will be one-half for each site, respectively. Finally, at a higher temperature than T m the walker will have a unit probability of remaining in state 2. In order to describe this process we assume that the time evolution of the random walker can be described by a master equation of = WP ; (8) where W represents the matrix transition whose elements w ij are the rates of transition from states i to j. In this work we use the following notation, the rate which identies the transition between state 1 and state 2 is =. The rate from state 2 to state 1 is written as =, where and represent the occupation probabilities, and, represent the frequency between state 1 and state 2, respectively. In this case, where the system has only two possible states, the master equation can be written as ( P1 P2 ) ( = )( P1 P 2 ) : (9) In this equation P represents the derivate in and P j is the probability to nd the random walker in the state j =1; 2. The explicit solution of Eq. (9) clearly depends on the initial condition assigned to the fraction of bonded base pairs maintained at initial temperature T 1. As a particular case, we analyze the solution of Eq. (9) assuming that the chain is immersed in an aqueous solution at temperature T 1 in such

7 218 O.R. Antonio, L.S. Garcia-Colin / Physica A 310 (2002) a way that the system is subject to the initial conditions P 1 (t =0)=0; P 2 (t =0)=1: (10) These conditions represent, physically, that at time t = 0 all the base pairs along the chain are unbonded. As a consequence of the conditions imposed on (10), the solution to Eq. (9) is that + exp ( + )t P 1 = ; (11) + exp ( + )t P 2 = ; (12) + where clearly P 1 + P 2 = 1. It is important to mention two points. First, these probabilities, (11) and (12), depend on the temperature in the aqueous solution and the thermodynamic parameters, which dene the stability between the complementary base pairs, through and. Second, a striking property of Eqs. (11) and (12) is that the system always arrives at a stationary state independent of the initial conditions. Indeed this stationary state is given by P1 e = + =1 Pe 2 : (13) On the other hand, according to the theory of stochastic processes, the stationary limit in time of Eqs. (11) and (12) should be consistent with the statistical mechanical properties in the equilibrium state [29,30]. According to this principle, in the next section, we present the treatment which shows the consistency between both the approaches developed before, namely the master equation and the SMM model. 4. Conditions for the statistical equilibrium In agreement with the stochastic processes theory [29,30] the time behavior of the probabilities P 1 and P 2 eventually reaches a stationary state as can be veried from Eqs. (11) and(12). Besides, in agreement with the theory of stochastic processes, a system far from equilibrium always has a tendency of reaching its equilibrium state in such a way that the probabilities in the stationary limit should be consistent with the probabilities obtained from equilibrium statistical mechanics, this means that Lim P 1 = P1 e = p 1 ; t Lim P 2 = P2 e = p 2 : (14) t Taking into account this important property we are able to nd a relation between the parameters involved in both formalisms. Substituting Eqs. (7) and (13) into Eq. (14), we obtain a relation between the rates used in the master equation and the probabilities dened in the SMM formalism ( ) = b + c a + b : (15)

8 O.R. Antonio, L.S. Garcia-Colin / Physica A 310 (2002) We now impose another condition namely, =, meaning that the frequency of transition between the states 1 2 has the same values as that of the frequency between the states 2 1. This hypothesis, well known as the principle of detailed balance, states that there is equal number of nearest-neighbor base pairs bonded unbonded and unbonded bonded. It is important to stress that this condition does not imply equality between the rates of probabilities and. Indeed, from Eq. (15) we see that = b + c a + b : (16) This expression is a simplied relation between the probabilities used in the SMM and in the master equation formalism. Further, Eq. (16) has no unique solution, there are innitely many possible solutions. In order to analyze the behavior predicted by our model we suggest a possible solution to Eq. (16) given by = b + c; = a + b: (17) Taking into account these relations the expressions given in Eqs. (11) and (12) can be written as P 1 = a + b a +2b + c + b + c a +2b + c exp ( (a +2b + c)t ) ; P 2 = b + c a +2b + c b + c a +2b + c exp ( (a +2b + c)t ) ; (18) where t = t is the unit of time. Using the denition of a; b and c given in Eq. (6), Eqs. (18) allows us to analyze the fraction of bonded and unbonded complementary base pairs in terms of the time and the temperature variations for a given initial bonded base pairs conditions. The behavior of P 1 and P 2 as a function of the temperature and time are presented in Figs. 2 and 3. In these gures we have taken t as the unit of time which depends on the transition probability and the real time t. In these gures, two time regimes that characterize the thermal relaxation are notorious. Fig. 2 shows a transient regime which, from the physical point of view, identies the time scale in which the system evolves towards the equilibrium state starting from the well-established initial conditions in the fraction of bonded base pairs at time zero. The equilibrium state in the denaturated chain is reached when the thermal energy in the chain provided by the aqueous solution is in equilibrium with the eective attractive potential between the complementary base pairs. The second regime, Fig. 3, is characteristic of the chain when it has reached the equilibrium state. The results expressed in Figs. 2 and 3 show qualitatively the theoretical behavior of the thermal relaxation predicted by our model. The explicit dependence on the real scale time is a function of the frequency transition, which determines the unit of time. In this work we do not calculate, which can be obtained from the experiment; however, we verify that the general behavior of the denaturation in a large time scale, as it is veried in Fig. 3, is to have a tendency to reach thermal equilibrium between the chain and the aqueous solution as observed in Fig. 1.

9 220 O.R. Antonio, L.S. Garcia-Colin / Physica A 310 (2002) Fig. 2. The relaxation process predicted by our model in a short-time scale (transient behavior) as a function of the temperature in the aqueous solution. The initial conditions are given by Eq. (10). The explicit unit of time depends on the frequency of transition of the random walk (see Eq. (18)) which can be obtained by experimental measurements. Fig. 3. The asymptotical behavior in time of the polyc:polyg relaxation as a function of the temperature in aqueous solution. The behavior predicted by our stochastic model is in agreement with the melting curve obtained by the SMM (Fig. 1).

10 O.R. Antonio, L.S. Garcia-Colin / Physica A 310 (2002) Conclusions and perspectives In this work we conclude that our model can be used as an alternative option to predict the melting temperature of simple chains of DNA, as polyc:polyg. The melting temperature obtained through our model is consistent with the experimental measurement for a polyc:polyg chain immersed in an aqueous solution of 1 M of NaCl. The melting temperature predicted by the SMM, 362:5 K, is favorably close to the experimental value, 364:2 K reported in the literature [22]. On other hand, we have suggested a model based on random walk which allows us to study the thermal relaxation of the chain with the aqueous solution. We have suggested this random walk in such a way that it is compatible with the results obtained by the SMM model; however, the quantitative behavior should be confronted with the experimental measurements. In this framework the theoretical description of this simple chain, polyc:polyg, is dependent on three physical parameters: the thermodynamic stability parameters in the nearest-neighbor interaction approach for the possible combination between pairs of nucleotides and the frequency of transition, which in principle can be measured at any temperature and physiological condition [24]. As a consequence, our model has no adjustable parameters. This approach to denaturation is in striking dierence with the conventional models reported in the literature [3 20]. Finally, the perspectives of this work can be outlined as follows. First, all the results presented here are valid for a polynucleotide chain immersed in a solution with ph 7 and concentration 1 M of Na +, these parameters are invariant at any temperature. In this stage, an immediate extension of this work consists in the prediction of the melting temperature for polyc:polyg when it is immersed in an aqueous solution at dierent phs and salt concentrations. Second, the homogeneity in the sequence of polyc:polyg has simplied the description of the process; however, this formalism can be applied to study the behavior of heterogeneous chains. As is well known, the heterogeneity in the sequences are a crucial factor to describe the denaturation of complex chains and should be taken into account in the description through the dierent Gibbs free energies for each of the congurations. In addition, the statistical description of the denaturation and thermal relaxation process of homogeneous and heterogeneous chains and the inuence that the physiological parameters, as ph and concentrations of Na + in the aqueous solution [11], have on this process are topics for future work. References [1] C.R. Cantor, P.R. Schimmel, Biophysical Chemistry, W.H. Freeman and Company, New York, [2] R. Sinden, DNA Structure and Function, Academic Press, San Diego, [3] D. Poland, H.A. Sheraga, Theory of Helix-coil Transition in Biopolymers: Statistical Mechanics Theory of Order-disorder Transition in Biological Macromolecules, Academic Press, New York, [4] R.M. Wartell, A.S. Benight, Phys. Rep. 126 (1985) 67. [5] Y. Kafri, D. Mukamenl, L. Peliti, Phys. Rev. Lett. 85 (2000) [6] J. Sponer, J. Leszczynski, P. Hobza, J. Phys. Chem. 100 (1996) [7] K. Drukker, G.C. Schatz, J. Phys. Chem. B 104 (2000) [8] M. Peryrad, A.R. Bishop, Phys. Rev. Lett. 83 (1989) [9] A. Campa, A. Giansanti, Phys. Rev. E 58 (1998) 3585.

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