phase transition as a mechanism of dna opening for

Phase transition as a mechanism of DNA openingfor replication and transcription

Y. Charles Li a,*, David Retzloff b

a Department of Mathematics, University of Missouri, Columbia, MO 65211, United Statesb Department of Chemical Engineering, University of Missouri, Columbia, MO 65211, United States

Received 9 December 2005; received in revised form 27 April 2006; accepted 31 May 2006Available online 30 June 2006

Abstract

We propose a dynamic model of DNA opening for replication and transcription. Numerical simulationson this model strongly indicate a phase transition mechanism of DNA opening.� 2006 Elsevier Inc. All rights reserved.

MSC: Primary 92; Secondary 35, 65

Keywords: DNA replication; DNA transcription; DNA unwinding; Enzyme; Phase transition

1. The model

DNA replication and transcription are dynamic processes in which enzymes are the actuators.For instance, for transcription, first enzymes search along the DNA strands for ‘weak region’, likethe so-called TATA-sequence region, to attach. Then the enzymes unwind the double helixstrands, and generate a bubble in that region. In the bubble region, the two strands are separated,transcription can take place, and RNA is generated. The bubble can propagate along the DNAstrands (in fact, often the enzyme sits there, and the DNA thread screws itself through the station-

0025-5564/$ - see front matter � 2006 Elsevier Inc. All rights reserved.doi:10.1016/j.mbs.2006.05.006

* Corresponding author. Tel.: +1 573 884 0622.E-mail addresses: [email protected] (Y.C. Li), [email protected] (D. Retzloff).

www.elsevier.com/locate/mbs

Mathematical Biosciences 203 (2006) 137–147

ary enzyme [3]). Inside the bubble, complicated enzyme activity takes place. Many transcriptionfactors are involved. For instances, the TATA box is gripped by the transcription factor TFIID,while TFIIH is the one that directly unwinds the two strands of DNA (via helicase action) to al-low the RNA polymerase to get access to the DNA bases [4].

For a replication, a Y-fork is developed by enzymes with a bubble propagating at the center ofthe Y-fork. The leading strand (3 0–5 0) can be directly read (replicated), while the Okazaki frag-ments are patched along the other strand (5 0–3 0). For prokaryotes, the place where the replicationstarts is quite well understood [9,13]. For instance, E. coli starts its replication at the oriC sequenceof 245 base pair long. For eukaryotes, the replication process is more complicated and poorlyunderstood [2,8]. For instance, eukaryotes only replicate during the synthesis phase of the cell cy-cle. A well studied eukaryotic system is the early embryo Xenopus [2,8].

Although the base pairs of the two DNA strands can make all sorts of dynamic motions: twist,slide and roll, we feel that the crucial quantity in characterizing enzyme-unwinding is the distancebetween the two bases in each base pair. One can denote this quantity by rn where n labels the basepairs. One can view rn’s as functions of time t. The dynamics of rn(t) is influenced by many factors.The sugar-phosphate chains act like springs offering a restoring force. The hydrogen bond be-tween the two bases in each base pair offers an attractive-repulsive electric force for which wechoose the 6–12 law. Each hydrogen bond is between a hydrogen (with positive electric charge)and an oxygen or nitrogen (with negative electric charge). The electric charge is of about 1/3 elec-tron or proton. In the Watson–Crick pairing, an A–T pair has two hydrogen bonds, and a G–Cpair has three hydrogen bonds. Even though the specific details of the enzyme forcing are notcompletely understood, in vitro DNA strands can be torn apart by external forces. In fact, hope-fully by studying dynamic models, one can understand the enzyme force better. In our model, thephase transition result seems insensitive to different forms of enzyme forces. Of course, other fac-tors also affect the dynamics of rn(t). But we consider the effects from other factors are minor.Finally, our dynamic model can be written in terms of a difference-differential equation

d2

dt2rn ¼ anðrnþ1 � 2rn þ rn�1Þ þ bn

1

r12n

� 1

r6n

� �þ fn;

where rn is the distance between the n-th base pair, an is the coefficient of the sugar-phosphatechain, its dependence on n models the stacking effects, bn is the coefficient of the hydrogen bonds,bn = 2b or 3b depending on T–A or G–C pair, and fn is the enzyme force. First we study the casethat an and bn are independent of n, an = a and bn = b. We have scaled the neutral distance be-tween the base pair to 1. Without the enzyme force fn, the free state rn = 1 for all n is neutrallystable. In this case, the DNA will not open (unwind). Basic modal solutions to the linearized equa-tion at the free state are

rnðtÞ ¼ cosðXt þ 2np=NÞ or sinðXt þ 2np=NÞ;where X ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2að1� cos 2p

N Þ þ 6bq

and N is a large integer.To solve the model numerically, we choose 12 base pairs and pose the Dirichlet boundary con-

dition r0 = r11 = 1. We choose the initial condition to be the free state plus a perturbation of order0.001. Our main conclusion is that we find a phase transition phenomenon for the DNA openingand closing. By fixing any two of the three quantities a, b and fn, we can find a critical value (athreshold) of the third quantity for the transition from opening to closing of DNA. Specifically,

138 Y.C. Li, D. Retzloff / Mathematical Biosciences 203 (2006) 137–147

when we fix a = 1.0 and b = 4.79, and set fn = c (n = 1,. . .,4) and fn = 0 (n = 5,. . .,10), we find thatthe critical c* is between 0.99 and 1.0 (see Figs. 1–3). When c is less than 0.99 (Fig. 1), the DNAdoes not open. When c is increased to 1.0 (Fig. 2), it takes about 120 time units for the DNA to

Fig. 1. The graphs of rn(t) when c = 0.99, a = 1.0 and b = 4.79. In all the figures, the color assignments are: r1 – red, r2 –green, r3 – blue, r4 – brown, r5 – yellow, r6 – magenta, r7 – cyan, r8 – coral, r9 – maroon, r10 – tan. (For interpretation ofthe references in colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. The graphs of rn(t) when c = 1.0, a = 1.0 and b = 4.79.

Y.C. Li, D. Retzloff / Mathematical Biosciences 203 (2006) 137–147 139

open. When c is greater than 1.1 (Fig. 3), the DNA immediately opens. Thus, the greater the c (thestronger the enzyme force) is, the quicker the DNA opens. We have tried other values of theparameters (a,b), and we found the same feature. Therefore, the critical c* acts like the criticaltemperature in thermal phase transition. When we fix a = 1 and fn = 1 (n = 1,. . .,4) and fn = 0(n = 5,. . .,10), we find that the critical b* is between 4.79 and 4.795 (see Figs. 2, 4 and 5). Whenb is less than 4.7 (Fig. 5), the DNA immediately opens. Thus, the smaller the b (the weaker the


Fig. 4. The graphs of rn(t) when b = 4.795, c = 1.0 and a = 1.0.


hydrogen bond) is, the quicker the DNA opens. When b is increased to 4.79 (Fig. 2), it takes about120 time units for the DNA to open. When b is 4.795 (Fig. 4), the DNA does not open. Of course,when b is greater than 4.795, the DNA does not open. Therefore, the critical b* also acts like thecritical temperature in thermal phase transition. When we fix b = 4.79 and fn = 1 (n = 1,. . .,4) andfn = 0 (n = 5,. . .,10), we find that the critical a* is between 1.0 and 1.5 (see Figs. 6–8). The smallerthe a (the weaker the sugar-phosphate chain) is, the quicker the DNA opens. When a is 0.5(Fig. 8), the DNA quickly opens. When a is increased to 1.0 (Fig. 6), it takes about 120 time units

Fig. 5. The graphs of rn(t) when b = 4.7, c = 1.0 and a = 1.0.

Fig. 6. The graphs of rn(t) when a = 1.0, b = 4.79 and c = 1.0.


for the DNA to open. When a is greater than 1.5 (Fig. 7), the DNA does not open. Therefore, thecritical a* also acts like the critical temperature in thermal phase transition. Finally, we fix a = 1.0and b = 4.79, and let fn to be time-dependent fn(t) = cH(t � 5(n � 1))[1 � H(t � 5n)],(n = 1,2,. . .,10) where H(t) is the Heaviside step function, H(t) = 0 (t < 0), and H(t) = 1 (t P 0).We find that the critical c* is between 1.1 and 1.2 (see Figs. 9–11). When c is less than 1.1(Fig. 9), the DNA does not open. The greater the c (the stronger the enzyme force) is, the quickerthe DNA opens. When c is increased to 1.2 (Fig. 10), it takes only about 25 time units for theDNA to open. When c is 1.5 (Fig. 11), the DNA almost immediately opens. Therefore, the criticalc* also acts like the critical temperature in thermal phase transition. Thus the time-dependent fn




and the time-independent fn lead to the similar phase transition phenomenon. We have tried otherforms of fn, and we found the similar transition phenomenon. We have run the numerics for timeunits up to 4000. The conclusions are the same.

2. Other parameter regimes

The phase transition phenomenon persists when we change the number of base pairs in the bub-ble, or include the dependence of an and bn upon n. We have simulated as few as 4 base pairs. The




dependence of bn upon n is rather simple bn = 2b or 3b depending on T–A or G–C pair, where b isthe electric charge of about 1/3 electron or proton scaled with rn. Therefore, for a specific DNApromoter, bn is quite well defined. The dependence of an upon n is very complicated. First of all, itdepends on the distribution of the T–A and G–C pairs. Second, it depends on the stacking effects.Expressions for these dependences are not clear. Nevertheless, the phase transition phenomenonseems very robust with respect to the change in the expression of an. We simulated the test exam-ple an = a(1 + �cosn/N) where � � 0.01. We found the similar transition phenomenon.

3. Predictions by the model

In principle, the critical value of the enzyme force predicted by the model provides an estimateon the minimal enzyme force necessary to unwind the DNA strands. For a specific system (e.g. E.coli), it might be possible to test the model by laboratory experiments. The parameters an and bn

can be estimated from the specific experimental setup e.g. at the T7A1 promoter [15,6,14]. TheT7A1 promoter contains 168 base pairs, and the RNA polymerase binds to it from base pair51 to base pair 140. This indicates that TFIIH unwinds the T7A1 promoter from base pair 51to base pair 140. Thus in our model we choose fn = c (n = 51,. . .,140) and fn = 0 (n the rest),bn = 2b or 3b depending on T–A or G–C pair, and an = a independent of n. From the estimate[6] by comparing experiment with model, b/a = 10�4 � 10�3. The 168 base pairs of the T7A1 pro-moter are given as follows (only one strand (3 0–5 0) is necessary) [15,6,14].

• T T G T C T T T A T T A A T A C A A C T C A C T A T A A G G A G A G A C A A C T T A A A GA G A C T T A A A A G A T T A A T T T A A A A T T T A T C A A A A A G A G T A T T G A C TT A A A C T C T A A C C T A T A G G A T A C T T A C A G C C A T C G A G A G G G A C A C GG C G A A T A G C C A T C C C A A T C G A C A C C G G G G T C A A



By choosing a = 1, we found that the critical enzyme force c* for unwinding, as a function of b,has the graph as plotted in Fig. 12. The unwinding criterion is given by maxnrn = 2. Using a least-square fit to the data, we found that

c� ¼ 0:2545139692 bþ 0:0001830570499:

The original distance between the two bases in a base pair (not open) is about 10�9 m. If wescale the 6–12 law back to the original distance, we find that the critical enzyme force c* is about10�12 N–pN [10]. These are the data that we recommend to experimentalists.

4. Stacking effects

Three major electrical forces affect stacking: (i) To avoid contact (electrical bond) of baseswith water, maximal neighboring base pairs overlap is necessary. (ii) To avoid repulsion ofnegatively charged surfaces of stacking bases, reducing base-to-base overlap by slide is needed.Slide, twist, and roll are often inter-related. (iii) Between partial charges on atoms in the basepair rings, maximizing attraction and minimizing repulsion are preferred. These stacking effectswill lead to the dependence of an upon n. One can also try to introduce angle variables be-tween the two bases in each pair [14]. Due to the stacking effects, such angles will also dependon n. By looking at the action of the hydrogen bond between the two bases in each pair, onerealizes that the angle variables have no effect on the rn equation. Thus, it does not seem to beimportant to include angle variables. The stacking effects change the configuration of theDNA strands, it seems that the best way to take this into account is through the dependenceof an upon n. an also takes into account thermal effects even though thermal opening itself is arare event under physiological conditions [7].

Fig. 12. The graph of the critical enzyme force c* as a function of b.


5. Discussion

The dynamics of DNA opening (unwinding) is an open question. Usually, such an opening re-lies on outside energy source – enzymes. Therefore, it is a non-equilibrium process [10]. Thermalopening is a rare event under physiological conditions [7]. Thus, the enzyme force is crucial forboth replication and transcription. The speculation that solitons play a role in DNA transcriptionis not founded [5,17,18,16,19,12]. Even though the DNA opening is a non-equilibrium process,researchers often treat it as an equilibrium thermodynamic process simply because only equilib-rium thermodynamic tools are available [1,10,11,7]. Through equilibrium thermodynamic studies,a phase transition can be identified and dynamical scalings can be calculated [1,10,11,7].

In this article, we study the kinetic equations to understand the dynamics of DNA opening asa non-equilibrium process. We always clearly observe a phase transition. When the enzymeforce is time-independent, the dynamics after the phase transition seems going toward thermalequilibrium. But when the enzyme force is time-dependent, the dynamics after the phase tran-sition seems to be far away from thermal equilibrium. Of peculiar interest is that we observesuch a phase transition for a very small system of 10 particles (10 base pairs). Often theDNA opening bubble indeed contains very few base pairs (�10 base pairs) [3]. For such a smallsystem, equilibrium thermodynamics should not be a good approach, and kinetic equationshave to be studied. Measuring the enzyme force can be a reality in the future. Techniques havebeen developed to study DNA opening by exerting a force on the piconewton scale to pull apartthe DNA strands [10,11].

References

[1] S. Bhattacharjee, Unzipping DNAs: towards the first step of replication, J. Phys. A 33 (2000) L423.[2] J. Blow, Control of chromosomal DNA replication in the early Xenopus embryo, EMBO J. 20 (2001) 3293.[3] C. Calladine, H. Drew, Understanding DNA: The Molecule and How It Works, second ed., Academic Press,

1997.[4] A. Dvir et al., Mechanism of transcription initiation and promoter escape by RNA polymerase II, Curr. Opin.

Genet. Dev. 11 (2) (2001) 209.[5] S. Englander et al., Nature of the open state in long polynucleotide double helices: Possibility of soliton

excitations, Proc. Natl. Acad. Sci. 100 (12) (1980) 7222.[6] V. Fedyanin, L. Yakushevich, Scattering of neutrons and light by DNA solitons, Stud. Biophys. 103 (1984)

171.[7] T. Hwa et al., Localization of denaturation bubbles in random DNA sequences, Proc. Natl. Acad. Sci. 100 (8)

(2003) 4411.[8] O. Hyrien et al., Paradoxes of eukaryotic DNA replication: MCM proteins and the random completion, BioEssays

25 (2003) 116.[9] A. Kornberg, T. Baker, DNA Replication, W. H. Freeman and Co., NY, 1992.

[10] D. Lubensky, D. Nelson, Pulling pinned polymers and unzipping DNA, Phys. Rev. Lett. 85 (7) (2000) 1572.[11] D. Marenduzzo et al., Dynamical scaling of the DNA unzipping transition, Phys. Rev. Lett. 88 (2) (2002)

028102.[12] M. Peyrard et al., Statistical mechanics of a nonlinear model for DNA denaturation, Phys. Rev. Lett. 62 (23)

(1989) 2755.[13] P. Russell, Genetics, Addison Wesley Longman, Inc., 1998.[14] M. Salerno, Discrete model for DNA-promoter dynamics, Phys. Rev. A 44 (8) (1991) 5292.


[15] B. Sclavi et al., Real-time characterization of intermediates in the pathway to open complex formation byEscherichia coli RNA polymerase at the T7A1 promoter, PNAS 102 (13) (2005) 4706.

[16] S. Takeno, S. Homma, Kinks and breathers associated with collective sugar puckering in DNA, Prog. Theor. Phys.77 (3) (1987) 548.

[17] S. Yomosa, Soliton excitations in deoxyribonucleic acid (DNA) double helices, Phys. Rev. A 27 (4) (1983) 2120.[18] S. Yomosa, Solitary excitations in deoxyribonucleic acid (DNA) double helices, Phys. Rev. A 30 (1) (1984)

474.[19] C.-T. Zhang, Soliton excitations in deoxyribonucleic acid (DNA) double helices, Phys. Rev. A 35 (2) (1987)

886.


phase transition as a mechanism of dna opening for

Documents