PRE-PRESS VERSION

The original publication is available athttp://www.springerlink.com/openurl.asp?genre=article&id=

doi:10.1007/s00249-004-0431-2Print version: Eur Biophys J (2005) 34: 13–18

A computer-generated supercoiled model of the pUC19 plasmid

E. A. Kummerle and E. Pomplun

Geschaftsbereich Sicherheit und StrahlenschutzForschungszentrum Julich,

52425 Julich, GermanyE-mail: E.Kuemmerle@fz-juelich.de

Abstract

DNA models have become a powerful tool in the simulation of radiation-inducedmolecular damage. Here, a computer code was developed which calculates the coordinatesof individual atoms in supercoiled plasmid DNA. In this prototype study, the known basepair sequence of the pUC19 plasmid has been utilised. The model was built in a three-stepprocess. Firstly, a Monte Carlo simulation was performed to shape a segment chainskeleton. Checks on elastic energy, distance and unknotting were applied. The temperaturewas considered in two different ways: (a) it was kept constant at 293K and (b) it wasgradually reduced from 350K to less than 10K. Secondly, a special smoothing procedurewas introduced here to remove the edges from the segment chain without changing thetotal curve length while avoiding the production of overshooting arcs. Finally, the basepair sequence was placed along the smoothed segment chain and the positions of all theatoms were calculated. As a first result, a few examples of the supercoiled plasmid modelswill be presented demonstrating the strong influence of appropriate control of the systemtemperature.

Keywords: DNA modelling, Monte Carlo simulation, polyline smoothing, supercoiledplasmid, track structure calculations

1 Introduction

Track structure calculations have become apowerful tool in computational microdosimet-ric research during the last two decades. Inthe meantime, various computer codes now ex-ist for simulating the transport of different ra-diation particles with a broad kinetic energyrange in liquid water and water vapour; see e.g.Terrissol and Vrigneaud (2001), Tomita et al.(1998) and Paretzke (1987), for an overviewsee Nikjoo and Uehara (1994). These codestrace the history of individual particles fromtheir starting points until the kinetic parti-cle energy has been completely absorbed orat least below a threshold (< 10 eV). Foreach interaction with the irradiated matter’smolecules (ionisation, excitation etc.) the spa-tial coordinates, the energy transfer, secondary

particles, etc. are recorded so that a trackstructure can be followed on the molecularlevel displaying the characteristics of differenttypes of radiation.

To utilise these tracks for the simulationof damage induction on the DNA molecule,in parallel to the code development targetmodels were generated. A parallel ‘dough-nut’ model of nucleosomal DNA was intro-duced by Charlton (1986) to investigate therange of low-energy Auger electrons releasedfrom incorporated 125I. Two ‘doughnuts’ inthe form of curved solid cylinders of 2 nmin diameter represent the DNA helix turn-ing around the histones. Furthermore, eachcylinder was sliced into 80 discs representingthe base pairs. The first linear DNA modelwhich allowed a differentiation between base

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and sugar-phosphate volumes was created bythe same author (Charlton and Humm 1988).A solid straight cylinder with a diameter of2.3 nm was divided into three compartments:an inner cylinder of 1.0 nm diameter alongthe central axis is occupied by the bases.The two sugar-phosphates of a base pair wererepresented by 0.34 nm wide arches each ro-tated by 36◦ for every successive base. Thenext step in this series was the linear helixmodel DNAMOD by Pomplun (1991) provid-ing the coordinates of the individual atomsof the DNA molecule and thereby enablingfor the first time the atomic resolution of atarget model. A modification of DNAMODled to a nucleosome structure with 144 basepairs turning the helix 1.75 times around thehistone volumes (Pomplun and Terrissol 1994,Terrissol and Pomplun 1994). Also the po-sitions of the water molecules in the innerand outer water sheath were included. Bothof these models have become the kernel forhigher-order models like chromatin fibre loops(Friedland et al. 1999).

The purpose of this paper is to provide asupercoiled plasmid DNA target model whichcan be used within track structure codes. Suchmolecules have often been used in radiobiolog-ical experiments, e.g. to investigate the dam-aging effectiveness of incorporated Auger elec-tron emitters like 125I (e.g. Kassis et al. 1999,Kassis et al. 2000) or 111In (Sahu et al. 1995).Supercoiled plasmid models have already beenused for strand-break induction calculationsfor γ-ray or electron irradiation (Tomita et al.1998, Watanabe and Saito 2002). Similar tothe plasmid modelling procedure used there,our work is also based on the Monte Carloalgorithm of Vologodskii et al. (1991) using asegment chain skeleton. Additionally, we de-veloped a subsequent smoothing algorithm tofinally obtain a model structure without anyedges. Furthermore, the trial changes of thesegment chain during the Monte Carlo pro-cedure were optimised on the basis of successstatistics.

2 Generation of the super-coiled DNA target model

The supercoiled DNA model was built inthree steps. (1) A closed chain consisting

of straight segments of equal length (i.e. anequilateral polygonal knot) was varied undertorsional and bending forces using a MonteCarlo simulation based on the method de-scribed by Vologodskii et al. (1991). (2) Theresulting segment chain was smoothed to aquasi-continuously bent curve. (3) This curvewas overlaid by a given DNA base pair se-quence and the positions of all the atoms werecalculated.

The programs were written in C++ us-ing the expression and meta-template librarytvmet (Petzold 2003) to achieve the highestperformance in vector and matrix calculations.The calculations were executed on a Linux PCwith 1.8GHz.

2.1 Monte Carlo simulation

2.1.1 Method

Following Vologodskii et al. (1991), the DNAmolecule was modelled by a closed chain con-sisting of N rigid segments with a length ofapproximately l = 10nm each (1/10 of theKuhn statistical length). The exact segmentlength resulted from the total chain lengthL = N · l . (In the case of pUC19, N = 91and l = 9.962 nm was chosen to obtain L =906.5 nm.)

The elastic energy of the chain is E = Eb+Et with the bending energy

Eb = kB T αN∑

i=1

Θ2i , (1)

where Θi is the angular displacement betweenthe directions of the segments i and i + 1, kB

is the Boltzmann constant, T = 293 K is thetemperature and α = 2.403 is the bending con-stant. The torsional energy is

Et = (2π2 C/L) · (∆Lk −Wr)2 (2)

where C = 3 ·10−28 Jm is the torsional rigidity.Wr is the writhe of the segment chain and

∆Lk = Lk − Lk0 = σ · Lk0 (3)

is the linking number difference between theactual linking number Lk of the closed DNAmolecule and the linking number Lk0 = L/z0

for relaxed DNA with the helix height z0 =3.375 nm (Chandrasekaran and Arnott 1989).σ is the specific linking number difference,

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also called superhelix density, which waschosen to be σ = −0.04 (typical σ val-ues for supercoiled DNA molecules are be-tween −0.05 and −0.07, see e.g. Bauer 1978).The writhe Wr was calculated according toKlenin and Langowski (2000), Method 2a.

The starting formation of the segmentchain was a plain, regular polygonal knot. Ateach Monte Carlo step, a trial change was ap-plied to the segment chain, the elastic energyof the new formation Ei+1 was calculated andcompared to the elastic energy of the previousformation Ei. If Ei+1 < Ei or

exp[(Ei − Ei+1)/(kB TM)] > ρ (4)

where TM is the Monte Carlo temperature(i.e. the temperature which controls the MonteCarlo simulation process) and ρ is a randomnumber between 0 and 1, the trial changewas accepted, otherwise it was rejected. Adistance check was performed to considerexcluded-volume effects. If the distance be-tween any two (non-adjacent) segments wasless than the effective DNA diameter d =3.5 nm, the trial change was rejected. Fi-nally, the Alexander polynomial was calculatedfor the segment chain using the program byHarris and Harvey (1999) to ensure that thenew formation was unknotted, otherwise it wasrejected.

Two types of trial changes were applied:(1) crankshaft rotation of an arbitrarily cho-sen sub-chain around the axis given by thesub-chain end points by an angle ϕ and (2)reptation move, i.e. the exchange of two ar-bitrarily chosen sub-chains of m and m + 1segments after appropriate deformation. (SeeVologodskii et al. (1991) for a more detailedexplanation.)

The following modifications were ap-plied to the procedure described byVologodskii et al. (1991):

(a) The crankshaft rotation angle ϕ wasnot chosen completely arbitrarily but on thebasis of success statistics. The ϕ range −π to πwas divided into 100 intervals of equal length.For each interval i the relative frequency f c

i

for accepting crankshaft rotation trial changeswas calculated and continuously updated dur-ing the Monte Carlo simulation using an ex-ponential forgetting algorithm. The a prioriprobability pc

i for trying a crankshaft rotation

with a ϕ value belonging to the interval i waschosen to be pc

i ∝ f ci ·√

ϕi where the factor√

ϕi

(with ϕi = max(|ϕ|) in the interval i) makesallowance for the fact that larger ϕ values aremore effective in changing the segment chainformation than small ones. At the beginningof the Monte Carlo simulation the pc

i valueswere set to the same start value for all inter-vals. Within the selected interval, the appliedϕ value was chosen randomly.

(b) The size m of the smaller reptationmove sub-chain was not fixed to 3 but variedbetween 2 and 9. Here too, success statisticswere built. The a priori probability pr

j for try-ing a reptation move with m = j was chosen tobe pr

j ∝ f rj with the relative frequency f r

j foraccepting such reptation move trial changes.At the beginning, all pr

j values were initialisedwith the same value.

(c) The a priori probabilities for crankshaftrotations and reptation moves were chosen tobe 2 : 1.

2.1.2 Progression of the Monte Carlosimulations

Two versions of the Monte Carlo procedurewere performed, (a) using a constant MonteCarlo temperature TM = 293K during the en-tire simulation and (b) starting with TM =350 K and reducing TM gradually to a finalvalue of typically less then 10 K. Here, TM

(cf. equation 4) was reduced by the factor0.9 whenever the elastic energy averaged overan interval of 1000 successful trial changeswas greater than 0.9 times the averaged elas-tic energy in the preceding interval. The fi-nal chains thus represent (a) a snapshot of aDNA molecule subjected to thermal deforma-tions and (b) a frozen-in DNA molecule. (N.B.:the temperature T = 293 K in equation 1 wasnever changed.)

Figures 1 and 2 show examples of the evo-lution of the elastic energy for both versionswith N = 91. The number of Monte Carlosteps was 106 each. In the case of constantTM, the elastic energy already reaches its finalfluctuation range after approximately 5 · 104

Monte Carlo steps (Figure 1). In the case ofTM being reduced, the decrease of the elasticenergy was controlled by the decrementationof TM (Figure 2).

Figure 3 shows the success statistics for

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Figure 1: Evolution of the elastic energy Eduring the Monte Carlo procedure for a seg-ment chain with N = 91 and σ = −0.04 withconstant TM = 293K.

Figure 2: Evolution of the elastic energy Eduring the Monte Carlo procedure for a seg-ment chain with N = 91 and σ = −0.04 whereTM (dashed line) was gradually reduced start-ing with TM = 350K.

crankshaft rotation and reptation move trialchanges for constant TM. The acceptanceprobability for crankshaft rotations decreasedstrongly with increasing angle |ϕ|. Themost frequently accepted reptation moves werethose with the smaller subchain sizes m = 4 orm = 5.

2.2 Smoothing of the segment chain

The requirements for the smoothing algorithmwere (1) to approximately conserve the totallength of the curve and (2) not to produce over-shooting arcs. The widely used spline func-tions, for example, do not fulfil any of these

Figure 3: Success statistics for a Monte Carloprocedure with N = 91, σ = −0.04 and con-stant TM = 293K. f c and f r are the relativefrequencies for accepting crankshaft rotation(angle ϕ) and reptation move (size m of thesmaller subchain) trial changes.

criteria. So the following procedure was ap-plied.

Smoothing was done iteratively and in ev-ery iteration cycle each chain segment was re-placed by two segments of approximately halfthe length. At every iteration, first the bisect-ing vectors ai of the angles between two adja-cent segments si−1 and si and the perpendicu-lar bisecting planes Bi of the segments si werecalculated (see Figure 4). Then the intersec-tion points P1

i and P2i of a plane Bi with the

two neighbouring vectors ai and ai+1 and thelines b1

i and b2i in the plane Bi through the

points P1i and P2

i , respectively, and crossingthe segment si were determined.

In the next step, provisional new segmentss2i and s2i+1 were constructed by calculatingthe base side of the isosceles triangles withthe equal sides ai and b1

i and b2i and ai+1,

respectively, and with the length of the baseside equal to half the length of the old seg-ment si. Generally, the end points of adjacentprovisional segments sj do not coincide, i.e. thesegments are not connected. The final new seg-ments s′j were therefore determined by takingthe centre points of each two end points of thesj which have to be connected as the cornersof the new segment chain.

In the case of a plane regular polygon, allthe provisional new segments sj are connectedso this procedure exactly conserves the lengthof the curve. In the general case of an irreg-

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i

i−1s

si+1

ai

ai+1bi

1iP

1i+1P2

iP =

bi+1

s2i+1

s

’s2i+1~

s2i~ s2i+2

~

Figure 4: The smoothing procedure illustratedin 2 dimensions. Contrary to the general 3-dimensional case, the points P2

i and P1i+1 co-

incide. Furthermore, the lines b1i and b2

i coin-cide in 2 dimensions, labelled bi in the figure.This line also represents the plane Bi.

ular polygonal knot, the total length of thecurve is typically reduced by less than 1% andthe length of individual segments may typi-cally vary from ∼93% to ∼102% after 5 it-eration steps. The small change of the totallength was finally compensated by rescalingthe length unit.

At every iteration step a distance check wasperformed and at the end of the smoothingprocedure a final knot check was made. After5 iteration steps the maximum angle betweentwo adjacent segments was reduced from typ-ically more than 90◦ to less than 10◦. A finalcalculation of the writhe showed that it waschanged by typically less than 0.5% due to thesmoothing procedure. Figure 5 shows as anexample a detail of a smoothed segment chain.

2.3 DNA sequence overlay

Now a given sequence of DNA basepairs, namely that of the pUC19 plasmid(Yanisch-Perron et al. 1985), was placedalong the smoothed segment chain. Whilefollowing that curve the coordinate system inwhich the positions of the DNA base atoms

Figure 5: Example of a detail of an initial seg-ment chain (grey) and the associated smoothedsegment chain (black).

were defined was moved ahead consideringthe twist ∆Tw = ∆Lk − Wr of the curve. Itsorientation was rotated continuously fromeach segment of the smoothed chain to thenext one by linear interpolation. Figure 6illustrates the result.

3 Results and conclusions

Figures 7 and 8 show examples of generated su-percoiled DNA model structures for the plas-mid pUC19 where the Monte Carlo tempera-ture was reduced from TM = 350 K to 1.5 Kor TM = 293 K was kept constant. In the firstcase, the resulting structure is formed in a veryorderly manner while in the second case thestructure is heavily distorted by thermal de-formations leading to an irregular shape. Thewrithe of the thermally deformed structureWr = −8.2 is somewhat smaller in its abso-lute value than that of the frozen-in structureWr = −9.9 . This clearly demonstrates thatthe temperature is a key parameter in the sim-ulation of realistic physiological plasmid forms.

A particular problem of using a DNA plas-mid model for track structure calculations isthe great variety of possible geometric con-formations at ambient temperature. The ac-tual form of a plasmid molecule is stronglyinfluenced by thermal motions. Track struc-ture calculations can be applied only on avery limited number of sample conformations.To find a small set of conformations whichcan be recognised as representative will be thenext step in providing models for DNA dam-

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Figure 6: Example of a detail of a DNA model structure generated by overlaying a DNA sequenceon a smoothed segment chain. (The same model structure as in Figure 7.) All molecule visuali-sations are generated using the molecular visualisation program VMD (Humphrey et al. 1996).

Figure 7: Supercoiled DNA model structurefor pUC19 (2686 base pairs) and the super-helix density σ = −0.04 generated by reduc-ing the Monte Carlo temperature graduallyfrom TM = 350K to 1.5K. The writhe isWr = −9.9 .

Figure 8: Supercoiled DNA model structurefor pUC19 (2686 base pairs) and the superhelixdensity σ = −0.04 generated with a constantMonte Carlo temperature TM = 293 K. Thewrithe is Wr = −8.2 .

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aging investigations. (Nevertheless, it is jus-tifiable to use static plasmid models in trackstructure calculations because the form of aplasmid molecule changes on a time scale ofabout 10−6 s (Jian et al. 1998) which is veryslow compared to the duration of the chemicalphase of the radiation action chain of about10−8 s.)

The successful modelling of the pUC19plasmid shown here now provides the abilityto also build models for other plasmids wherethe base pair sequence is known. As a fur-ther candidate, the Escherichia coli PlasmidpBR322 with 4361 base pairs is under consid-eration, being a biomolecule which is also veryfrequently used in radiation biology.

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