First-passage problems in DNA replication: effects of template tension on stepping and exonuclease activities of a DNA polymerase motor
aa r X i v : . [ q - b i o . S C ] J u l First-passage problems in DNA replication:effects of template tension on stepping andexonuclease activities of a DNA polymerase motor
Ajeet K. Sharma and Debashish ChowdhuryDepartment of Physics, Indian Institute of Technology,Kanpur, 208016October 29, 2018
Abstract
A DNA polymerase (DNAP) replicates a template DNA strand. Italso exploits the template as the track for its own motor-like mechanicalmovement. In the polymerase mode it elongates the nascent DNA by onenucleotide in each step. But, whenever it commits an error by misincor-porting an incorrect nucleotide, it can switch to an exonuclease mode. Inthe latter mode it excises the wrong nucleotide before switching back toits polymerase mode. We develop a stochastic kinetic model of DNA repli-cation that mimics an in-vitro experiment where a single-stranded DNA,subjected to a mechanical tension F , is converted to a double-strandedDNA by a single DNAP. The F -dependence of the average rate of repli-cation, which depends on the rates of both polymerase and exonucleaseactivities of the DNAP, is in good qualitative agreement with the corre-sponding experimental results. We introduce 9 novel distinct conditionaldwell times of a DNAP. Using the methods of first-passage times, we alsoderive the exact analytical expressions for the probability distributionsof these conditional dwell times. The predicted F -dependence of thesedistributions are, in principle, accessible to single-molecule experiments. A linear molecular motor is either a macromolecule or macromolecular complexthat moves along a filamentous track [1, 2, 3, 4, 5]. In spite of its noisy steppingkinetics, on the average, it moves in a directed manner. Its mechanical work isfuelled by the input energy which, for many motors, is chemical energy. Thedistributions of the dwell times of a motor at discrete positions on its trackas well as the duration of many complex motor-driven intracellular processeshave been calculated [6, 7, 8, 9, 10, 11, 13] using the methods of first-passagetimes [14], a well-known formalism in non-equilibrium statistical mechanics.Experimentally measured distributions of dwell times of a motor can be utilizedto extract useful information on its kinetic scheme [11, 15].For motors which can step both forward and backward on a linear track,four distinct conditional dwell times can be defined; distributions of these four1onditional dwell times have been calculated for some motors [8, 9, 12, 13].In this paper we consider a specific molecular motor called DNA polymerase(DNAP) and argue that its movements on the track is characterized by nine distinct conditional dwell times because of the coupling of its dual roles duringits key biological function. We define these nine conditional dwell times andcalculate their distributions analytically treating each of these as an appropri-ate first-passage time. As a byproduct of this exercise, we obtain an importantresult, namely, its mean velocity, that characterizes one of its average proper-ties; the theoretically predicted behaviour is consistent with the correspondingexperimental observations reported earlier in the literature. The distributionsof the nine conditional dwell times are new predictions which, we believe, canbe tested by single-molecule experiments.Deoxyribonucleic acid (DNA) is a polynucleotide, i.e., a linear heteropoly-mer whose monomeric subunits are drawn from a pool of four different species ofnucleotides, namely, A (Adenine), T (Thymine), C (Cytosine) and G (Guanine).In this heteropolymer the nucleotides are linked by phosphodiester bonds. Thegenetic message is chemically encoded in the sequence of the nucleotide species.DNA polymerase (DNAP) [16], the enzyme that replicates DNA, carries outa template-directed polymerization [17]. During this processes, repetitive cy-cles of nucleotide selection and phosphodiester bond formation is performed topolymerize a DNA strand. In every elongation cycle, hydrolysis of the substratemolecule supplies sufficient amount of energy to the DNAP for performing itsfunction. Therefore, DNAPs are also regarded as molecular motor [1, 2, 3, 4, 5]that transduce chemical energy into mechanical work while translocating step-by-step on the template DNA strand that serves as a track for these motors.In an in-vitro experiment, Wuite et al. [18] applied a tension on a ssDNA.The two ends of this DNA fragment were connected to two dielectric beads;one end was held by micro-pipette, while the other end, trapped optically by alaser beam, was pulled. This DNA fragment also served as a template for thereplication process carried out by a DNAP. Replication converted the ssDNAinto a dsDNA. The average rate of replication was found to vary nonmono-tonically with the tension applied on the template strand [18]. Similar resultswere obtained also in the experiments carried out by Maier et al. [19], wheremagnetic tweezers were used to apply the tension on template DNA. The ob-served nonmonotonic variation of the average rate of replication was explained[18, 19, 20, 21, 22] as a consequence of the difference in the force-extensioncurves of ssDNA and dsDNA [23].Fidelity of replication carried out by a DNAP is normally very high [24]. Itachieves such high accuracy by discriminating between the correct and incor-rect nucleotides by kinetic proofreading . The mechanism of kinetic proofread-ing enables the DNAP to reduce the error ratio to values far lower than thethermodynamically allowed value of exp ( − ∆ F k B T ), where ∆ F is the free energydifference of enzyme substrate complex for correct and incorrect nucleotides.Thus, DNAP is capable of correcting most of its own error during the ongoingreplication process itself.A DNAP performs its normal function as a polymerase by catalyzing theelongation of a new ssDNA molecule using another ssDNA as a template. How-ever, upon committing a misincorporation of a nucleotide in the elongatingDNA, the DNAP can detect its own error and transfer the nascent DNA to2nother site where it catalyses excision of the wrongly incorporated nucleotide.The distinct sites, where the polymerisation (pol) and exonuclease (exo) reac-tions are catalyzed, are separated by 3-4 nm on the same DNAP [25]. Thenascent DNA is transferred to the pol site from the exo site after the wrongnucleotide is cleaved from its tip by the DNAP. Thus, the transfer of the DNAbetween the pol and exo sites couples the polymerase and exonuclease activitiesof the DNAP.In the next section we develop a microscopic model for the replication of assDNA template that is subjected to externally applied tension F , a situationthat is very similar to the in-vitro experiment reported in refs. [18, 19]. Therates of both pol and exo activities of the DNAP enter into the expression thatwe derive for the average rate of elongation of the DNA. The F -dependence ofthis rate is consistent with the experimental observations reported in [18, 19].We then define 9 distinct conditional dwell times of the DNAP and identifyingeach of these with an appropriate first-passage time [14], we calculate theirdistributions analytically. We believe that experimental measurements of thesedistributions are likely to elucidate the nature of the interplay of the pol andexo activities of DNAP. The nucleotides on the template DNA are labelled sequentially by the integerindex j ( j = 1 , , ..., L ) which also serves to indicate the position of the DNAPon its track. The chemical (or conformtional) state of the DNAP is denoted bya discrete variable µ ( µ = 1 , ..., j, µ . The kinetic scheme used for our model is adaptedfrom that proposed originally by Patel et al. [26] and subsequently utilized byvarious other groups [27, 21]. The kinetic scheme of our model is shown in figure(1), where the four different values 1, 2, 3 and 4 of µ are the allowed chemicalstates in the polymerase-active mode of the enzyme, while in chemical state 5the exonuclease catalytic site is activated.The structure of DNA polymerase resembles a “cupped right hand” of ahuman, where its sub domains are recognized as palm, thumb and finger subdomains [28]. Template DNA enters from the finger sub-domain and takes exitfrom thumb sub-domain. The catalytic site where the binding occurs is locatedbetween finger and palm domain. Transitions between polymerase activated ki-netic states of the enzyme (i.e., chemical states 1,2, 3 and 4) can be summarizedas [29, 30] E o D j + dN T P k ⇀↽ k − E o D j dN T P k ⇀↽ k − E c D j dN T P k ⇀↽ k − E c D j +1 P P i k ⇀↽ k − E o D j +1 (1)where E c and E o represent the closed and open finger configuration DNAP,respectively, while D j denotes the length of the nascent DNA strand.Let us start with the state E o D j , labelled by µ = 1, in which the fingerdomain of DNAP is open and the DNAP is located at the site j on its tem-plate. Now a substrate molecule (dNTP) binds with the DNAP and resultingstate E o D j dN T P is labeled by “2” . The transition 1 → k , while corresponding reverse transition 2 → k − .Binding energy of dNTP switches the open finger configuration of DNAP into3losed finger configuration and the corresponding transition 2 ( E o D j dN T P ) → E c D j dN T P ) take place at the rate k . The reverse transition 3 → k − . This new closed finger configuration of DNAP catalyzesthe formation of phosphodiester bond between dNTP and nascent DNA strandthereby elongating the nascent DNA from length j to j + 1; this process isrepresented by the transition 3 ( E c D j dN T P ) → E c D j +1 P P i ) that occurs atthe rate k ( k − being the rate of the reverse transition). Finally, the transition4( j ) → j + 1) completes one elongation cycle; the corresponding rates of theforward and reverse transitions are k and k − , respectively. The transition4( j ) → j + 1) captures more than one sub-step which includes opening of thefinger domain, release of P P i and the forward movement of the DNAP to thenext site on the template. k k k k k -4 k -3 k -2 k -1 k k -1 k k -2 k k -3 k k -4 k k -1 k k -2 k k -3 k exo k x k p k p k p k x k x k exo Figure 1: A pictorial depiction of 5 state kinetic model for DNA polymerase(see the text for a detailed explanation).Immediately after completing one elongation cycle, the DNAP is normallyready to bind with a new substrate molecule and initiate the next elongationcycle. However, if a wrong nucleotide is incorporated in an elongation cycle, theDNAP is likely to transfer the nascent DNA from the pol site to the exo site.This switching from pol to exo activity is represented by the transition 1 → k x ; the reverse transition, without cleavage, takes placeat the rate k p . In the exo mode the cleavage of the last incorporated nucleotide,at the rate k exo , effectively alters the position of the DNAP from j + 1 to j . External load force tilts the free energies and alters the barriers for the forwardand reverse transitions [31]. But, not all the rate constants change significantly4ith the tension F applied on the template. We hypothesize that only thefollowing transitions are affected by the tension F : (I) 3 →
4, i.e., the polymer-ization step, where new dNTP subunit is incorporated into nascent DNA chainand a single stranded nucleotide is converted into a double stranded DNA. (II)1 → β hairpin [32, 33, 34]. Moreover, polymerase to exonucleaseswitching causes local melting of the dsDNA.Suppose ∆Φ( F ) is the change in the free energy barrier so that k ( F ) = k (0) exp ( − θ ∆Φ /k B T ) , k − ( F ) = k − (0) exp ((1 − θ )∆Φ /k B T ) k p ( F ) = k p (0) exp ( − θ ′ ∆Φ x /k B T ) , k x ( F ) = k x (0) exp ((1 − θ ′ )∆Φ x /k B T )(2)where k B is the Boltzmann constant, T is the absolute temperature and k (0), k − (0), k x (0), k p (0) are the values of the corresponding rate constants in theabsence of external force. The symbols θ (0 ≤ θ ≤
1) and θ ′ (0 ≤ θ ′ ≤ F ) and ∆Φ x ( F ) are derived in appendix A by relating these to ∆Φ ′ ( F )which is the change in the stretching free energy when a ssDNA is converted intodsDNA. As we show in the next section, following force dependence of k ( F )and k x ( F ), k ( F ) = k (0) exp ( − ∆Φ( F ) /k B T ) and k x ( F ) = k x (0) exp (∆Φ x ( F ) /k B T ) , (3)together with k − ( F ) = k − (0) , k p ( F ) = k p (0), i.e., θ = 1 , θ ′ = 0, shows a goodqualitative agreement with the experimental data. In this subsection we derive the force-velocity curve for our model DNAP motorand compare it with those reported earlier in the literature. Let P µ ( j, t ) be theprobability of finding DNAP in chemical state µ , at the position j on its track,at time t . The probability to finding the DNA polymerase in chemical state µ ,irrespective of its position, is P µ ( t ) = L X j =1 P µ ( j, t ) (4)where L is the total number of nucleotides in template DNA strand. Normali-sation of the probability imposes the condition X µ =1 P µ ( t ) = 1 (5)5t all times. The time evolution of the probability P µ ( t ) is governed by followingequations dP ( t ) dt = − ( k x + k + k − ) P ( t ) + k − P ( t ) + k P ( t ) + k p P ( t ) (6) dP ( t ) dt = k P ( t ) − ( k − + k ) P ( t ) + k − P ( t ) (7) dP ( t ) dt = k P ( t ) − ( k − + k ) P ( t ) + k − P ( t ) (8) dP ( t ) dt = k − P ( t ) + k P ( t ) − ( k − + k ) P ( t ) (9) dP ( t ) dt = k x P ( t ) − k p P ( t ) (10)Now we solve these equations in steady state and calculate the probabilityof finding the DNA polymerase in µ th chemical state ( P stµ ). P stµ = x µ x + x + x + x + x (11)Expressions for x µ ’s are given in Appendix B.Now we define the average rate of polymerization V p and the average rate ofexcision V e as V p = P st k − P st k − and V e = k exo P st (12)Therefore, the average velocity of the DNAP on its track is V = V p − V e (13)In figure (2) the average velocity of the DNAP is plotted against the tensionapplied on DNA track. Rate constants used for this plot are collected from theliterature [26] and listed in table 1.Because of the F -dependence of the form assumed in (3), at lower tension tran-sition 2 → → poly → exo switching cause the significant increasein the exonuclease cleaving at higher forces. Observed trend of variation of theaverage velocity is the direct consequence of the nonmonotonic behavior of the∆Φ( F ), shown in figure (5). The average velocity of a DNAP and its dependence on the tension applied onthe corresponding template does not provide any information on the intrinsicfluctuations in both the pol and exo activities of these machines. Probing fluc-tuations in the kinetics of molecular machines have become possible because ofthe recent advances in single molecule imaging, manipulation and enzymology.In this section we investigate theoretically how the fluctuations in the pol andexo activities of a DNAP would vary with the tension applied on the templateDNA. For this purpose we use the same kinetic model introduced in section 1,6
10 20 30 40 50 60
F (pN) -50-250255075100125 V ( F ) dNTP=10 µ MdNTP=30 µ MdNTP=60 µ MdNTP=100 µ M Figure 2: Velocity of DNA polymerase is plotted against the force applied ontemplate strand for a few different values of dNTP concentration. The numericalvalues of the parameters used for this plot are listed in table 1.Rate constant Numerical value k µM − s − k − s − k s − k − s − k (0) 9000 s − k − s − k s − k − s − k x (0) .2 s − k p s − k exo s − Table 1: Numerical values of the rate constants used for graphical plotting ofsome typical curves obtained from the analytical expressions derived in thispaper. 7hat we have used in subsection 2.1 for calculating the average properties ofDNAP.The variable chosen to characterize the fluctuations in replication processis the time of dwell of DNAP at a single nucleotide on the template, which isnothing but the effective duration of its stay in that location. While moving onthe one dimensional template strand three different mechanical steps are takenby DNAP, which are(1) Forward step in the pol mode: 4( j ) → j + 1).(2) Backward step in the pol mode: 1( j + 1) → j ).(3) Backward step (caused by cleavage) in the exo mode: 5( j + 1) → j ).If a molecular motor takes more than one type of mechanical step then thefluctuations in the durations of its dwell at different locations cannot be char-acterized by a single distribution; instead, distributions of more than one typeof conditional dwell times can be defined [10]. So, in the context of our modelof DNAP, three different types of mechanical step would generate nine differ-ent distribution of conditional dwell times. We denote the forward, backwardand cleavage steps are by the symbols +, − and x , respectively. Ψ mn ( t ) is theconditional dwell time of the DNA polymerase when step m is followed by n,where the three allowed values of each of the subscripts m and n are + , − , x .For the convenience of calculation of the distributions Ψ mn ( t ), first we assumethat the DNAP is already at the j th site on the template strand and that therate constants for all the transitions leading to this special site j are equated tozero. In other words,(1) k = 0 only for the transition 4( j − → j ) (and not for any i = j ),(2) k − = 0 only for 1( j + 1) → j ) (and not for any i = j ),(3) k exo = 0 only for 5( j + 1) → j ) (and not for any i = j ).Now appropriate initial conditions will ensure the type of previous step takenby DNAP.If P µ ( j, t ) is the probability of finding the DNA polymerase in µ th chemicalstate at site j at time t , then time evolution of these probabilities are governedby following master equation. dP ( j, t ) dt = − ( k − + k + k x ) P ( j, t ) + k − P ( j, t ) + k p P ( j, t ) (14) dP ( j, t ) dt = k P ( j, t ) − ( k − + k ) P ( j, t ) + k − P ( j, t ) (15) dP ( j, t ) dt = k P ( j, t ) − ( k − + k ) P ( j, t ) + k − P ( j, t ) (16) dP ( j, t ) dt = k P ( j, t ) − ( k + k − ) P ( j, t ) (17) dP ( j, t ) dt = k x P ( j, t ) − ( k p + k exo ) P ( j, t ) (18)These equation can be re-expressed in the following matrix form. ddt P ( t ) = MP ( t ) (19)8ere P(t) is a column matrix, whose elements are P ( j, t ), P ( j, t ), P ( j, t ), P ( j, t ) and P ( j, t ). And M = − ( k − + k + k x ) k − k p k − ( k − + k ) k − k − ( k − + k ) k −
00 0 k − ( k + k − ) 0 k x − ( k p + k exo ) (20)Now introducing the Laplace transform of the probability of kinetic states by,˜ P µ ( j, s ) = Z ∞ P µ ( j, t ) e − st dt (21)Solution of equation (19) in Laplace space is, ˜P ( j, s ) = ( s I − M ) − ˜P ( j,
0) (22)Here ˜P ( j, s ) is the vector of the probability of individual chemical state inLaplace space and ˜P ( j,
0) is the column vector of initial probabilities.Determinant of matrix s I − M is a fifth order polynomial det ( s I − M ) = αs + βs + γs + δs + ǫs + ζ ; (23)full expressions for α , β , γ , δ , ǫ and ζ in terms of the primary rate constantsare given in Appendix C. Ψ ++ , Ψ + − , Ψ + x Following set of initial conditions guarantees that previous step taken by DNApolymerase is a forward step. P ( j,
0) = 1 and P ( j,
0) = P ( j,
0) = P ( j,
0) = P ( j,
0) = 0 (24)So three different distribution of dwell time, where first step is forward, aredefined as follows:Ψ ++ ( t ) = P ( j, t ) k | [ P ( j, ,P ( j, P ( j, P ( j, P ( j, (25)Ψ + − ( t ) = P ( j, t ) k − | [ P ( j, ,P ( j, P ( j, P ( j, P ( j, (26)Ψ + x ( t ) = P ( j, t ) k exo | [ P ( j, ,P ( j, P ( j, P ( j, P ( j, (27)By applying the initial condition (24) in equation (22), we get˜ P ( j, s ) = a + a sαs + βs + γs + δs + ǫs + ζ (28)˜ P ( j, s ) = b s + b s + b s + b s + b αs + βs + γs + δs + ǫs + ζ (29)˜ P ( j, s ) = c s + c s + c s + c αs + βs + γs + δs + ǫs + ζ (30)Mathematical expressions for a , a , b , b , b , b , b , c , c , c and c are givenin Appendix D. 9 t (s) Ψ ( ++ ) (t) Simulation F = 0 pNFormula F = 0 pN Simulation F = 10 pNFormula F = 10 pNSimulation F =30 pNFormula F = 30 pN t (s) Ψ ( +− ) (t) Simulation F = 0 pNFormula F = 0 pNSimulation F = 30 pN Formula F = 30 pN t (s) Ψ ( − + ) (t) Simulation F = 0 pNFormula F = 0 pNSimulation F = 30 pNForumula F = 30 pN t (s) Ψ ( −− ) (t) Simulation F = 0 pNFormula F = 0 pNSimulation F =10 pNFormula F = 10 pNSimulation F = 30 pNFormula F = 30 pN
Figure 3: Ψ ++ ( t ), Ψ + − ( t ), Ψ − + ( t ), and Ψ −− ( t ) are plotted for a few differentvalues of F. t (s) Ψ ( + x ) ( t) Formula F = 0 pNFormula F = 30 pN t (s) Ψ ( − x ) (t) Formula F = 0 pNFormula F = 30 pN t (s) Ψ ( x + ) (t) Formula F = 0 pNFormula F = 10 pNFormula F = 30 pN t (s) Ψ ( x -) (t) Formula F = 0 pNFormula F = 30 pN
Figure 4: Ψ + x ( t ), Ψ − x ( t ), Ψ x + ( t ), and Ψ x − ( t ) are plotted for a few differentvalues of F. 10y inserting the inverse Laplace transforms of the expressions (28), (29) and(30) into the equations (25), (26) and (27), respectively, we getΨ ++ ( t ) = (cid:20) ( a − a ω ) k ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( a − a ω ) k ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( a − a ω ) k ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( a − a ω ) k ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( a − a ω ) k ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t (31)Ψ + − ( t ) = (cid:20) ( b − b ω + b ω − b ω + b ω ) k − ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( b − b ω + b ω − b ω + b ω ) k − ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( b − b ω + b ω − b ω + b ω ) k − ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( b − b ω + b ω − b ω + b ω ) k − ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( b − b ω + b ω − b ω + b ω ) k − ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t (32)Ψ + x ( t ) = (cid:20) ( c − c ω + c ω − c ω ) k exo ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( c − c ω + c ω − c ω ) k exo ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( c − c ω + c ω − c ω ) k exo ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( c − c ω + c ω − c ω ) k exo ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( c − c ω + c ω − c ω ) k exo ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t (33)where ω , ω , ω , ω and ω are the roots of following equation αω − βω + γω − δω + ǫω − ζ = 0; (34)the explicit expressions of α, β, γ, δ, ǫ and ζ in terms of the primary rate con-stants of the kinetic model are given in appendix C. The coupled nature ofthe pol and exo activities is revealed by the mixing of the corresponding rateconstants in the expressions of Ψ + , ± and Ψ + x .11 .2.2 Calculation of Ψ − + , Ψ −− , Ψ − x Following initial conditions ensures that DNA polymerase has reached to site j by making a backward step: P ( j,
0) = 1 and P ( j,
0) = P ( j,
0) = P ( j,
0) = P ( j,
0) = 0 (35)So three different distributions of dwell time, where first step is backward, aredefined as follows:Ψ − + ( t ) = P ( j, t ) k | [ P ( j, ,P ( j, P ( j, P ( j, P ( j, (36)Ψ −− ( t ) = P ( j, t ) k − | [ P ( j, ,P ( j, P ( j, P ( j, P ( j, (37)Ψ − x ( t ) = P ( j, t ) k exo | [ P ( j, ,P ( j, P ( j, P ( j, P ( j, (38)After applying the above initial condition in equation (22), we get˜ P ( j, s ) = d s + d s + d s + d s + d αs + βs + γs + δs + ǫs + ζ (39)˜ P ( j, s ) = k x k − k − k − αs + βs + γs + δs + ǫs + ζ (40)˜ P ( j, s ) = e + e sαs + βs + γs + δs + ǫs + ζ (41)Full expressions for d , d , d , d , d , e and e in terms of the primary rateconstants of the kinetic model are given in Appendix D. Inverse transform ofequation (39), (40) and (41) gives the mathematical expression for P ( j, t ), P ( j, t ) and P ( j, t ).Substituting the inverse Laplace transforms of (39), (40) and (41) into theequations (36), (37) and (38) respectively, we get the following distributions ofthe conditional dwell time:Ψ − + ( t ) = (cid:20) ( d − d ω + d ω − d ω + d ω ) k ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( d − d ω + d ω − d ω + d ω ) k ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( d − d ω + d ω − d ω + d ω ) k ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( d − d ω + d ω − d ω + d ω ) k ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( d − d ω + d ω − d ω + d ω ) k ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t (42)12 −− ( t ) = (cid:20) k x k − k − k − k − ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) k x k − k − k − k − ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) k x k − k − k − k − ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) k x k − k − k − k − ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) k x k − k − k − k − ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t (43)Ψ − x ( t ) = (cid:20) ( e − e ω ) k exo ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( e − e ω ) k exo ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( e − e ω ) k exo ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( e − e ω ) k exo ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( e − e ω ) k exo ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t (44)where ω , ω , ω , ω and ω are the roots of the equation (34). Ψ x + , Ψ x − , Ψ xx Now we consider the case where DNA polymerase has arrived at site i by makingan exonuclease cleavage. The initial condition P ( j,
0) = 1 and P ( j,
0) = P ( j,
0) = P ( j,
0) = P ( j,
0) = 0 (45)ensures that previous mechanical step is an exonuclease cleaving. Now we definefollowing distributions of conditional dwell timeΨ x + ( t ) = P ( j, t ) k | [ P ( j, ,P ( j, P ( j, P ( j, P ( j, (46)Ψ x − ( t ) = P ( j, t ) k − | [ P ( j, ,P ( j, P ( j, P ( j, P ( j, (47)Ψ xx ( t ) = P ( j, t ) k exo | [ P ( j, ,P ( j, P ( j, P ( j, P ( j, (48)After applying the above initial condition in equation 22, we get˜ P ( j, s ) = f s + f s + f s + f s + f αs + βs + γs + δs + γs + ζ (49)˜ P ( j, s ) = k k k k p αs + βs + γs + δs + ǫs + ζ (50)13 P ( j, s ) = g s + g s + g s + g αs + βs + γs + δs + ǫs + ζ (51)The expressions for f , f , f , f , f , g , g , g and g are given in AppendixD. The values of P ( j, t ), P ( j, t ) and P ( j, t ) are obtained from the inverseLaplace transform of the (49), (50) and (50). After inserting the values of P ( j, t ), P ( j, t ) and P ( j, t ) in equations (46), (47) and (47), we get the exactanalytical expression for Ψ xx ( t ), Ψ x +( t ) and Ψ x − ( t ).Ψ xx ( t ) = (cid:20) ( f − f ω + f ω − f ω + f ω ) k exo ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( f − f ω + f ω − f ω + f ω ) k exo ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( f − f ω + f ω − f ω + f ω ) k exo ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( f − f ω + f ω − f ω + f ω ) k exo ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( f − f ω + f ω − f ω + f ω ) k exo ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t (52)Ψ x + ( t ) = (cid:20) k k k k k p ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) k k k k k p ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) k k k k k p ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) k k k k k p ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) k k k k k p ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t (53)Ψ x − ( t ) = (cid:20) ( g − g ω + g ω − g ω ) k − ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( g − g ω + g ω − g ω ) k − ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( g − g ω + g ω − g ω ) k − ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( g − g ω + g ω − g ω ) k − ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t + (cid:20) ( g − g ω + g ω − g ω ) k − ( ω − ω )( ω − ω )( ω − ω )( ω − ω ) (cid:21) e − ω t (54)where ω , ω , ω , ω and ω are the roots of the equation (34).The distributions of the conditional dwell times Ψ mn , except Ψ xx , are plot-ted for a few typical values of the parameters in figs.3 and 4. Since Ψ xx is14ndependent of the tension F , it has not been drawn graphically. We have alsopresented our numerical data, obtained from direct computer simulation, forthe distributions plotted in fig.3. Each of these distributions is a sum of severalexponentials. Therefore, in general, these distributions are expected to peakat a nonzero value of time t . However, some of the distributions in fig.3 and4 appear as a single exponential. This single-exponential like appearance is anartefact of the parameters chosen for plotting these curves although, in reality,the full distributions are sum of several exponentials.An interesting feature of the distributions plotted in figs.3 and 4 is a non-monotonic variation of the probability of the most probable conditional dwelltimes with increasing F (see, for example, Ψ ++ and Ψ −− ). This trend of vari-ation is a consequence of the nonmonotonic variation of ∆Φ with F (see fig.5). The distributions of the conditional dwell times ψ ±± have been extracted forsome motors in the last decade from the data obtained from single-moleculeexperiments. But, to our knowledge, none of the distributions Ψ ±± , Ψ x ± , Ψ ± x and Ψ xx have been measured experimentally so far specifically for the DNAPmotor. In this section we first mention a few recently developed single-moleculetechniques that probe some aspects of DNAP kinetics during replication.In a landmark paper Eid et al. [38] reported a single-molecule method formonitoring replication exploiting fluorescently labelled nucleotide monomers.The fluorophores are “Phospholinked” (i.e., linked to the phosphate group ofthe nucleotide monomer) [38]. Since DNAP-catalyzed phosphodiester bond for-mation releases the fluorophore from the nucleotide, the temporal sequence ofthe color of the fluorescence provides the sequence of the nucleotides that areincorporated in the elongating DNA. Christian et al.[37] have developed a single-molecule technique for monitoring replication by a DNAP with base-pair reso-lution. This method is based on F¨orster resonance energy transfer (FRET). Useof this technique also makes it possible to discriminate between the polymer-ization activity and exonuclease activity of the DNAP. It is likely that in nearfuture appropriate adaptations of these or some combination of force-based andfluorescence-based single molecule techniques may achieve sufficiently high res-olution required for measuring the nine distributions of conditional dwell timesintroduced in this paper.Next, we propose a reduced description of the stochastic pause-and-translocationof the DNAP in terms of fewer conditional dwell times which, as we explain be-low, may be measurable with the currently available single molecule techniquesbecause these do not distinguish between chemical and mechanical backwardsteppings. Let us defineΨ + ( t ) = Ψ ++ ( t ) + Ψ + − ( t ) + Ψ + x ( t ) , (55)Ψ − ( t ) = Ψ − + ( t ) + Ψ −− ( t ) + Ψ − x ( t ) (56)and Ψ x ( t ) = Ψ x + ( t ) + Ψ x − ( t ) + Ψ xx ( t ) (57)15s the distributions of conditional dwell times in which, regardless of the natureof the next step, the step taken by the DNAP is a forward polymerase step (+),backward polymerase step ( − ) and backward exonuclease step ( x ), respectively.For a given set of initial conditions, overall probability to leave the j th siteshould be unity. Therefore, following conditions must be satisfied: Z ∞ ( k − P ( t )+ k P ( t )+ k exo P ( t )) | [ P ( j, P i ( j, dt = Z ∞ Ψ + ( t ) dt = 1(58) Z ∞ ( k − P ( t )+ k P ( t )+ k exo P ( t )) | [ P ( j, P i ( j, dt = Z ∞ Ψ − ( t ) dt = 1(59) Z ∞ ( k − P ( t )+ k P ( t )+ k exo P ( t )) | [ P ( j, P i ( j, dt = Z ∞ Ψ x ( t ) dt = 1(60)i.e., Ψ + ( t ), Ψ − ( t ) and Ψ x ( t ) are probability distributions normalized to unity.Therefore, the overall distribution of dwell time, irrespective of the type of stepstaken by DNAP, is the weighted sumΨ( t ) = q + Ψ + ( t ) + q − Ψ − ( t ) + q x Ψ x ( t ) . (61)where q + , q − and q x denote the probabilities of taking forward polymerase step(+), backward polymerase step ( − ) and backward exonuclease step ( x ) by aDNAP, respectively. The explicit expressions for q + , q − and q x are given by q + = P st k P st k + P st k − + P st k exo = x k x k + x k − + x k exo (62) q − = P st k − P st k + P st k − + P st k exo = x k − x k + x k − + x k exo (63) q x = P st k exo P st k + P st k − + P st k exo = x k exo x k + x k − + x k exo (64)where x µ s, in terms of the rate constants, are given in Appendix B.We now recast eqn (61) in a form that would facilitate direct contact withexperiments that are feasible with the currently available techniques. WritingΨ( t ) = Ξ ++ ( t ) + Ξ + − ( t ) + Ξ − + ( t ) + Ξ −− ( t ) (65)we identify the four new distributions of conditional dwell times Ξ ±± ( t ) to beΞ ++ ( t ) = q + Ψ ++ ( t ) (66)Ξ + − ( t ) = q + [Ψ + − ( t ) + Ψ + x ( t )] (67)Ξ − + ( t ) = q − Ψ − + ( t ) + q x Ψ x + ( t ) (68)Ξ −− ( t ) = q − [Ψ −− ( t ) + Ψ − x ( t )] + q x [Ψ x − ( t ) + Ψ xx ( t )] (69)where the symbols ”+” and ” − ” denote forward and backward movements ofthe DNAP irrespective of the mode of movement. For example, the DNAP canmove backward either by polymerase or exonuclease activity; however, the newly16efined conditional dwell times Ξ − ( t ) does not discriminate between these twomodes of backward movement. For the purpose of comparison with experimentaldata, expressions (62),(63) and (64) for q + , q − and q x and the expressions derivedin section 3 for the conditional dwell times Ψ ±± , Ψ ± x , Ψ x ± should be substitutedinto the eqns.(66)-(69).For a DNAP with the data set given in table 1, the probabilities for apolymerase-dependent forward step (+), polymerase-dependent backward step( − ) and exonuclease activity ( x ) are 0.9736, .0261 and .0003, respectively. Thus,under normal circumstances back-stepping and exonuclease events are very un-likely. Moreover, two consecutive exonuclease steps would be extremely rare.However, the frequency of exonuclease activity of the DNAP can be increased byusing mutants of the same DNAP. By increasing the concentration of pyrophos-phate far above the equilibrium concentration, back-stepping events can bemade more frequent [21]. Besides, transfer-deficient mutants and exonuclease-deficient mutants [25] can be used to test the effects of variation of the corre-sponding rate constants on the various dwell time distributions. DNA replication is carried out by DNAP which operates as a molecular motorutilizing the template DNA strand as its track. In this paper we have presented atheoretical model for DNA replication that allows systematic investigation of thepol and exo activities as well as their coupling. More specifically, the situationconsidered here mimics an in-vitro experiment where a tension is applied onthe template strand throughout the replication process. We have calculated theeffect of the tension on the average speed of replication, capturing the effects ofboth the pol and exo activities of the same DNAP. Our theoretical results insection 3.1 are in good qualitative agreement with the results of single moleculeexperiments reported in the literature [18, 21].However, the intrinsic fluctuations in the pol and exo processes contain someadditional information which cannot be extracted from average properties. It iswell known that the fluctuation in the dwell times provides a numerical estimatethe number of kinetic states [11, 15]. More specifically, one defines a “random-ness parameter” r = ( < τ > − < τ > ) / < τ > where τ is the dwell time andthe symbol < . > indicates average; 1 /r provides a lower bound on the num-ber of kinetic states in each mechano-chemical cycle of the motor. Moreover, if r is larger than unity in any parameter regime, it would indicate existence ofbranched pathways. Furthermore, conditional dwell times can reveal existenceof correlations between individual steps of the mechano-chemical pathways of amolecular motor [11]. Besides, hidden substeps may be missed in the noisy datarecorded in a single-motor experiment; the distributions of conditional dwelltimes are quite useful in detecting such substeps. Exact analytical expressionsfor the distributions of conditional dwell times that we report here may find usein the analysis of the experimental data for extracting these information [9].Although both the pol and exo activities of the DNAP have been studiedextensively [35], the distributions of dwell times of DNAP have not been mea-sured so far in any single molecule experiment [36]. In this paper we havealso mentioned a few recently developed single-molecule techniques for DNAP[38, 37] which, after minor alteration, might be the appropriate tool for measur-17ng the conditional dwell times introduced in this paper. We have also proposeda reduced description of the pause-and-translocation of DNAP in terms of thedistributions of fewer conditional dwell times which, in principle, can be mea-sured by the currently available single-molecular techniques. We hope our modeland results will motivate experiments to study the unexplored stochastic fea-tures of the kinetics of one of the most important genetic processes, namelyDNA replication driven by DNAP. Understanding this kinetics will throw lighton the propagation of life from one generation to the next. Acknowledgements : DC thanks Berenike Maier for useful discussions. Wealso thank the anonymous referees for constructive criticism and suggestionswhich helped in significant improvement of the manuscript. This work has beensupported by the Dr. Jagmohan Garg Chair Professorship (DC), J.C. Bosenational fellowship (DC), Department of Biotechnology (DC) and Council ofScientific and Industrial Research (AKS).
Appendix A
Here the parameters with subscripts “1” and “2” correspond to ssDNA anddsDNA, respectively. Let b i ( F ) ( i = 1 ,
2) denote the average equilibrium pro-jections of base pair in the direction of the applied force F . Suppose, Φ i ( F )( i = 1 ,
2) are the corresponding free energies. Then, for a given force F , thefree energy difference between single base-pair of dsDNA and ssDNA can beexpressed as [23]∆Φ ′ ( F ) = Φ ( F ) − Φ ( F ) = − Z F ( b ( F ′ ) − b ( F ′ )) dF ′ (70)where the right-hand side can be evaluated if the functions b i ( F ) are known.For the freely jointed chain (FJC) model of DNA, is established. b i ( F ) = (cid:20) coth (cid:18) F A i k B T (cid:19) − k B T F A i (cid:21)(cid:18) FK i (cid:19) b maxi (71)where K i , A i and b maxi are, respectively, the elastic modulus, the persistencelength and the average length of a base pair in the absence of any force.Inserting the expression (71) into the equation (70) we numerically computethe free energy difference between single base pair of dsDNA and that of ssDNAfor the given force F . In figure 5 we plot ∆Φ ′ against the tension F . Thenumerical values of the parameters that we use for this computation are givenin the table 2.We now assume that change in the barrier height ∆Φ( F ) that enters into theequation (3) is equivalent to n ∆Φ ′ ( F ) where n > ′ ( F ) is the stetching free energy difference between ssDNAand dsDNA i.e., between the initial state 3 and final state 4 of the transitionfor which the barrier height, i.e., the free energy difference between state 3and the transition state, contains the force-induced extra term ∆Φ( F ). Ourassumption ∆Φ( F ) = n ∆Φ ′ ( F ) ( n >
1) is similar, in spirit, but not identical tothe assumptions made by Wuite et al. [18] and Maier et al.[19]. The physicalmeaning of this assumption is that the tension-induced change ∆Φ( F ) of thebarrier arising from the legnth mismatch between the ssDNA and dsDNA basepairs is equivalent to n times the free energy difference ∆Φ ′ ( F ). Atomisticexplanation of the physical origin of the tension-induced change n ∆Φ ′ ( F ) of18arameter values b max .58 nm b max .34 nm A .7 nm A
50 nm K
900 pN K ′ using equation (70) and (71). F (pN) -2.5-2-1.5-1-0.50 ∆ Φ ’( F ) / k B T Figure 5: Free energy difference ∆Φ ′ is plotted against the tension F .19he activation energy would require more fine-grained modeling of the localneighborhood of the catalytic site [22] which is beyond the scope of the Markovkinetic models of the type developed in this paper. In our numerical calculationswe use n = 3 which is consistent with the best fit values reported in refs.[18, 19].The parameter value n = 3 should not be confused with the step size of theDNAP which is 1 nucleotide.The polymerase and exonuclease catalytic sites are separated by about 3.5nm. A DNA molecule migrating from the polymerase site to the exonucleasesite of DNAP would cause local melting of more than one termial base pairs[21, 32, 33, 34]. Therefore, based on arguments similar to those used earlier forthe rate constant k ( F ), we now expect ∆Φ x ( F ) = m ∆Φ ′ ( F ). Since no furtherinformation is available to fix the numerical value of m , we use m = 3 becausethis choice provides the best fit to the experimenta data [18].WLC model provides a slightly better quantitative estimate of the forceextension curve of dsDNA in the range of 0 to 10 pN force [41]. However,given the uncertainties of the other parameters used for plotting our resultsgraphically, the simpler FJC model is good enough. Indeed, it produces the nonmonotonicity of ∆Φ( F ) as a function of force (F) which, in turn, can be usedto estimate the mean rate of elongation as well as the conditonal dwell times. Appendix B x = 1 (72) x = k + x k − k − + k (73) x = k k ( k + k − ) + k − k − ( k − + k )( k + k − )( k − + k )( k − + k ) − k − k ( k − + k ) − k k − ( k + k − )(74) x = k − + x k k + k − (75) x = k x k p (76) Appendix C α = 1 (77) β = k + k + k + k + k − + k − + k − + k − + k p + k x + k exo (78) γ = k k + k k + k k + k k + k k + k k + k exo ( k + k + k + k + k x + k − + k − + k − + k − ) + k p ( k + k + k + k + k − + k − + k − + k − ) + k x ( k + k + k + k − + k − + k − ) + k − k − + k − k − + k − k − + k − k − + k − k − + k − k − + k ( k − + k − )+ k ( k − + k − ) + k ( k − + k − ) + k ( k − + k − + k − ) (79)20 = k k k + k k k + k k k + k k k + k − k − k − + k − k − k − + k − k − k − + k − k − k − + k exo ( k k + k k + k k + k k + k k + k k + k − k − + k − k − + k − k − + k − k − + k − k − + k − k − )+ k exo k x ( k + k + k + k − + k − + k − ) + k exo ( k k − + k k − + k k − + k k − + k k − + k k − + k k − + k k − + k k − ) + k p ( k k + k k + k k + k k + k k + k k + k − k − + k − k − + k − k − + k − k − + k − k − + k − k − + k k − + k k − + k k − + k k − + k k − + k k − + k k − + k k − + k k − ) + k x ( k k + k k + k k + k − k − + k − k − + k − k − + k k − + k k − + k k − + k k − ) + k k k − + k k k − + k k − k − + k k k − + k k − k − + k k k − + k k k − + k k k − + k k − k − + k k − k − + k k − k − + k k − k − (80) ǫ = k k k k + k − k − k − k − + k exo ( k k k + k k k + k k k + k k k + k − k − k − + k − k − k − + k − k − k − + k − k − k − ) + k exo k x ( k k + k k + k k + k k − + k k − + k k − + k − k − + k k − ++ k − k − + k − k − ) + k exo ( k k k − + k k k − + k k − k − + k k k − + k k − k − + k k k − + k k k − + k k k − + k k − k − + k k − k − + k k − k − + k k − k − ) + k p ( k k k + k k k + k k k + k k k + k − k − k − + k − k − k − + k − k − k − + k − k − k − + k k k − + k k k − + k k − k − + k k k − + k k − k − + k k k − + k k k − + k k k − + k k − k − + k k − k − + k k − k − + k k − k − ) + k x ( k k k + k − k − k − + k k k − + k k − k − ) + k k − k − k − + k k k − k − + k k k k − (81) ζ = k exo ( k k k k + k k k k x + k k k x k − + k k x k − k − + k x k − k − k − + k k k k − + k k k − k − + k k − k − k − + k − k − k − k − )+ k p ( k k k k + k k k k − + k k k − k − + k k − k − k − + k − k − k − k − ) (82) Appendix D a = k k k ( k exo + k p ) (83) a = k k k (84) b = ( k exo + k p )( k k k + k k k − + k k − k − + k − k − k − ) (85) b = k k k + k k k − + k k − k − + k − k − k − + k exo ( k k + k k + k k + k − k − + k − k − + k − k − + k k − + k k − + k k − + k k − )+ k p ( k k + k k + k k + k − k − + k − k − + k − k − + k k − + k k − + k k − + k k − ) (86)21 = k k + k k + k k + ( k exo + k p )( k + k + k + k − + k − + k − )+ k k − + k k − + k k − + k − k − + k k − + k − k − + k − k − (87) b = k + k + k + k exo + k p + k − + k − + k − (88) b = 1 (89) c = k x ( k k k + k − k − k − + k k k − + k k − k − ) (90) c = k x ( k k + k k + k k + k k − + k k − + k k − + k − k − + k k − + k − k − + k − k − ) (91) c = k x ( k + k + k + k − + k − + k − ) (92) c = k x (93) d = k exo ( k k k + k k k x + k k x k − + k x k − k − + k k k − + k k − k − + k − k − k − ) + k p ( k k k + k k k − + k k − k − + k − k − k − ) (94) d = k k k + k − k − k − + k exo ( k k + k k + k k + k k − + k k − + k k − + k k − + k − k − + k − k − + k − k − ) + k exo k x ( k + k + k − + k − ) + k p ( k k + k k + k k + k k − + k k − + k k − + k k − + k − k − + k − k − + k − k − ) + k x ( k k + k k − + k − k − )+ k k k − + k k − k − (95) d = k k + k k + k k + k exo ( k + k + k + k x + k − + k − + k − )+ k p ( k + k + k + k − + k − + k − ) + k x ( k + k + k − + k − )+ k k − + k k − + k − k − + k k − + k − k − + k − k − + k k − (96) d = k + k + k + k exo + k p + k x + k − + k − + k − (97) d = 1 (98) e = k − k − k − ( k exo + k p ) (99) e = k − k − k − (100) f = k k k k + k − k − k − k − + k x ( k k k + k k k − + k k − k − + k − k − k − ) + k k k − k − + k k − k − k − + k k k k − (101) f = k k k + k k k + k k k + k k k + k k k − + k k k − + k k − k − + k k k − + k k − k − + k − k − k − + k k k − + k k k − + k k k − + k k − k − + k k − k − + k k − k − + k − k − k − + k k − k − + k − k − k − + k − k − k − + k x ( k k + k k + k k + k k − + k k − + k k − + k − k − + k k − + k − k − + k − k − ) (102)22 = k k + k k + k k + k k + k k + k k + k k − + k k − + k k − + k k − + k − k − + k k − + k k − + k − k − + k − k − + k k − + k k − + k k − + k − k − + k − k − + k − k − + k x ( k + k + k + k − + k − + k − ) (103) f = k + k + k + k + k x + k − + k − + k − + k − (104) f = 1 (105) g = k p ( k k k + k k k − + k k − k − + k − k − k − ) (106) g = k p ( k k + k k + k k + k k − + k k − + k k − + k − k − + k k − + k − k − + k − k − ) (107) g = k p ( k + k + k + k − + k − + k − ) (108) g = k p (109) References [1] Howard J 2001
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