Accelerated search kinetics mediated by redox reactions of DNA repair enzymes
aa r X i v : . [ q - b i o . S C ] M a y Accelerated search kinetics mediated by redox reactions of DNA repair enzymes
Pak-Wing Fok ∗ Applied and Computational MathematicsCalifornia Institute of Technology,Pasadena, CA 91125 andDept. of BiomathematicsUniversity of California, Los Angeles, CA 90095-1766 Tom Chou † Depts. of Biomathematics and Mathematics,University of California, Los Angeles, CA
Abstract
A Charge Transport (CT) mechanism has been proposed in several papers ( e.g. , Yavin et al.
PNAS
Key words:
Repair Enzymes; Charge Transport; Target Search; Lesion; DNA
The genomes of all living things can be damaged by ionizing radiation and oxidative stress. These factors can causemismatches in the DNA strand resulting in localized lesions. The role of Base Excision Repair (BER) enzymesis to locate and remove these lesions. If the lesions are allowed to persist, they can give rise to mutations andultimately diseases such as cancer.The localization of BER enzymes to lesions is physically related to the binding of transcription factors topromoter regions that regulate gene expression. In 1970, experiments by Riggs et al. [1, 2] showed that theassociation rate of the LacI repressor to its operator is about 100 times faster than the maximum rate predictedby Debye-Smoluchowski theory. This theory assumes that LacI is transported to its target on DNA via 3Ddiffusion. To explain the experimental observations, the theory was modified to account for facilitated diffusion[3, 4, 5]. In this process, the LacI repressor can spend part of its time attached to the DNA and perform a 1Drandom walk before detaching and diffusing in 3D again (see Fig. 1). Provided the protein spends about half itstime on the DNA and half its time in solution and the diffusivities in 1D and 3D are comparable, the predictedsearch time can be reduced by as much as 100 fold [6]. However, these conditions are very restrictive as theprotein can spend up to 99.99% of its time associated to the DNA [7] and the diffusion constant along DNA (in1D) is in general much smaller than the one in the cytoplasm (in 3D) [8]. Therefore, many modifications of thebasic facilitated diffusion theory have been proposed, including intersegmental transfers [9], the effect of DNAconformation [10], directed sliding [11], and finite protein concentration [12].A series of recent papers [13, 14, 15] have revealed a special kind of long-ranged interaction for certain BERenzymes based on charge transport (CT) along DNA. MutY, a type of DNA glycosylase, contains a [4Fe-4S] ∗ Dept. of Biomathematics, UCLA, Los Angeles, CA 90095-1766 † Dept. of Biomathematics, UCLA, Los Angeles, CA 90095-1766, Tel: 310-206-2787 ccelerated Search of DNA Repair Enzymes cluster which is very sensitive to changes in its environment. Specifically, its redox potential is modified dependingon whether it is in a polar environment (when the enzyme is in solution) or in a more hydrophobic one (when theenzyme is attached to DNA). In solution, the [4Fe-4S] cluster is resistant to oxidation. However, when attachedto DNA, the cluster is more easily oxidized through the reaction [4Fe-4S] → [4Fe-4S] + e − . Furthermore, the3+ form has a binding affinity about 10 ,
000 times greater than the 2+ form [15].A model for the “scanning” of BER enzymes along DNA, aided by CT, was proposed in [13, 14, 15], and isdepicted in Fig. 2. When a BER enzyme adsorbs to DNA, it oxidizes and releases an electron along the strand(see Fig. 2(a)). Distal enzymes, already adsorbed onto the DNA can absorb these electrons, become reduced anddesorb. Hence, binding and unbinding of enzymes are associated with oxidation and reduction of their iron-sulfurclusters. CT along DNA can be disrupted by the presence of defects that affect electron transport. For example,guanine radicals (“oxoGs”), formed under oxidative stress, can absorb electrons: see Fig. 2(b). By acting assites of reduction, they promote the adsorption of BER enzymes [13, 16]. Once the radical has absorbed anelectron, it converts to a normal guanine base and no longer participates in CT. However, “permanent” defects,or lesions, can also exist on DNA which can absorb more than a single electron (see Fig. 2(c)). For example,oxoGs can erroneously pair with adenine bases when the DNA replicates. Such lesions may continuously absorbelectrons with a certain probability, or otherwise reflect them. In contrast to the oxoG-cytosine case, the removalof oxoG-adenine lesions require MutY to be present at the damaged site.In this paper, we develop a model of CT-mediated BER enzyme kinetics that includes enzyme diffusion alongDNA, a binding rate that depends on electron dynamics, and the effects of finite enzyme copy number. Ourkey finding is that the proposed charge transfer mechanism employed by BER enzymes accelerates their searchfor targets along DNA in real finite enzyme copy number systems. In the next section, we derive the governingequations of enzyme kinetics. These equations are rendered non-dimensional and key parameters are defined andestimated. In Section 3, we numerically solve our model equations under various conditions and estimate the timefor the binding of an enzyme to a localized lesion. We conclude with a discussion of our results in Section 4.
Consider the diffusion and adsorption-desorption kinetics of repair enzymes in a bacterium such as
E. coli : seeFig. 1. The chromosome in bacteria is circular but tightly coiled up into a nucleoid that has an effective volumeof about 8 × nm . If a repair enzyme is associated with the DNA strand, it can diffuse freely along the DNAto find lesions. These associated enzymes can spontaneously desorb from the strand, but they can also becomeoxidized, leading to tighter binding to the DNA. If later on, the enzyme is reduced, its association with the DNAweakens and it can quickly dissociate from the DNA. Localized lesions prevent the passage of electrons (releasedalong the DNA by oxidation of associated repair enzymes) by either reflecting or absorbing them.We write mass-action equations for the reactions occuring in Fig. 3, coupled to equations that determine theelectron dynamics. We assume that the enzyme density in the bulk, R b ( t ) (where t is time), is well mixed andhas no spatial dependence. The density of DNA-adsorbed BER enzymes in the reduced and oxidized state aredenoted by R a ( x, t ) and Q ( x, t ) respectively, where 0 ≤ x ≤ L is the coordinate along the DNA and lesions arelocated at x = 0 and x = L . The density of guanine radicals is g ( x, t ) and the density of rightward and leftwardelectrons is N + ( x, t ) and N − ( x, t ). Note that R b ( t ) has units of inverse volume, while R a ( x, t ), Q ( x, t ), N ± ( x, t ),and g ( x, t ) carry units of inverse length. The governing equations corresponding to the processes depicted in Figs.2 and 3 are ccelerated Search of DNA Repair Enzymes ∂Q ( x, t ) ∂t = D + ∂ Q∂x − v ( N + + N − ) Q + mR a , (1) ∂R a ( x, t ) ∂t = D − ∂ R a ∂x + v ( N + + N − ) Q − k off R a + k on (cid:18) Ω L (cid:19) R b − mR a , (2) dR b ( t ) dt = − k on R b + k off Ω Z L R a dx, (3) ∂N + ( x, t ) ∂t + v ∂N + ( x, t ) ∂x = f N − − f N + − vN + ( Q + g ) + mR a , (4) ∂N − ( x, t ) ∂t − v ∂N − ( x, t ) ∂x = − f N − + f N + − vN − ( Q + g ) + mR a , (5) ∂g ( x, t ) ∂t = − v ( N + + N − ) g. (6)These equations must be solved subject to the boundary conditions N + (0 , t ) = rN − (0 , t ) , N − ( L, t ) = rN + ( L, t ) ,Q (0 , t ) = Q ( L, t ) = 0 , R a (0 , t ) = R a ( L, t ) = 0 , (7)and initial conditions Q ( x,
0) = 0 , R a ( x,
0) = 0 , R b (0) = n / Ω ,N + ( x,
0) = 0 , N − ( x,
0) = 0 , g ( x,
0) = g /L. (8)In Eqs. 1-6, D + is the diffusivity of adsorbed MutY along the DNA, D − is the diffusivity of adsorbed MutY , v is the speed of electrons along DNA, m is the electron release (oxidation) rate of adsorbed MutY , k off is theintrinsic desorption rate of MutY , k on is the intrinsic adsorption rate of MutY to the DNA from solution, Ωis the cell volume, L is the arclength of the DNA, and f is the electron flip rate (see below). In Eqs. 7, r is theelectron reflectivity of lesions, which we describe in more detail later. In Eqs. 8, n is the copy number of MutY,and g is the initial number of guanine radicals on the DNA. The definitions of all constants are summarized inTable 1.We now give a brief justification of equations 1-6 and conditions 7 and 8. The form of the first three equationscan be understood from Fig. 3(b) which summarizes the reactions among the three species R b , R a , and Q . Eq.1 describes the time rate of change of adsorbed MutY due to oxidation of adsorbed MutY (+ mR a ) andreduction by incoming electrons ( − v ( N + + N − ) Q ). The first term on the right hand side represents diffusion alongthe DNA. Eq. 2 describes the evolution of adsorbed MutY in terms of the reduction of MutY (+ v ( N + + N − ) Q ),spontaneous desorption into solution ( − k off R a ), adsorption of aqueous MutY ( k on (Ω /L ) R b ) and oxidation intoMutY ( − mR a ). Since MutY binds to DNA less strongly than MutY , it is possible that D + is appreciablysmaller than D − . Eq. 3 is an equation for the concentration of MutY in solution which can decrease by enzymesbinding to the DNA ( − k on R b ) and increase by enzymes unbinding from the DNA (represented by the integralterm). Because we assume enzymes in the bulk solution are well mixed, any increases in bulk concentration aredue to an integrated DNA-adsorbed density which does not distinguish between enzymes that are released fromdifferent positions along the DNA, but only sees the total number of enzymes that desorb.Eqs. 4 and 5 describe the electron dynamics. In our model, right and left-moving electrons (see Fig. 3(a))propagate along the DNA with speed v ; this process is represented by the two convective terms on each of the lefthand sides. Also, electrons are lost when they are absorbed by MutY or by guanine radicals, and produced whenreleased by adsorbed MutY . These processes are represented by the third and fourth terms on the right handside of Eqs. 4 and 5, respectively. Finally, leftward and rightward electrons can inter-convert [16] by scattering offinhomogeneities and thermally induced conformational changes in the DNA [25, 26]. This process is representedby the first and second terms on the right hand side. The flip rate f characterizes how frequently a travelingelectron changes direction. If f is large, the electron move diffusively, but if f is small, it moves in a more ballisticmanner. Finally, Eq. 6 represents the evolution of the guanine radical population. OxoGs are annihilated when The mean free path of an electron is estimated to be λ ∼ −
10 base pairs and the flip rate approximated as v/λ . Estimated using the time taken for the restriction endonuclease BsoBI to unbind from DNA [23], t off =150s and taking k off = 1 /t off .This value of t off may not be an accurate value for the unbinding time for MutY. Assumes that about 1 in 40,000 guanine bases are oxoGs [24] and the length of
E. coli
DNA is L = 5 × bp. ccelerated Search of DNA Repair Enzymes D ± Diffusivity of adsorbed enzymes 5 × bp /s [17] v Electron velocity 10 bp/s [18] f Electron flip rate 10 − s − m Electron release rate ∼ s − [19]Ω Bacterium volume 3 . × nm [20] L Length of DNA 5 × bp k on MutY attachment rate 2000 s − [16] k off MutY detachment rate 7 × − s − n Copy number of MutY in
E. coli r Electron reflectivity of lesions 0 – 1 - g Number of oxoGs on
E. coli
DNA ∼ Table 1: Key constants for used for repair enzyme model Eqs. 1- 6 and conditions 7,8.Parameter Definition Calculated Value η D + / ( k on L ) . − U v/ ( k on L ) 5 σ m/ ( m + k off ) ∼ F f /k on × − × Table 2: Dimensionless parameters in Eqs. 10-14 ccelerated Search of DNA Repair Enzymes they absorb electrons as represented by the − v ( N + + N − ) g term. Radicals might also be spontaneously generatedand modeled by a source term on the right hand side of Eq. 6. In this paper, we neglect spontaneous oxoGsgeneration.Equations 1-8 use a mean field approximation that neglects stochastic fluctuations in enzyme, electron andguanine number. The effect of noise in the system could be included through the use of a chemical master equation[27]; however, generalizing the equation to account for spatial variations along the DNA is beyond the scope ofthis paper [28]. Nonetheless, we expect our results for lesion targeting by enzymes will be qualitatively accurate.Lesions at x = 0 and x = L (see Fig. 3(a)) define the domain of solution for Eqs. 1-6 which are subject to theboundary conditions 7. This domain can represent a circular DNA with circumference L containing a single lesion.In the first equation of 7, leftward traveling electrons are converted to rightward traveling ones by the lesion thatreflects leftward electrons with probability r . If r = 0, leftward electrons are absorbed by the lesion. On the otherhand, if r = 1, the lesion is fully reflective and the rightward and leftward electron densities are equal. Similarconsiderations apply to the lesion at x = L . Since we will eventually use our mean-field mass action equations toestimate the mean time for a repair enzyme to find a lesion, we assume that the lesions are perfectly “absorbing”and set Q = R a = 0 at the lesion positions. Our simulations are performed on a domain with g oxoG radicalsand a bulk solution that contains n enzymes (see Eqs. 8); hence the adsorbed oxoG density is g /L and the bulkconcentration is n / Ω. Before non-dimensionalizing Eqs. 1-6 we can make one important simplification. On the right hand side of Eq. 2,the sizes of the second, third, fourth and fifth terms are approximately v/L , k off /L , k on /L and m/L (in units ofbp − s − ) respectively. Guided by Table 1, we assume the term mR a dominates. Since the oxidation rate is large,adsorbed MutY quickly oxidizes into the 3+ form upon adsorption onto DNA. For times t ≫ /m and rates k off + m ≫ v/L, k on Eq. 2 gives R a ( x, t ) ≪ ≤ x ≤ L and we can neglect spatial gradients in R a as wellas ∂R a /∂t . Therefore, we approximate Eq. 2 with R a ( x, t ) ≈ m + k off (cid:18) v ( N + + N − ) Q + k on (cid:18) Ω L (cid:19) R b (cid:19) . (9)Upon substitution of Eq. 9 into Eqs. 1, 4 and 5, we eliminate the equations for R a and find a reduction analogousto one commonly used in deriving the steady-state limit of Michaelis-Menten kinetics [29].We now non-dimensionalize our equations by measuring time in units of k − , length in units of L , concentrationof adsorbed species in units of 1 /L and concentration of bulk species in units of 1 / Ω. Our final set of reducedand nondimensionalized equations that describe the transport and kinetics of MutY repair enzymes, right andleft-moving electrons, and guanine radicals is ∂Q ( x, t ) ∂t = − U (1 − σ )( N + + N − ) Q + η ∂ Q∂x + σR b , (10) dR b ( t ) dt = U (1 − σ ) Z ( N + + N − ) Qdx − σR b , (11) ∂N + ( x, t ) ∂t + U ∂N + ( x, t ) ∂x = F ( N − − N + ) − gU N + + σR b − (cid:16) − σ (cid:17) U N + Q + σ U N − Q, (12) ∂N − ( x, t ) ∂t − U ∂N − ( x, t ) ∂x = − F ( N − − N + ) − gU N − + σR b σ U N + Q − (cid:16) − σ (cid:17) U N − Q, (13) ∂g ( x, t ) ∂t = − U ( N + + N − ) g, (14)where we have defined the dimensionless quantities η = D ± k on L , U = vk on L , F = fk on , (15) ccelerated Search of DNA Repair Enzymes and σ ≡ mm + k off , (16)which can be estimated using Table 2. As we discuss later, the parameter σ represents the effective binding ratein terms of the competition between the electron release rate m and the desorption rate of DNA-bound MutY k off , and lies between 0 and 1. The dimensionless boundary and initial conditions are N + (0 , t ) = rN − (0 , t ) , N − (1 , t ) = rN + (1 , t ) , Q (0 , t ) = Q (1 , t ) = 0 , (17)and Q ( x,
0) = 0 , R b (0) = n ,N + ( x,
0) = N − ( x,
0) = 0 , g ( x,
0) = g . (18)Our model can approximate the case of “infinite” enzyme copy number when the transport of bulk enzymes isdiffusion-limited. Although most of the enzymes cannot immediately adsorb onto the DNA as they are too faraway, we assume that a certain number, R b , are in the vicinity of the nucleoid, say within a volume Ω ′ (see Fig.1), and are able to directly engage in adsorption. However, instead of being depleted over time, R b is continuouslyreplenished by far enzymes that diffuse into Ω ′ ⊂ Ω to keep R b fixed. Therefore, to obtain the infinite copy numberlimit, we hold R b constant in Eqs. 10, 12 and 13 and Eq. 11 no longer applies. To summarize, we model theinfinite copy number case by holding R b constant. In the finite copy number case, R b ( t ) is allowed to vary in timethrough Eq. 11. Finally, note that equations describing a simple diffusing enzyme that does not undergo CT canbe recovered from Eqs. 10-14 by setting U = 0. In this case, the equations for Q ( x, t ) and R b ( t ) decouple fromthe rest. σ In Eqs. 10-14, the rate of creation of reduced, adsorbed enzyme R a from reduced bulk enzyme R b is exactly R b since we measure time in units of 1 /k on . However, the overall rate of the compound reaction R b ⇋ R a → Q is σR b . Consider a MutY that is adsorbed onto the DNA. If it absorbs an incoming electron, it can either desorbinto the bulk or it can release an electron back along the DNA and remain oxidized. The parameter σ in Eq. 16 isthe probability of electron release. When k off ≫ m , a MutY that absorbs an electron will preferentially desorb( R a → R b ), but when k off ≪ m a MutY will simply release the electron it just absorbed to stay adsorbed ontothe DNA ( R a → Q ). These limiting behaviors are realized by taking σ → σ → σ ∼
0, a bulk reduced enzyme that adsorbs onto the DNA quickly desorbs back into the bulk, while if σ ∼ on the DNA prefers to oxidize and stay adsorbed rather than go into solution. Once it is oxidized, anyfurther electrons that are absorbed will be re-emitted in a random direction. Hence the electron changes directionwith probability 1 / : when σ = 1, the terms with prefactors (1 − σ/ σ/ F ( N − − N + ) terms to yield an effective flip rate of F + U Q/
2. The seedingof oxidized enzymes on the DNA increases the effective electron flipping rate because these enzymes can absorbelectrons and immediately release them back along the DNA in the direction they came from or in the directionthey were going.We end this section with the comment that the model for the CT redox process in Fig. 2 is not exactly equivalentto the reaction scheme in Fig. 3(b). In Fig 2, a bulk MutY ( R b ) adsorbs onto a DNA and immediately oxidizes,releasing an electron along the DNA. DNA-bound MutY ( Q ) remains adsorbed until it absorbs an incomingelectron, whereupon it reduces and immediately desorbs into the bulk. For this model to hold, the reaction kineticsin Fig. 3(b) must be non-Markovian. Specifically, consider the intermediate quantity R a in Fig. 3(b). An R a enzyme oxidizes to a Q enzyme ( R a → Q ) only if it ‘remembered’ that it was originally created via a R b → R a reaction. Likewise, an R a enzyme desorbs ( R a → R b ) only if it ‘remembered’ that it was originally created througha Q → R a reaction. We now compute and analyze solutions to Eqs. 10-14 for the infinite and finite copy number cases. The equationsare solved numerically using second order finite differences on a non-uniform grid that clusters grid points nearthe boundaries and a trapezoidal rule to approximate the integrals. MATLAB’s stiff solver ode15s was used tointegrate the equations in time. In the infinite case R b is held at the value n and in the finite case, R b ( t ) is ccelerated Search of DNA Repair Enzymes included in the dynamics with initial condition R b (0) = n . Furthermore in each case we consider the dynamicsassociated with CT enzymes where U >
0, and the dynamics associated with “passive”, non-CT enzymes where U = 0. Setting U = 0 decouples the equations for electron and guanine radical dynamics (Eqs. 12-14) from theequation for Q ( x, t ), the density of DNA-bound enzymes (Eq. 10).We shall explore the behavior of Eqs. 10-14, and the associated search times defined below, with respect to: • σ , the effective binding affinity. Generally we have 0 < σ <
1. From the values of m and k off in Table 1, wehave σ ≈ − . This value of σ renders the desorption term − U (1 − σ )( N + + N − ) Q in Eq. 10 insignificant,making the effect of CT negligible. Therefore, a necessary requirement for an effective CT mechanism is that σ <
1. In our simulations for the MutY system, we take σ = 0 . k off inTable 1 is for BsoBI and not MutY. • η , the diffusivity of MutY along DNA. The value in Table 2 of η = 10 − is based on the diffusive slidingof a human glycosylase, hOgg1, which has a diffusivity of about 5 × bp /s [17]. However, this value maynot necessarily be an accurate value for MutY. Therefore we will explore a range of diffusivities η near 10 − . • g , the initial guanine radical density: There are about 30 oxoGs at any given time on E. coli
DNA, but thisnumber depends on environmental conditions. Hence we explore a range of values centered around g = 30. • r , the lesion reflectivity: the interaction between electrons and lesions depends on unknown molecular factorsat the lesion and in the bulk cytoplasm. Hence, we explore a full range of r values between 0 and 1. • F , the electron flip rate: the precise dynamics of electrons on DNA is a very complicated process; ourestimate for F in Table 2 makes many simplifications and may not be accurate. We will explore a range of F values centered around 10 .Figure 4 shows snapshots of adsorbed enzyme, guanine and electron density profiles for a finite enzyme copynumber ( n = 30) system. The profiles are shown near the lesion at x = 0 at times t = 2 and t = 5. Theelectron density is generally smaller at the lesions and larger in the middle of the domain, resulting in a largerenzyme desorption rate away from lesions (the desorption rate in Eq. 10 is proportional to the total electrondensity N + + N − ). Thus, the CT enzyme density is smaller than that for passive enzymes away from lesions. Theenhanced desorption of CT enzymes from the interior continuously replenishes the number of enzymes in solutionso that R b ( t ) decreases less rapidly than for passive enzymes. For intermediate times, the net deposition rate islarger for CT enzymes, the enzyme density near the lesions is also larger (Fig. 4(a)) and grows in time (Fig. 4(b)).For long times, the density vanishes everywhere: this is the trivial steady state solution to Eqs. 10-14.Figure 5 shows the DNA-bound enzyme density at t = 40. In (a), there is a sharp spike in the enzyme densitynear the lesion at x = 0, but otherwise the enzyme density is relatively small. Note that all densities are symmetricabout x = 1 /
2. In Eq. 10, CT enzymes desorb with a rate proportional to the total electron density N + + N − . Asseen in (a), this density is smallest at the lesions. Therefore the enzyme density near x = 0 and x = 1 grows morequickly compared to the interior density. The inset shows a rapid variation in Q of about 600 within a boundarylayer of width ∼ − . Using a non-uniform grid that clusters the mesh points near the boundaries, we are ableto resolve these boundary layers to calculate the flux of enzymes through the lesions. In Fig. 5(b), CT ( U = 1)and passive ( U = 0) enzyme densities are compared when the copy number is finite. The CT-enzyme density hassharp maxima near the lesions, while the passive enzyme density does not. Compared to the infinite copy numbercase, the size of the maxima is smaller since the number of enzymes in the bulk (and hence the deposition rate)decreases with time. Because of the maxima, the flux of enzymes into the lesion, J ( t ) = η (cid:20) ∂Q ( x, t ) ∂x + ∂R a ( x, t ) ∂x (cid:21) x =0 (19)is greater compared to the passive case. Figure 5(c) compares the current for CT ( U = 1) and passive enzymes( U = 0) when the copy number is infinite. The passive enzyme current is always greater than the CT enzymecurrent because for a constant deposition rate, any desorption reduces the number of enzymes on the DNA and theflux of enzymes into the lesion. Therefore, for infinite copy number systems, search by passive enzymes will alwaysbe faster than CT enzymes. In contrast, when the bulk contains a finite number of enzymes, Fig. 5(d) shows thatthe CT current is always greater than the passive current. This is due to free electrons on the DNA that determinethe local desorption rate. In the CT mechanism, enzymes are knocked off the DNA by incoming electrons and onaverage, desorb from lesion-free portions of the DNA and re-adsorb at positions closer to the lesion . The result isthat for intermediate times ( t & ccelerated Search of DNA Repair Enzymes for the passive case. Ultimately, we have J ( t ) → t → ∞ for both passive and CT enzymes, but CT ensuresthat this behavior occurs at a much later time.Next, we consider the typical time for the first enzyme to reach a lesion. Since the enzyme density is symmetricabout x = 1 /
2, the total flux can be found by using twice the enzyme flux to one lesion defined in Eq. 19. Thetypical search time τ s is then approximated by integrating 2 J ( t ) until one enzyme has diffused into the lesion: Z τ s J ( t ) dt ≈ . (20)From solving the full set of equations 1-6 numerically, we find that the gradients in R a ( x, t ) at the lesions are neg-ligible compared to those of Q ( x, t ), verifying the validity of eliminating R a and using J ( t ) ≈ η ( ∂Q ( x, t ) /∂x ) x =0 as the total enzyme current. In the mean field limit, an alternative definition of the search time is τ s ≈ R ∞ t exp h − R t J ( t ′ )d t ′ i d t . We have computed τ s using this mean field approximation and find negligible qualita-tive differences from τ s computed using Eq. 20.Figure 6(a) shows that the search times are extremely sensitive to the initial number of oxoGs g . In particular,there is a rapid increase in τ s as g increases past the enzyme copy number n = 30. The CT mechanism relies onthe presence of free electrons that cause enzymes to desorb from lesion-free portions of the strand and re-adsorbnear lesion sites, while oxoGs suppress CT by absorbing free electrons. When g > n , all enzymes from the bulkadsorb onto the DNA and any released electrons are absorbed by nearby guanine radicals. Instead of participatingin CT-mediated redistribution and localization, the enzymes cannot desorb and must rely on slow diffusive slidingalong the DNA strand to find their targets. When g < n , at least one enzyme is always in solution and istransported through the cytoplasm. Since 3D transport is assumed to be fast, the search time is correspondinglysmall. Also in this plot, τ s increases as r increases but the search time is much more sensitive to g : the searchtime changes by about 20% for g ≈ .
05% for g ≈
50 over the whole range of r . In our model,the search time is not greatly affected by whether lesions reflect or absorb electrons; what is important is that thelesions prevent their passage along the DNA.Figure 6(b) shows that for the range of η values explored, there is a value 0 < σ ∗ < τ s is minimum. To understand why there is an optimal σ ∗ , consider the CT mechanism’s dependence onthe binding affinity σ . If σ = 1, enzymes strongly bind onto the DNA. Even when they absorb electrons, theywill re-emit them to stay adsorbed on the strand. Hence, there is no desorption, no fast transport through thecytoplasm and acceleration of the search. On the other hand, if σ = 0, enzymes do not stay on the DNA longenough to even slide into lesions and the search is correspondingly slow. Our results show that the search isoptimal when the enzyme is weakly associated with the DNA i.e. < σ ≪ η to clearly show the minimum with respect to σ . For smaller η values we have σ ∗ → + , but the dependence of τ s on σ does not change qualitatively. For larger η values, enzymes do not rely on CT to localize to lesions andcan find their targets quickly using diffusive sliding. In this case, the search is most efficient if as many enzymesas possible adsorb on the DNA; this situation is realized by taking σ = 1 and τ s monotonically increases as σ getssmaller.Figure 7(a) shows how the search time varies as a function of 1D enzyme diffusivity along the DNA. Noticethat the search time for CT enzymes ( U = 1) is much smaller than that for passive enzymes ( U = 0). Indeed, τ s can be reduced by several orders of magnitude when the effects of CT are included. If fewer oxoGs are initiallypresent, the search occurs more quickly. Consistent with Fig. 6(a), the search time is extremely sensitive to theinitial number of guanine radicals on the DNA. For passive enzymes, τ s scales as O ( η − ). For CT enzymes, the O ( η − ) behavior switches to τ s = O ( η − / ) for sufficiently large η with the crossover dependent on g .For finite copy number Fig. 7(b) again shows that the search time decreases if CT is included, but this timefor different flip rates. For the large values of F used in Fig. 7, one can show that the effective diffusion coefficientof the electron density scales as 1 /F [16]. Therefore, as F increases the electron density dissipates more slowlythrough the partially absorbing lesions. A greater density of free electrons implies more enzyme desorption, moretransport through the cytoplasm and faster search times. In the F → ∞ limit, we expect the enzymes to self-desorb independently of the oxoG density. This can be seen from Eqs. 12 and 13 where the dominant terms onthe right hand side are ± F ( N − − N + ) and σR b /
2. In principle, as F → ∞ , one can approximate N ± in terms of R b ( t ) and substitute into − U (1 − σ )( N + + N − ) Q in Eq. 10 to further reduce the system to only two equationsfor Q ( x, t ) and R b ( t ). ccelerated Search of DNA Repair Enzymes Although both plots in Fig. 7 are for the finite copy number case, we also performed analogous simulationsfor the infinite copy number limit. We found that including the effects of CT by taking U = 1 always led to anincrease in the search time compared to the passive case: for fixed η and F , increasing U always increased τ s regardless of the value of g . Our key finding is that charge-transport (CT) dynamics mediated by redox reactions can significantly reducesearch times of repair enzymes in real cells where the copy number is finite and the diffusivity along the DNA issmall. In theoretical systems where the copy number is infinite, CT actually slows down the search. The speed-upin finite systems arises because of a spatially dependent desorption rate. Specifically, the desorption is greateralong intact portions of the DNA but smaller near lesions. As a result, CT-induced enzyme-enzyme interactions“recycle” enzymes so that they desorb from lesion-free parts of the DNA and reattach closer to lesion sites. Ourproposed mechanism is illustrated in Fig. 8. A related mechanism has been implicated in mRNA translationwhere ribosomes are recycled, enhancing protein production rates. [30]If we re-dimensionalizing the search times by using an estimated value of k on = 2000 s − , we find that passiveenzymes with diffusivity η ∼ − have long search times τ s of about 15 minutes (see Fig. 7), comparable to thethe life cycle time of E. coli . With the CT mechanism and g = 28 initial oxoGs, the search time drops to a fewseconds. For smaller values of η , the difference in search times between passive and CT enzymes becomes greater.Using g = 20, we calculate τ s to be about 30 hours and 2 seconds respectively for passive and CT enzymesthat have diffusivity η ∼ − . Therefore for realistic enzyme diffusivities, we think that CT is an indispensablemechanism that allows enzymes such as MutY to locate lesions on the DNA in a reasonable amount of time.When the initial number of oxoGs exceeded the enzyme copy number, we found a large increase in the searchtime. In this case, search takes place mostly through slow diffusive sliding along the DNA. However, when thecopy number (number of potential electron emitters) exceeds the number of electron absorbers, we find that thesearch time decreased drastically, with the search taking place mainly through the transport of enzymes throughthe cytoplasm. Therefore, we predict that the spontaneous generation of electron absorbing defects (such as oxoG)would significantly slow down the search and conversely, the presence of other redox-active proteins (such as thetranscription factor SoxR [31]) would speed up the search. Although such proteins may not be directly involved inlesion search, they may be upregulated when the cell is oxidatively stressed, increasing the population of electronemitters in the system. The iron-sulfur cluster responsible for CT in MutY is also found in other repair enzymeslike EndoIII [32]. Hence EndoIII could also participate in CT, emit electrons to promote the desorption of MutYand speed up the search.Recall that classical facilitated diffusion theory [3, 4, 5] predicts a large reduction in the search time of proteinsproviding equal amounts of time are spent in 1D and 3D. However, most proteins are strongly associated withDNA so that the speed up is not achieved in practice. The passive enzyme system considered in this study can bethought of as a suboptimal search by facilitated diffusion: with U = 0, a MutY that oxidizes and binds to the DNAcannot desorb back into the cytoplasm and the protein spends much more time diffusing in 1D. However, when U >
0, bound oxidized MutY can be knocked off the strand by electrons. CT therefore provides a mechanism forMutY to spend more time in 3D than it otherwise would. In other words, CT-aided MutY could be one systemwhere the conditions required for speed up are actually satisfied. In addition, when MutY binds near lesions, itmay diffusively slide along the DNA into its target: the target size is effectively increased with the DNA actinglike an antenna [10]. This antenna effect is enhanced by enzymes preferentially oxidizing and adsorbing onto partsof the DNA that are near lesion sites.Extensions to our model may include spatial gradients in the bulk enzyme concentration, more careful treatmentof electron dynamics, and adding fluctuations in copy number. Nonetheless, our simple deterministic modeldescribes mechanisms and yields results qualitatively consistent with findings in [13, 14].
References
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Structure ccelerated Search of DNA Repair Enzymes Figure Captions
Figure 1: (a) Target search on prokaryotic DNA, which is tightly coiled up into a nucleoid. Proteins in the bulk candiffuse to the DNA through the cytoplasm to locate their targets. (b) Searching proteins (hexagons) locate targets(diamonds) by sliding along DNA, punctuated by attachment and detachment. Ω represents the cell volume whileΩ ′ represents all points in the vicinity of the nucleoid. Enzymes within Ω ′ can engage in direct adsorption ontothe DNA.Figure 2: Charge Transport (CT) mechanism proposed in [13, 14, 15]. (a) A repair enzyme (in solution) is in the2+ state and adsorbs onto the DNA. Its iron-sulfur cluster oxidizes in the process, releasing an electron along theDNA. A repair enzyme (already adsorbed on the DNA) is in the 3+ state and accepts an incoming electron. Itsiron-sulfur cluster reduces and the enzyme desorbs. (b) Guanine radicals (“oxoGs”) can absorb free electrons onthe DNA. These radicals are annihilated upon absorbing an electron. (c) Lesions can partially reflect and absorbelectrons.Figure 3: (a) Summary of the CT model, described by Eqs. 1-6. Bulk enzymes, with density R b ( t ), can attach tothe DNA and oxidize to release rightward and leftward electrons with densities N + ( x, t ) and N − ( x, t ) respectively.Guanine radicals with density g ( x, t ) act as electron absorbers. Upon adsorption, oxidized enzymes with density Q ( x, t ) are formed with R a ( x, t ) as a transient, intermediate quantity. Fixed lesions are located at x = 0 and x = L . (b) Redox reaction diagram for the MutY repair enzyme. MutY in solution is represented by R b ( t ),MutY adsorbed onto DNA is represented by R a and MutY adsorbed onto DNA is represented by Q .Figure 4: Density profiles for enzyme, guanine, and electrons on DNA in a finite enzyme copy number system( R b (0) = n = 30) at time (a) t = 2 and (b) t = 5. Dashed lines show density profiles of passive enzymes in whichthe CT mechanism is absent. Parameters used were η = 10 − , σ = 0 . , F = 10 , and g = 28.Figure 5: Enzyme profiles and currents for infinite ((a) and (c)) and finite ((b) and (d)) copy numbers. In eachfigure, the profile or current is plotted for passive (dashed) enzymes where U = 0 and CT (solid) enzymes where U = 1. Insets show the large gradients in enzyme density within a thin boundary layer near the lesions. Parametersused were σ = 0 . η = 10 − , g = 28, r = 0 . F = 10 and n = 30.Figure 6: Search time τ s for Charge-Transport Enzymes, for copy number n = 30, electron flip rate F = 10 and electron speed U = 1. The actual search time in seconds can be recovered by dividing by k on , whose value isestimated in Table 1. (a) Search time as a function of initial guanine density g and lesion electron reflectivity r .Parameters used were σ = 0 . η = 10 − . (b) Search time τ s as a function of enzyme binding affinity σ andenzyme diffusivity along DNA, η . Parameters used were g = 30 and r = 0 . τ s of passive enzymes ( U = 0) compared with CT enzymes ( U >
0) as a function ofenzyme diffusivity η for various initial guanine densities g . Parameters used were σ = 0 . r = 0 . F = 10 and n = 30. (b) Search time of passive enzymes compared with CT enzymes for different electron flip rates F .Parameters used were σ = 0 . r = 0 . η = 10 − and n = 30. For both plots, the actual search time in secondscan be recovered by dividing by k on , whose value is estimated in Table 1. ccelerated Search of DNA Repair Enzymes Figure 8: Recyling of enzymes via the Charge-Transport mechanism. In a finite copy number system, the mecha-nism increases the enzyme desorption rate for intact portions of DNA but decreases it near lesions. Therefore onaverage, enzymes are “recycled” to lesion sites by 3D transport through the cytoplasm. In many cell systems, thismethod of finding lesions is faster than a 1D search by diffusive sliding. ccelerated Search of DNA Repair Enzymes Figures
Figure 1: ccelerated Search of DNA Repair Enzymes ccelerated Search of DNA Repair Enzymes ccelerated Search of DNA Repair Enzymes lesionlesion re−adsorption re−adsorptiondesorptionlesionlesion re−adsorption re−adsorptiondesorption