The spread of viruses in growing plaques predicted by classical models is greater than that measured experimentally. There is a widespread belief that this discrepancy is due to biological factors. Here we show that the observed speeds can be satisfactorily predicted by a purely physical model that takes into account the delay time due to virus reproduction inside infected cells. No free or adjustable parameters are used

D. Jou

D. Stollar

H.-W. Ackermann

J. Crank

J. Warnatz

J. Yin

J. C. Maxwell

J. D. Murray

Joaquim Fort

K. Shishido

M. Delbrück

P. Menozzi

T. D. Brock

U. Ebert

Vicenç Méndez

Y. Lee

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Time-Delayed Spread of Viruses in Growing Plaques

The spread of viruses in growing plaques predicted by classical models is greater than that measured experimentally. There is a widespread belief that this discrepancy is due to biological factors. Here we show that the observed speeds can be satisfactorily predicted by a purely physical model that takes into account the delay time due to virus reproduction inside infected cells. No free or adjustable parameters are use

Fort, Joaquim

Méndez López, Vicenç

DUGiDocs – Universitat de Girona

American Physical Society

Diposit Digital de Documents de la UAB

VOLUME 89, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 21 OCTOBER 2002Time-Delayed Spread of Viruses in Growing PlaquesJoaquim Fort1 and Vicenc¸ Me´ndez21Departament de Fı´sica, Universitat de Girona, Campus de Montilivi, 17071 Girona, Catalonia, Spain2Facultat de Cie`ncies de la Salut, Universitat Internacional de Catalunya, Gomera, s/n,08190 Sant Cugat del Valle`s, Catalonia, Spain(Received 11 March 2002; published 8 October 2002)178101-1The spread of viruses in growing plaques predicted by classical models is greater than that measuredexperimentally. There is a widespread belief that this discrepancy is due to biological factors. Here weshow that the observed speeds can be satisfactorily predicted by a purely physical model that takes intoaccount the delay time due to virus reproduction inside infected cells. No free or adjustable parametersare used.DOI: 10.1103/PhysRevLett.89.178101 PACS numbers: 87.18.Hfcontaining agar-immobilized, stationary-phase host bac-teria [13–15]. We deal with this case. Then, it is possible Vt  Bt  k1VB  It; (4)I. Introduction.—Reaction-diffusion fronts arise insolidification [1], superconductors [2], liquid crystals[3], and combustion [4]. They also appear in ecology[5], archaeology [6], genetics [7], and virology [8]. Buttime-delayed front speed predictions have been so farcompared to observations only in two applications,namely, the Neolithic transition in Europe [6] and theBlack Death plague [9]. Up to now, the delay time hasnever previously been taken into account in predicting thespeed of fronts that can be experimentally reproducedin the laboratory. This is done here for the first time. Adelay effect is used below to explain the replication andspread of viruses in growing plaques. This work is alsorelevant to physics, because wave-front solutions are im-portant in many fields where a delay time must also beincluded [10–12].We investigate the spreading dynamics of viruseswhich infect bacteria. It is shown that the present typeof models are inadequate as they do not take into accountthe time delay between the arrival of a virus into a celland the reproduction of the virus. We extend the presentmodels by including this effect. No free or adjustableparameters are used at all. Experiments independent tothose we want to explain are used to determine the valuesof all parameters. With the values thus obtained, we areable to explain the dependence for the front velocity as afunction of the bacterial concentration. This is the firsttype of evidence for the relevance of delay effects in frontpropagation experiments which can be repeated in thelaboratory.A plaque is a region of dead (i.e., lysed) cells. It formsdue to the adsorption of viruses to host cells, their repli-cation within, and the spread of the new generation afterlysis. Many virus phages can be successful only in amedium of growing cells. In such cases, plaques willstop growing after the bacteria reach an stationary phase.On the other hand, for viruses such as T7, it has beenobserved that plaques grow without bound in a medium0031-9007=02=89(17)=178101(4)$20.00 to measure the speed over a long period, but it does notagree with that predicted by existing models [16–18].II. Virus-cell interactions.—In a growing plaque, thereare three species. Their interactions can be summarizedby the reactionsV  B!k1 I !k2 Y  V; (1)where Y is the production (or yield) of new viruses (V) perinfected host (I), k1 is the rate constant of adsorption ofviruses (V) to uninfected bacteria (B), and k2 is the rateconstant for death (or lysis) of infected bacteria (I).Instead of presenting the equations of our model di-rectly, it will be much clearer to introduce them afterconsidering several very simple situations.Consider first a homogeneous medium, composed ini-tially of infected cells (I) and a few free viruses. As timegoes on, cells will die and viruses (V) will appear at arate determined by the second of the reactions (1). Thisprocess can be modeled by means of the usual logisticequations,Vt  YIt  Yk2I1 IImax; (2)where    denotes concentration, and the subindex  t  @=@t stands for the time derivative. The solutionto Eqs. (2) under the appropriate boundary conditions( limI!ImaxV  0 and limI!0V  Vmax) is easilyfound,V  YImax1 c1 exp	k2t ; (3)which describes well the corresponding experimentaldata and makes it possible to determine the values ofk2 and the yield Y (see Fig. 1). c1 is an integration con-stant which depends on the arbitrary definition of themoment t  0.Another very simple case corresponds to the homoge-neous, nonreproductive adsorption of viruses [first reac-tion in (1)]. Then, the usual equations are2002 The American Physical Society 178101-10 5 10 15 20 25020406080[ V]     ( 10 8  FPU / ml )τ   = 18.4 mint  (min)0 2 4 60.00.51.01.5  g   ( [V ] )t  ( min )FIG. 1. One-step growth of virus T7 on E. coli versus the timeelapsed after adsorption (t  0). In this experiment, the con-centration is uniform and adsorption is negligible (Bt  0),attained by dilution. Full line: least-square fit of the experi-mental data points [14] to Eq. (3), which yields k2 1:39 min1 and   18:4 min. Before the rise in V, theexperimental dots refer in fact to infected bacteria [19], andthe yield is the ratio of the final value Vmax to this initialconcentration [19], in this case Y  34:5. Inset: T7 adsorptionon E. coli, based on the data in Ref. [20]. The assay was carriedout in the presence of KCN, which is known to inhibit the virusreproduction. Then Eqs. (4) hold; thus, B  V  const.From the initial concentrations (taking care of the multiplicity)in Ref. [20], we find const 1:39108 ml1. The solution forV satisfies g	V lnf	Vconst=Vg  lnf	Vt0const=Vt0g constk1t. Full line: least-square fit of theexperimental data [20] to this equation. The slope yields k1	1:290:59109 ml=min. This value is rather uncertain,but it turns out that the predictions are essentially the samethroughout this range of values for k1 (Fig. 2).0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.50.6τ  = 0B max = 10 7  ml −1, k 1 = 1.88·10 − 9 ml/minB max = 10 7  ml −1, k 1 = 0.70·10 − 9 ml/min(a)c    ( mm / h )f0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.01.21.41.6B max = 10 8 ml −1 ,  k 1 = 1.88·10 −9  ml / minτ  = 0B max = 10 8 ml −1 ,  k 1 = 0.70·10 −9  ml / min(b)c    ( mm / h )fFIG. 2. Speed of the growth of T7 virus plaques on E. coli asa function of the bacterial relative concentration f. Uppercurves: predictions of the classical model [17], i.e., Eqs. (7)with   0. Lower curves: time-delayed model (  18:4 min,from Fig. 1). Symbols: experimental data [17]. The open andclosed data symbols refer to initial bacterial concentrations of107 and 108 ml1, respectively. The mean value and error forf  0:2 has been computed by taking into account that thereare some additional experimental data available for f  0:2 inRefs. [14,15]. In (b), the two lower curves are not distinguish-able from each other at this scale, showing that the uncertaintyin the value of k1 (Fig. 1, inset) has very little effect on thepredicted speed.VOLUME 89, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 21 OCTOBER 2002which, comparing to the appropriate experiment, allowus to determine the value of k1 (Fig. 1, inset).III. Plaque growth.—Consider now a situation inwhich both adsorption and reproduction take place. Ifthe virus concentration is not uniform, virus diffusionhas also to be considered and the simplest generalizationof Eqs. (2) and (4) is8>><>>:Vt  DeffVrr  FV	V; B; IBt  FB	V; B; IIt  FI	V; B; I;(5)where Deff is the effective diffusion coefficient (explainedin detail in Sec. V),   rr  @2=@r2 stands for the sec-ond spatial derivative in the radial direction from theplaque center, and8>>><>>>:FV	V; B; I  k1VB  Yk2I	1 IImaxFB	V; B; I  k1VBFI	V; B; I  k1VB  k2I	1 IImax:178101-2IV. Time-delayed diffusion: approximate treatment.—Equations (5) are Yin’s classical model [16–18] of plaquegrowth with logistic dynamics. However, this model isunable to explain experiments [16–18]. Our idea is verysimple. We will modify the classical model above by tak-ing into account that there is a delay time between themoment when a virus adsorbs into a cell and that in whichthe cell dies and the new generation of viruses begins tospread. Therefore, in agreement with the derivation inRef. [6], we propose to take into account this delay inthe diffusion process by modifying the first of Eqs. (5) byits corresponding hyperbolic generalization, namely [6],Vt  2 Vtt  DeffVrr  FV 2	FVt: (6)178101-2VOLUME 89, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 21 OCTOBER 2002We stress that, for the moment, we are building the simplest possible model with a time delay by considering the well-known hyperbolic approximation to the full time-delayed evolution equation (see ‘‘note added’’ at the end of thisLetter). In the simple, hyperbolic approximation considered, our model isVt  2 Vtt  DeffVrr  k1VB  2	VBtYk2I1 IImax 2	I1 IImaxt;Bt  k1VB; It  k1VB  k2I1 IImax:(7)In the first of Eqs. (7),  is the rest time, i.e., the timeinterval during which particles (viruses in our case) donot move [see Ref. [6] for the microscopic derivation ofEq. (6), including the factor 12 appearing in its terms with]. Therefore, in the case considered here,  is simply themean time interval from the adsorption of a virus by anuninfected host until the death of the thus infected cell.Thus, it may be inferred from Fig. 1 to be   18:4 min.Note that the interpretation of the delay time above isanalogous to that in the single-species, time-delayedtheory of the Neolithic transition in Europe, where  isthe time interval between two successive migrations [6].Let B0 stand for the initial concentration of bacteria,previous to the arrival of viruses. We introduce the di-mensionless variables BB=B0, VV=B0, II=B0, tk2t, k2, r rk2=Deffp, and 1k1B0=k2. Welook for front solutions [17,21] by assuming the concen-trations depend only on z r ct, where cc= Deffk2pand c is the front speed, and linearizing the system (7) inthe wave-front edge z!1, 	 V; B; I 	"V;1"B;"I	0;1;0, with "	"V;"B;"I"0 exp	z. For a solu-tion 	"V;"B;"I 	0;0;0 to exist, the determinant of thematrix corresponding to the linearized form of Eqs. (7)must vanish. This yields the characteristic equation forour problem, namely,01 2c2c31	1	11 2c221 c	1 2	Y1c1	Y1: (8)If the values of the parameters are known, this equationcan be solved numerically in order to find out the dimen-sionless front speed c. Numerical simulations [22] for avariety of cases have shown that the front travels with theminimum possible speed. Therefore, the minimum speedis calculated from cmin>0 c	, where c	 is givenby characteristic Eq. (8). This is the marginal stability orlinearization speed selection mechanism, which appliesto the so-called pulled fronts [23,24].V. Hindered diffusion.—We now have a time-delayedmodel and the values of the reactive parameters. Concern-ing diffusion, the effective coefficient for a solute (vi-ruses in our case) dispersing through a continuous me-dium (agar) in the presence of a suspension of spheroids(host bacteria) which adsorb the solute is given by Fricke’sequation [25],Deff  1 f1 fxD; (9)178101-3where f  B0=Bmax is the concentration of bacteria rela-tive to its maximum possible value. In the experiments[14,15,17], f was measured indirectly by noting that thebacterial concentration B0 (and thus Bmax) depends on theinitial nutrient concentration, and typical bacterial con-centrations were estimated in the range 107–108 ml1[17]. In Eq. (9), D is the diffusion coefficient in theabsence of suspended particles, and x takes care of theirshape according tox  11 13 	 21M2 1M; (10)whereM  1sin20 12cos20sin30ln1 sin01 sin0; (11)and cos0  b=a, with a the dimension of the suspendedspheroids (cells) along their axis of symmetry and b theirsize along the other axis. Fricke’s results have been foundto agree very well with experimental observations ofblood cell suspensions [25,26]. For spherical particles(a  b), x  2 and one recovers Maxwell’s formula[27], which is the approximation previously used in virusdiffusion by some authors [16,17]. For E. coli, a=b  4:0[28], which yields x  1:67, and we use this more accu-rate value. Also in Eq. (9), D is the diffusion coefficientof the virus through agar in the absence of bacteria,which can be approximated to that of P22 [17] becauseit is very similar to T7 in size and shape [29], i.e., D 4 108 cm2=s [30]. Now that we have estimations ofthe values for all of the parameters in the model (Y, k2, ,k1, Bmax, x, and D) from independent experiments, we cancompare the predictions both from our new, time-delayedmodel (  18:4 min) and from the classical one (  0)to the observed infection speeds, without making use ofany free or adjustable parameters.VI. Theory versus experiment.—They are compared inFig. 2. The classical approach (upper curves in Fig. 2)predicts speeds that are clearly inconsistent with theobserved ones, as previously noted [16–18]. In contrast,the new time-delayed model (lower curves in Fig. 2)yields good quantitative agreement with the observations.This is the first time that the time-delay effect is used toexplain wave-front propagation experiments that can bereproduced under controlled conditions.VII. Conclusions.—Our model provides a satisfactoryexplanation for the growth of virus plaques. This is abasic problem which is also important because it providesan alternative way to predict and recognize virus strains,178101-3VOLUME 89, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 21 OCTOBER 2002as well as their evolution [14,15]. Previous attempts to usereaction-diffusion models were unsuccessful. This led tothe misbelief that physical models cannot explain theseexperiments [17,31]. Agreement to experiment has beenattained here, after taking proper care of (i) the role ofthe delay time, (ii) the fact that the volume occupied bycells hinders the diffusion of viruses, and (iii) a carefulreevaluation of the values of all of the parameters fromindependent experiments.On the theoretical side, the present paper is the first onein which a system of reaction-diffusion equations, one ofwhich is time delayed, has been derived and solved toobtain the front speed. The time-delayed approach re-ported should thus be useful in physical applicationssuch as combustion [4] and superconducting [2] fronts,since the delay time arises naturally from the kinetictheory [10,11]. On the other hand, physicists can contrib-ute to further topics related to that reported here, e.g., thecharacterization of mutant virus strains (which are rec-ognized because of their front speeds [13,14]), the mod-eling of the front shapes in virus infections [15], etc.The authors thank J. Yin for discussions, as well asJ. Barbe´, who also provided us with Ref. [20]. This workwas supported in part by the Generalitat de Catalunyaunder Grant No. SGR-2001-00186, and by the MCYTunder Grants No. BFM2000-0351 and No. REN2000-1621 CLI (J. F.).Note added.—Our analysis above makes use of Eq. (6),which is in fact an approximation to the full time-delayedequation. It is an approximation because it keeps only afew terms in some Taylor expansions, e.g., V 	r; t  V 	r; t ’  @V@t  22@2V@t2[6,22]. But since V changes rap-idly at a time scale smaller than  (Fig. 1), this approxi-mation may, in principle, break down. To see how thisaffects the results, we consider the full time-delayedequation, namely [see Ref. [32], Eqs. (16) and (21)],V	r; t   V	r; t X1k1	2Deffk	2k!@2kV@r2kX1k1kk!@k1FV@tk1; (12)instead of its hyperbolic (or second-order) approximation(6). Then, it is easily seen by repeating the same steps asin Sec. IV that Eq. (8) is replaced by	expc  1	1 c	c 1  1Y cosh	 2p   1c	1 c: (13)We have repeated the calculations leading to Fig. 2 byusing this equation instead of (8), and the results changeat most by a few percent (typically, below 1%). Thechanges are so small that they cannot be seen in Fig. 2.This strengthens our conclusion that, for the first time, an178101-4experiment which (i) was previously unexplained and(ii) can be reproduced in the lab, has been explained interms of time-delayed wave-front propagation.[1] R. Kupferman, O. Shochet, E. Ben-Jacob, and Z. Schuss,Phys. Rev. B 46, 16 045 (1992).[2] S. J. Di Bartolo and A. T. Dorsey, Phys. Rev. Lett. 77,4442 (1996).[3] W. van Saarloos, M. van Hecke, and R. Holyst, Phys. Rev.E 52, 1773 (1995).[4] J. Warnatz, U. Maas, and R.W. Dibble, Combustion(Springer, Berlin, 2001).[5] O. Jousson et al., Nature (London) 408, 157 (2000).[6] J. Fort and V. Me´ndez, Phys. Rev. Lett. 82, 867 (1999).[7] P. Menozzi, A. Piazza, and L. L. Cavalli-Sforza, Science201, 786 (1978).[8] Y. Lee and J. Yin, Nature Biotech. 14, 491 (1996).[9] V. Me´ndez, Phys. Rev. E 57, 3622 (1998).[10] G. Chen, Phys. Rev. Lett. 86, 2297 (2001).[11] D. Jou, J. Casas-Va´zquez, and G. Lebon, ExtendedIrreversible Thermodynamics (Springer, Berlin, 2001).[12] P. K. Galenko and D. A. Danilov, J. Cryst. Growth 216,512 (2000); Phys. Lett. A 272, 207 (2000).[13] Y. Lee and J. Yin, Nature Biotech. 14, 491 (1996).[14] J. Yin, J. Bacteriol. 175, 1272 (1993).[15] J. Yin, Biochem. Biophys. Res. Commun. 174, 1009(1991).[16] L. You and J. Yin, J. Theor. Biol. 200, 365 (1999).[17] J. Yin and J. S. McCaskill, Biophys. J. 61, 1540 (1992).[18] J. Fort, J. Theor. Biol. 214, 515 (2002).[19] M. Delbru¨ck, Biol. Rev. Proc. Cambridge Philos. Soc. 21,30 (1946).[20] K. Shishido, A. Watarai, S. Naito, and T. Ando,J. Antibiot. (Tokyo) 28, 676 (1975).[21] J. D. Murray, Mathematical Biology (Springer, Berlin,1993).[22] V. Me´ndez, J. Fort, and J. Farjas, Phys. Rev. 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Time-delayed spread of viruses in growing plaques

Time-delayed spread of viruses in growing plaques

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