error catastrophe for viruses infecting cells: analysis of...

Phil. Trans. R. Soc. A (2010) 368, 5569–5582doi:10.1098/rsta.2010.0274

Error catastrophe for viruses infecting cells:analysis of the phase transition in terms

of error classesBY JULIA ALONSO1 AND HUGO FORT2,*

1Instituto de Física, Facultad de Ingeniería, Universidad de la República,Julio Herrera y Reissig 565, 11300 Montevideo, Uruguay

2Instituto de Física, Facultad de Ciencias, Universidad de la República,Iguá 4225, 11400 Montevideo, Uruguay

RNA viruses offer a very exciting arena in which to study evolution in ‘real time’ owingto both their high replication rate—many generations per day are possible—and theirhigh mutation rate, leading to a large phenotypic variety. They can be regarded as aswarm of genetically related mutants around a dominant or master genetic sequence.This system is called a ‘viral quasi-species’. Thus, a common framework to describeRNA viral dynamics is by means of the quasi-species equation (QSE). The QSE is infact a system of a very large number of nonlinear coupled equations. Here, we considera simpler formulation in terms of ‘error classes’, which groups all the sequences differingfrom the master sequence by the same number of genomic differences into one populationclass. From this, based on the analogies with Bose condensation, we use thermodynamicinspired observables to analyse and characterize the ‘phase transition’ through the so-called ‘RNA virus error catastrophe’.

Keywords: viral quasi-species; error classes; error catastrophe; finite-size scaling

1. Introduction

RNA viruses have become especially popular for experimental evolution, owingto their rapid replication rates (Domingo & Holland 1994), which allow studiesto run for hundreds of generations (Burch & Chao 2000). Examples of RNAviruses are HIV, influenza A (H1N1), foot and mouth disease, hepatitis C andpoliovirus. In addition, the high mutation rates of RNA viruses generate a largegenetic variation. Thus, the quasi-species (QS) concept introduced by Eigen(1971) is helpful to address the process of the Darwinian evolution of RNAviruses (Domingo 1992; Eigen & Biebricher 1998). A QS refers to the equilibriumspectrum of closely related mutants, dominated by a master sequence, that isgenerated by a mutation–selection process (Domingo 2006).

*Author for correspondence ([email protected]).

One contribution of 13 to a Theme Issue ‘Complex dynamics of life at different scales: from genomicto global environmental issues’.

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5570 J. Alonso and H. Fort

On the one hand, the heterogeneous structure of the population can act asa reservoir of phenotypical variants potentially useful against environmentalchanges. On the other, there appear to be limits on how advantageous a highmutation rate can be, and there is evidence that, if an RNA virus QS goes beyonda mutation threshold, the population will no longer be viable. The phenomenonthat occurs when the loss of genetic fidelity results in a lethal accumulationof errors has been termed the error catastrophe (Domingo & Holland 1994;Eigen & Biebricher 1998). This originated the controversy of survival of the fittestversus survival of the flattest (Wilke et al. 2001): a fast-replicating organism thatoccupies a narrow peak versus one that occupies a lower but flatter peak.

Here, we propose a framework to formulate problems like the above-mentionedtrade-off between mutation and fitness and the error catastrophe. We try tomaintain a balance between realism and simplicity. Unlike many models found inthe literature, where viruses are modelled as free particles without any interactionwith a host, here we explicitly consider viruses to be parasite binding organisms.In virology, the number of mutations per genome and replication round is oftenreferred to and interpreted as an error class in the mathematical sense. We usethis concept in our model, which greatly reduces the number of equations to besolved. Moreover, among the many possible distributions of fitness among theerror classes, we choose the simplest one or Eigen landscape. This consists inassuming that all the classes have the same fitness except for a single masterclass that has a higher fitness (see §3).

The use of physical tools for the analysis of the dynamics of various biologicalsystems is becoming increasingly common. RNA viruses in particular represent avery exciting challenge in this regard because (i) they exhibit an ‘order parameter’that becomes zero at the error catastrophe, and (ii) the diversity of a QS can bestraightforwardly quantified by means of the entropy. These two facts suggest ananalogy with the phenomenon of Bose condensation. This, in turn, allows us toanalyse the critical behaviour of thermodynamic-like observables like an orderparameter and specific heat (see §4), and to characterize the phase transition ofthe system (see §5).

2. From the molecular quasi-species model to the quasi-species basicmodel of viral dynamics

QS theory was proposed by Eigen (1971) to describe the self-replication ofbiological macromolecules with the aim of understanding the origin of life.Since then, the QS concept has been applied to RNA virus populations,reflecting their large genetic variability (Domingo et al. 1985). For sometime, some researchers regarded this approach as incompatible with the moreconventional approach of population dynamics, namely population genetics.However, later it was realized that they are simply two equivalent approachesto describe the evolution (mutation–selection balance) of RNA viruses (seeWilke 2005): while population genetics involves a top-down approach to theproblem, QS theory, based on chemical kinetics from its birth, corresponds toa bottom-up approach.

QS theory enables virologists to describe Darwinian evolution even when RNAreplication occurs in a cell-free system. Given a population of organisms, with n

Phil. Trans. R. Soc. A (2010)



Error catastrophe for viruses in cells 5571

different types of sequences, the genomic structure of the population is given bythe vector:

v = (v1, v2, . . . , vn), (2.1)

where vi is the population of those organisms whose genome sequence is of type i.The quasi-species equation (QSE), describing the selection–mutation balanceamong sequences and then governing the evolution of v, is

dvdt

= Wv − A(v)v, (2.2)

where W is the selection–mutation matrix whose elements are Wij = AjQij , Ajis the replication rate of sequence j , Qij is the probability that a j sequencemutates to an i sequence and A(v) is the mean replication rate of the population.The eigenvector associated with the largest eigenvalue of W is the steady-statesolution of the previous equation and is known as the QS.

The population of a given variant, vi , depends not only on its intrinsicreplication rate Ai but also on the rate of replication of the other variants (whichmay occur due to errors in the replication of these) and their populations. As aconsequence, the target of the evolutionary process (selection–mutation) is not theindividual sequence i, with its intrinsic or own replication rate Ai , but the QS as awhole entity. What matters is the effective replication rate of a particular genomei, which is the fitness of that genome. The set of fitness values for sequences thatcomprise the QS is known as the fitness landscape. The most common fitnesslandscape is the Eigen landscape or single-peak landscape (mentioned in §1): asingle fitness peak at the master sequence emerging from a plateau of constantheight. In this context, the QSE model predicts an error threshold (criticalmutation rate) above which the master sequence is extinguished. To make thispoint more clear, it is helpful to reduce the vector equation (2.2) to only twoscalar equations by a mean-field approximation: one corresponding to the mastersequence’s population vm (defined as the most frequent in the population) and theother describing the mutant cloud composed by all the remaining sequences. LetAm be the replication rate of the master sequence and Ac the replication rate ofthe cloud. A common additional simplification is to neglect the small probabilityof mutations from the cloud to the master sequence (Nowak 1992). Under thissimplifying assumption, if M is the mutation rate of the master sequence, thefractional population of the master sequence becomes

vMFm = 1 − M

Mc, M < Mc,

and vMFm = 0, M > Mc,

⎫⎪⎬⎪⎭ (2.3)

where Mc = 1 − Ac/Am and the superscript MF is to remind us of the mean-fieldapproximations we are considering. Figure 1 shows the behaviour of vMF

m as afunction of M/Mc as given by equations (2.3).

Nevertheless, a virus is an obligatory intracellular parasite, which means thatit needs to enter into a cell to use its functions, and this fundamental biologicalaspect is not taken into account in the QSE. A simple model that considers cells,and distinguishes between infected and non-infected cells, is the so-called basic





0 0.5 1.0 1.5 2.0

0.2

0.4

0.6

0.8

1.0

M/Mc

vMF

m

Figure 1. The fractional population undergoes a transition once M reaches the error threshold:Mc = 1 − Ac/Am.

model of viral dynamics (BMVD), proposed by Nowak & May (2000). They alsoextended the model to include the concept of QS in it, yielding a QS-BMVDgiven by the following equations:

dxdt

= l − dx − x∑

i

bivi ,

dyi

dt= x

∑j

Qijbjvj − aiyi ,

anddvi

dt= kiyi − uivi ,

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

(2.4)

where the time-dependent variables in this model are: the population ofuninfected cells x , the population of free viral particles with the ith viral genomeor sequence and yi the population of infected cells that produce the ith sequence.The dynamics of infection is described by chemical kinetics using the followingcoefficients/terms:

— bi is the infectivity of virus particles of type i,— xbivi is the rate at which uninfected cells react with viral particles of type i,— l is the immigration rate,— dx is the rate of normal mortality (in the absence of virus),— kiyi is the rate at which infected cells produce viral particles,— uivi is the rate at which viral particles die,— aiyi is the rate at which infected cells die,— xQijbjvj represents the mutational gain of type i: when j = i then the

j-genome (an i-genome actually) is replicated without error in the cell,while if j �= i the j-genome is replicated with error, becoming a genome oftype i.

Notice that the equation for virus evolution is different from QSE (2.2).





This model contains parameters that usually are not easy to determine. Toovercome this obstacle, we rewrite it in a dimensionless form. We assume thesame intrinsic rate of death for all infected cells, ai = a, and then we measuretime in units of 1/a. So we rename t/(1/a) as t, x/(l/d) as x , yi/(uid/kibi) as yiand vi/(d/bi) as vi . In this way, equations (2.4) can be rewritten as

dxdt

= da

(1 − x − x

∑i

vi

),

dyi

dt= Rix

∑j

Qijvj − yi ,

anddvi

dt= ui

a(yi − vi).

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(2.5)

At equilibrium it turns out that yi = vi , so we obtain the following equations forthe equilibrium points:

x = 11 + ∑

i vi

and Ri

∑j

Qijvj −(

1 +∑

k

vk

)vi = 0.

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(2.6)

The free parameters left—besides mutation rates—are known as the basicreproductive ratios Ri = (lbiki)/(daui) associated with each genome. The setof different reproductive ratios of each sequence takes the role of a fitnesslandscape in our model. Incidentally, when analysing RNA viruses, virologistsoften consider the number of mutations in a genome with respect to anothertaken as a reference, e.g. the master sequence (see Crotty et al. 2001;Vignuzzi et al. 2006).

3. Viral dynamics in error classes

For the sake of simplicity, we consider binary sequences (bit sequences) so thatthe number of errors that lead to the transformation of one sequence into anotheris equal to the Hamming distance dH between these two sequences. Swetina &Schuster (1982) were the pioneers in taking a reference sequence (e.g. the mastersequence) and putting together all sequences with the same Hamming distanceto the reference sequence into what is mathematically known as an error class.Following this recipe, we reformulate the QS-BMVD model of Nowak & May(2000) in terms of error classes. This reduces the number of resulting equationsand makes it easier to carry out numerical simulations: in bit sequences of lengthL this number is reduced from 2L to L + 1.

Let us denote by GK the set of different elements that make up the error classK (dH = K ) and by VK its population. Then

VK =∑i∈GK

vi .





Assuming that all sequences in the same error class K have the same basicreproductive ratio RK , we obtain instead of equations (2.6)

x∗ = 1

1 + ∑LJ=0 V ∗

J

and RK

L∑J=0

Q ′KJ V ∗

J −(

1 +L∑

J=0

V ∗J

)V ∗

K = 0.

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(3.1)

Here, the elements of mutation matrix Q ′ for error classes are given by

Q ′KJ =

min{K , J }∑P=max{K+J−L, 0}

(JP

) (L − JK − P

)(1 − mb)L

(mb

1 − mb

)K+J−2P

, (3.2)

where mb is the single-bit error rate. Let us consider the Eigen single-peaklandscape, where the master class (K = 0) has the higher fitness, R0, and allthe other classes have a lower fitness, R = R0/s, where s is known, in the jargonof virology, as the superiority parameter of the master sequence (s > 1). Thepreviously considered MF approximation reduces the problem even further tojust two different error classes, K = 0 and K = C , for all the sequences in thecloud of mutants. In addition, if, as we did before, mutations from the class C tothe class 0 are neglected, then one gets:

Q ′0C = 0 (no mutations from cloud to master class),

Q ′CC = 1 (mutations from cloud remain in the cloud)

and Q ′C0 = (1 − mb)L = 1 − M (mutation rate for master class is M ).

Substituting the above relations into equations (3.1) yields a simple expressionfor the fractional population of the master class (class 0), depending on whetherM is above or below an error threshold Mc = 1 − s−1:

V MF0 = Mc − M

(1 − M )Mc, M < Mc,

and V MF0 = 0, M > Mc.

⎫⎪⎬⎪⎭ (3.3)

Figure 2 shows the behaviour of the fractional population of the master class V MF0

versus M/Mc for s = 10 and the fractional population of the cloud (1 − V MF0 ).

In figure 3, we show the corresponding ‘phase diagram’. The threshold curveseparates the QS phase from the drift phase (Solé et al. 2006) in which the virusis no longer viable.

4. Observables

Clearly, the phenomenology of the transition experienced by viral populationsfrom the QS to the drift phase is reminiscent of Bose condensation or the phasetransition in the Landau two-fluid model describing the behaviour of superfluidhelium or He II. Below the critical temperature Tc there coexist two components:a normal fluid and a superfluid component. As the temperature is decreased, more





0 0.5 1.0 1.5 2.0

0.2

0.4

0.6fr

actio

nal p

opul

atio

ns 0.8

1.0

M/Mc

Figure 2. The fractional populations of the master class V MF0 (full line) and of the cloud (dashed

line) as a function of M/Mc, for s = 10. The resemblance to the Bose condensation is clear.

1 3 5 7 9 11 13 150

0.2

0.4

0.6

0.8

1.0

s

M

Figure 3. Mean-field phase diagram: the QS (drift) phase is below (above) the curve 1−M/Mc.Solid line, Mc = 1 − s−1.

and more normal fluid is transformed into the Bose condensate or superfluid. Inthe absolute zero temperature limit, 100 per cent of the system is condensate inHe II. Comparing figure 2 with a graph of the superfluid and normal fractionsof helium atoms, ns and nn, versus the temperature T , the analogy is clear:the superfluid (normal) component corresponds to the master sequence (cloud ofmutants) and the temperature with the mutation rate M . Based on this analogy,we take the fractional population of the master class, V0, as the order parameterand M plays a role analogous to temperature in physical systems.





0.95 1.00 1.05

20

30

40

50

60

70

S

M/Mc

Figure 4. Entropy as a function of M/Mc for L = 100 and s = 10.

Furthermore, this transition involves a change in the ‘biodiversity’ of the viralpopulation, which is usually measured by its entropy (see, for example, Domingo2007). Thus, from the usual definition of the entropy S ,

S = −2L∑i=1

pi ln(pi), (4.1)

where pi is the probability of finding a given sequence i in the populationof 2L different types of sequences of viral QS. That is, pi coincides with thefractional population of the sequence i. If i ∈ GK , then vi = VK/CL

K , where CLK is

the degeneracy of the K class, and equation (4.1) can be expressed as

S = −L∑

K=0

VK ln(

VK

CLK

). (4.2)

Numerical results for the entropy can be seen in figure 4 for L = 100 and s = 10.Entropy increases as the presence of the master sequence decreases, i.e. it increaseswith M , which then plays the role of temperature. At the threshold Mc, theentropy reaches a maximum value ln(2L − 1) and then remains constant forM > Mc. Following the analogy with thermal physics, a specific heat can beobtained from the entropy as

c = MdSdM

. (4.3)

It is worth mentioning that the analogy between the error threshold transitionfor QS and the phase transition taking place in other physical systems like thetwo-dimensional Ising model has been explored in many papers (see Leüthausser1986, 1987; Tarazona 1992; Pastor-Satorras & Solé 2001). Additionally, this phasetransition approach has been set in the context of cancer seen as a QS problem(see Solé 2003; Solé & Deisboeck 2004).





5. Critical exponents from finite-size scaling

To characterize a phase transition that is presumably continuous, it is necessaryto analyse the singular behaviour that the relevant physical quantities of thesystem have at the critical point. These singularities take the form of power lawsin the reduced temperature t = (T − Tc)/Tc. The finite-size scaling (FSS) methodis a straightforward way to extract the critical exponents by analysing howmeasured observables depend on changes in the size parameter L (see Newman &Barkema 1999). The following relations are only valid when the critical reduced‘temperature’ or reduced mutation rate e = (M − Mc)/Mc is close to 0:

c = Lg/nc(L1/ne), (5.1)

c = La/nc(L1/ne) (5.2)

and m = Lb/nm(L1/ne), (5.3)

where n, a, b and g are, respectively, the critical exponents for the correlationlength x, the specific heat c, the order parameter m and the susceptibility c. Thescaling functions c, c and m in the above relations, by construction, should beindependent of the size L, but strongly vary with the parameters e, n, a, b and g.If the correct values for the parameters are chosen, the data obtained for differentsystem sizes will collapse onto a single curve. Before continuing, we want to makea remark concerning the critical exponent for the correlation length n appearing inequations (5.1)–(5.3). In our model, as in the combinatorial problems discussed byKirkpatrick & Selman (1994) or the case of QSE studied by Campos & Fontanari(1998), there is no geometric criterion to define the correlation length x. However,it is expected that the region where the transition occurs is reduced to 0 as L−1/n

when L −→ ∞; then the exponent n can be associated with this characteristicbehaviour. In this way, we can still use data collapsing to estimate the criticalexponents a and b.

The results of numerical simulations for V0 for L = 50, 70, 100, 120, 150 and 200with s = 10, together with the approximation by mean field V MF

0 (neglecting inthe approximation mutations from the cloud to the master sequence) are shownin figure 5 in terms of M/Mc. From this plot, it is possible to estimate the criticalexponents b (corresponding to the behaviour of the order parameter in the vicinityof the transition) and n.

As can be seen in figure 5, higher values of L give better similarity between V0and V MF

0 . Hence, mean field is a good approximation in the thermodynamic limit(L −→ ∞) and the critical temperature can be taken as Mc = 1 − 1/s. Near thetransition, the order parameter V0 is well approximated by V MF

0 and, therefore, wecan assume that b = 1 as follows. Rewriting equation (3.3) for M ≈ Mc (M < Mc)in terms of reduced temperature e = (M − Mc)/Mc:

V MF0 ≈ − 1

1 − Mc

M − Mc

Mc∝ −e1.

On the other hand, as seen in the inset of figure 5 for each L, V0 linearlyapproaches 0, vanishing for an ML value that we can estimate by adjusting thevalues of V0 by a straight line in the region where M/Mc is close to 1. Once we





0.95 1.00 1.050

0.05

0.10

0.15

0.20

0.25

0.30

0.995 1.000 1.005

0

0.02

0.04

0.06

0.08

M/Mc

v 0

Figure 5. V0 for various lengths of sequences versus M/Mc. Open circles, L = 50; inverted triangles,L = 70; open squares, L = 100; open triangles, L = 120; crosses, L = 150; diamonds, L = 200;dot-dashed line, V MF

0 .

have the points ML/Mc of intersection with the axis M/Mc we can use a resultfrom finite-size scaling:

ln(

ML

Mc− 1

)= −1

nln(L) + constant.

Then, as we have ML/Mc values for each L, to obtain the slope (−1/n) it is enoughto make a linear fit on the pairs of values ln(L) and ln(ML/Mc − 1) as shown infigure 6. Finally, the exponent is:

n = 0.98 ± 0.06.

Figure 7 shows that for n = 0.98, b = 1 and Mc = 1 − 1/s, the collapse of data isvery good.

The specific heat c of the system can be obtained from the entropy byequation (4.3). Numerical results for c for different values of L are shown infigure 8. As we know from FSS, the maximum of the specific heat behaves as

cmax ∼ La/n.

Taking the logarithm on both sides, we obtain a straight line:

ln(cmax) = a

nln(L) + constant.

Then a linear fit on the pairs of values ln(L) and ln(cmax) gives the slope of the linea/n (figure 9). As n is known, a can be determined and the error in its estimate:

a = 1.9 ± 0.1.

Once we have determined a/n, we can plot c(eL1/n) = c(e)L−a/n versus eL1/n.Figure 10 shows that the data collapsing is quite good so the estimations for the





3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4−8.0

−7.8

−7.6

−7.4

−7.2

−7.0

−6.8

−6.6

−6.4

−6.2

ln(L)

ln(M

L/M

c −

1)

Figure 6. Estimated exponent n: n = 0.98 ± 0.06.

−10 −5 0 5 100

10

20

30

40

50

60

70

L1/v

V0L

b/v

−0.5 0 0.5−1

0

1

2

3

4

5

6

Figure 7. Collapse of the scaling functions V0Lb/n versus the scaling variable eL1/n. Open circles,L = 50; inverted triangles, L = 70; open squares, L = 100; open triangles, L = 120; crosses, L = 150;diamonds, L = 200.

critical exponents b, n, a and the critical temperature are reliable. Summarizingwe get

n = 0.98 ± 0.06,

b = 1.0

and a = 1.9 ± 0.1.

⎫⎪⎬⎪⎭ (5.4)





0.95 1.00 1.050

2

4

6

8

10

12

14

16

18

c (×

103 )

M/Mc

Figure 8. Specific heat for different L values versus M/Mc. Open circles, L = 50; inverted triangles,L = 70; open squares, L = 100; open triangles, L = 120; crosses, L = 150; diamonds, L = 200.

3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.46.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0

ln(L)

ln(c

max

)

Figure 9. Estimated exponent a: a/n = 1.97 ± 0.01.

6. Final comments

There are a large number of theoretical results based on analytic approachesto the full system of QSE that have led to relevant conclusions about manyviruses. In this work, we attempt to simplify the insight into QS by takinginto account a dimensionless form of the equations while including the hostcells in which the viruses replicate. We have proposed a reformulation of the





−10 −5 0 5 100

0.1

0.2

0.3

0.4

0.5

eL1/v

cL−

a/v

Figure 10. Collapse of the scaling functions cL−a/n versus the scaling variable eL1/n. Open circles,L = 50; inverted triangles, L = 70; open squares, L = 100; open triangles, L = 120; crosses, L = 150;diamonds, L = 200.

Nowak & May (2000) QS-BMVD model, which incorporates cells, in terms oferror classes. The utilization of error classes allows a great simplification of thesystem of differential equations to be solved.

We use this new framework to analyse the error catastrophe phase transition.Exploiting analogies with widely studied models from physics, like, for example,Bose condensation, we identify a useful ‘order parameter’ to characterize the twophases separated by the error catastrophe. In addition, by taking the entropy asa diversity parameter for viral QS, we were able to calculate a ‘specific heat’as another quantity to characterize the phase transition. As an applicationof statistical mechanics techniques, using the FSS method, we obtained thecorresponding critical exponents.

Here, we considered the simplest Eigen landscape. In future work it should beinteresting to explore what happens in the case of more realistic landscapes.

To conclude, we think that formulating QS in terms of physical observablesprovides new insights as well as alternative powerful quantitative techniques toapproach RNA viruses.

H.F. thanks Juan Arbiza and Esteban Domingo for stimulating discussions and valuable comments.This work was supported in part by PEDECIBA and ANII (Uruguay).

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error catastrophe for viruses infecting cells: analysis of...

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