Environ. Sci. Technol. 1994, 28, 928-933
Virus Coagulation in Aqueous Environments Stanley B. Grant
Department of Civil and Environmental Engineering, University of California, Irvine, Irvine, California 927 17
A mathematical model is presented for the temporal decline in total infectious units caused by simultaneous first-order inactivation and Brownian coagulation of viruses in an aqueous environment. On the basis of published physicochemical and biological constants for poliovirus, human immunodeficiencyvirus, and indigenous marine and freshwater bacteriophage, the model predicts that virion-virion coagulation is negligible in most aquatic systems. This analysis provides a framework for investigating the effect of coagulation and inactivation on viral infectivity and for developing more sophisticated models of virus survival outside the host cell. Introduction Viruses are found in most, if not all, aquatic environments where they exist as natural inhabitants ( 1 , 2 )or as contaminants introduced from waste treatment facilities ( 3 , 4 ) . Virus survival in these environments depends on a number of biological and physicochemical processes that are not clearly understood at the present. Individual virus particles lose their ability to infect host cells with time through a process called inactivation. Inactivation usually involves an irreversible alteration in the protein or nucleic acid components of the virus and occurs a t a rate characteristic of the specific virus and its environment. However, viral infections are not always caused by single virus particles. Host cell infections arise from infectious units (IU), which include individual virus particles (virions), viral aggregates, and in some cases, unencapsulated viral nucleic acid (DNA or RNA) (5, 6). Because an aggregate consisting of many virions may give rise to only a single infection, the overall level of infectivity of a virus suspension depends both on the number of virions present and on the degree to which these viruses are aggregated (5-8). Because both aggregation and inactivation can affect viral infectivity, any progress made in quantifying these processes is of keen interest to many fields of environmental and biological science. In this paper, a kinetic analysis is presented for the relatively simple case of simultaneous inactivation and Brownian coagulation of an initially monodisperse suspension of virus particles. Closed-form expressions are presented for the total particle concentration and the concentration of each aggregate size class, based on the von Smoluchowski equations for Brownian coagulation, modified to account for first-order virus inactivation. These solutions indicate that the magnitude of a single dimensionless parameter ( Q ) can be used to determine if coagulation affects the number of IU present in a virus suspension. The impact of coagulation on viral infectivity is evaluated by calculating ranges of Q values for poliovirus, human immunodeficiency virus (HIV), and indigenous marine and freshwater bacteriophage. Model Development Aggregates of virus particles can develop during virus assembly in the cytoplasm of host cells (6), or they may 928
Environ. Sci. Technol., Vol. 28, No. 5, 1994
develop outside the host cell by virion-virion coagulation. Sharp and co-workers (9)demonstrated that the dynamic coagulation of reovirus and poliovirus particles follows the classic von Smoluchowski equations for Brownian coagulation. When these equations are appropriately modified to include first-order virus inactivation, the following governing differential equation is obtained for an aggregate of j viruses (nj)
where k is a first-order inactivation constant, and Kij is a parameter called the coagulation kernel that accounts for the collision and adhesion mechanisms operative between particles of size i and j . The first summation on the right-hand side of eq 1 represents the rate a t which aggregates of size j are formed by collision of aggregates of size i and size j - i ; the second summation represents the rate a t which aggregates of size j are lost by collision with all other particles; the third term represents the loss of aggregates of size j due to first-order inactivation; and the fourth term represents the increase in aggregates of size j due to the inactivation of aggregates of size j + 1. Assuming that Kij is equal to a constant K , which is independent of aggregate size (see discussion), eq 1 can be rewritten as an infinite set of differential equations
which, when summed, yield the following differential equation (3)
where N', = Nm/Nm(0), N , is the total number of particles (singlets plus aggregates) at any given time; N,(O) is the initial number of virus particles; n: = ni/N,(O) equals the normalized concentration of the ith aggregate size class; T = kt represents real time ( t ) scaled by the first-order inactivation rate constant (k);and Q = l/(kTb)represents the ratio of the time scale for inactivation ( l l k )to the time The scale for Brownian coagulation [ T b = 2/(KNm(0))3. 0013-936X/94/0928-0928$04.50/0
0 1994 American Chemical Society
assumptions used to develop these equations are addressed later in the paper. Because f2 premultiplies the coagulation terms in eqs 2 and 3, its magnitude will determine the relative importance of inactivation and coagulation processes in a given system. If coagulation is slow relative to inactivation, then Q 0 and the above equations reduce to simple first-order inactivation. Above some value of Q, coagulation strongly affects the total number of virus particles (and thus IU) present in the suspension. Solutions to these equations are needed in order to determine the critical Galue of Q below which coagulation effects can be neglected. Equations 2 and 3 represent an infinite set of coupled nonlinear differential equations for which the existence of a simple analytical solution would seem unlikely. However, by seeking solutions similar to those derived by von Smoluchowskifor Brownian coagulation, the following two closed-form expressions were found for the number concentration of the ith aggregate
1 oo
-
lo-'
1 o-2
(4)
1o
-~
1o
-~
and the total particle concentration --7
m
(5) 2
0
given an initially monodisperse suspension (see supplementary material section). Equation 5 is plotted semilogarithmically against dimensionless time 7 for several different choices of Q in Figure 1 (thin solid lines). Also shown in the figure are the curves corresponding to first-order inactivation (heavy dashed line)
and simple Brownian coagulation (10) (dotted lines) 1
"-'E
(7)
When the effective time scale for coagulation is short relative to that for inactivation (a> 1in the figure), curves calculated from eq 5 overlay the curves corresponding to Brownian coagulation initially, but eventually join the firstorder inactivation line with increasing dimensionless time. When the effective time scale for coagulation is long relative to the time scale for inactivation (f2 < 1 in the figure), eq 5 is indistinguishable from first-order inactivation. Thus, the critical value of Q in this system is Q = 1. Inactivation controls viral infectivity when Q < 1,while both coagulation and inactivation processes are important when fl > 1. Application to Aqueous Systems The magnitude of Q depends on biological factors that influence the rate of virus inactivation and on physicochemical characteristics of the system that determine the rate of virus coagulation. The rate a t which viruses inactivate is influenced by a wide array of factors, most notably temperature (11-131, solution pH and ionic strength (13), the presence of virucidal agents (14-161, and the intrinsic lability of the virus particle. Given a
4
6
8
1 0 1 2 1 4
z = kt
-
Flgure 1. Decline in total vlrus particle concentratlon caused by firstorder inactivation (heavy dashed line, eq e), Browniancoagulation(. ., eq 7), and a mixture of these two processes (-, eq 5). Values of a are indlcated. Equation 5 overlays the line corresponding to firstorder inactivation for all Q < 1.
particular set of conditions, a single empirical inactivation coefficient k can usually be determined. The time scale for Brownian coagulation depends inversely on the initial particle concentration and the coagulation kernel [Tb = 2/(KN,(O))].Experimental data suggest that virus coagulation occurs more rapidly a t higher particle concentrations (17,18), in agreement with this definition for 7b. The coagulation kernel can be written as K = 2kb/ Wwhere k b is the rate constant for diffusion-controlled coagulation and W is the stability ratio which accounts for the electrostatic, hydrodynamic, dispersive, steric, and bridging forces that develop between two particles on close approach. As repulsive forces dominate particle-particle interactions W m; for strongly attracting particles W 1. The value of W can be determined de novo for a few idealized cases (19,201or experimentally by comparing observed and diffusion-limited coagulation rates (20). Upper-limit ranges in the value of Q for a given system can be calculated by assuming W = 1 (Le., all collisions result in aggregate formation) and using measured inactivation rates and plausible ranges in the virus particle concentration. 0 values calculated for poliovirus, HIV, and indigenous marine and freshwater bacteriophage are shown in Figure 2, panels A-C, respectively. The vertical dimension of the bars in this figure represents the approximate concentration range for each virus in the environment indicated. The position of the bars with respect to the critical coagulation value of Q = 1(indicated by the horizontal line) is determined by the corresponding
-
-
Envlron. Sol. Technol., Vol. 28, No. 5, 1994 029
A.
0
loo
B.
lo'or 1 o5
Flgure 2. Range of 0 values calculated for (A) poliovirus, (B) human Immunodeficiencyvirus, and (C) indigenousbacteriophage In the envlronments indicated. s2 values were estimated from 0 = kb&(o)/(kw), assuming W = 1 and kb = 4kT/3p where k is Boltrmann's constant (1.38 X erg/K), Tis temperature, and p is dynamic viscosity. The inactivation rate constant (k)was calculated(or taken directly) from the studies referenced at the bottom of each panel. The virus particle concentration was assumed to range from an arbitrarily chosen low of 101/mL to a high value which depended on the virus and the environment: (A) 1012/mLfor lab stocks (59), 10s/mL for ail other environments (60);(B) 101*/rnL for lab stocks (53),1 03/mLfor waste water, tap water, and seawater (57), 106/mLfor clinicaldisinfection environments(54);(C) 10e/mLfor all environments ( 1, 2). The width of each bar in this figure represents the temperature at which the correspondlng lnactlvatlon rate was measured: 2-6 O C (I), 10-16 O C and 20-35 OC (I). Arrows indicate that the Inactivation rate reported in the reference was a minlmum (1) of maximum (t)value. Numbers below bars in panel A indicate the serotype of the poliovirus used in the inactivation experiment.
(I),
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inactivation rate reported in the references a t the bottom of each panel. Because the position of each bar is calculated assuming a stability ratio W = 1,the ranges of Q shown in Figure 2 must be regarded as upper limit estimates. While virion-virion coagulation is unlikely to occur for values of Q less than 1, Q values greater than 1 do not necessarily implicate coagulation because large values of W can reduce Q below the critical value. Also note that the kinetic analysis presented in this paper does not address the possibility of “heterogeneous” coagulation between viruses and inorganic or organic colloidal material. Thus, the Q values in Figure 2 cannot be used to infer anything about this potentially important process. Coagulation is most likely to occur in environments where the virus particle concentration is high. This is reflected in the large values of Q calculated for laboratory stocks (panels A and B) and, to a lesser extent, for poliovirus in wastewater and for indigenous bacteriophage in marine and freshwater environments (panels A and C, respectively). Laboratory studies indicate that the coagulation of poliovirus is minimal a t neutral and alkaline pH (21,22)implying that W is much greater than 1under these conditions. Because the pH of typical stock buffers and wastewater are also near 7, and Q values shown for these environments in Figure 2A are probably too high. The Q values corresponding to indigenous bacteriophage may also be too high (Figure 2C), since the concentration ranges used to calculate Q in this case are based on total counts of bacteriophage-like particles in marine and freshwater environments as determined by transmission electron microscopy ( 1 , 2 ) . Recent evidence suggests that a sizable fraction of these bacteriophage-like particles may be inactivated (23),and it is unclear how many different types of viruses are represented in these total population counts (24). In most of the other cases shown in Figure 2, coagulation is not predicted to be an important process either because the virus concentrations are too low or because virus inactivation occurs too quickly. Even for relatively resistant enteroviruses like poliovirus, Q is less than 10 in most freshwater systems. Disinfection processes result in With the exception of laboratory very low Q values stocks, il values for HIV are all considerably less than 1 due to the lability of this virus outside its host and the low concentrations anticipated for most environments. Overall, the ranges in Q values shown in Figure 2 indicate that virion-virion coagulation is probably a rare event in most engineered and natural systems. Coagulation is predicted to affect viral infectivity only if the virus concentration is high, the stability ratio is close to one (i.e., the viruses are destabilized), and the inactivation rate is slow.
Discussion In the analysis presented here, virus inactivation is assumed to follow first-order kinetics and to occur a t a rate that is independent of the physical location of the virus; i.e., whether the virus is present in an aggregate or as a single particle. While virus inactivation often does follow first-order kinetics, deviations from this rate law have been reported in natural (25)and disinfection (6,26) systems. Several causes of non-first-order inactivation have been identified over the years, including natural variations in the lability of individual virions within a viral population (27, 281, and the association of viruses with
other viruses (viral aggregates) ( 6 )or with nonviral organic or inorganic colloidal material (29-32). Aggregates of virus particles appears to inactivate more slowly than single virus particles (61,although it is not clear whether this is due to the “protective” environment conferred by the aggregate or due to the increased number of viruses that must be inactivated before the aggregate as a whole is noninfective (6). If it is the former process, then the same inactivation rate should not be applied to all viruses regardless of their state of aggregation as is done here. This analysis also assumes that when viruses inactivate they are immediately eliminated from the system. Such an assumption is warranted in cases where the inactivation event is accompanied by a physical disruption of the virus particle, as has been observed for the thermal inactivation of some bacteriophage (27,28,33). In other cases, virus inactivation may be more subtle involving, for example, a mutation in the viral genome (34,351 or the cleavage of a viral protein (14,36). If so, then inactive virus particles may persist intact for some period of time, and these inactivated viruses may cause host cell infection through a process called multiplicity reactivation, particularly if the viral genome is segmented or multipartite (37). In developing eq 5, a number of assumptions were also made about the nature of virus coagulation. By using the von Smoluchowski equations as a starting point for this analysis, it is assumed that viral aggregates result only from binary collisions between particles and that stochastic effects caused by local fluctuations in the virus concentration are negligible (38). Simultaneous collisionsamong three or more particles would not be expected a t concentrations typically encountered in natural systems (39),and stochastic effects can generally be neglected if the particle concentration is not too low (40). This analysis also neglects the breakup or deaggregation of virus aggregates. Experimental data indicate that viral aggregates can be dispersed by treatment with sonic or ultrasonic waves (5, 7 , 8 , 4 1 )or by the addition of protein-rich nutrient broth (42)or proteolytic enzymes ( 5 ) . In natural environments, it is possible that viral aggregates may also be dispersed by changes in solution chemistry. For example, in one study, poliovirus aggregates were completely dispersed by dilution into clarified secondary sewage plant effluent (43). In such cases eq 5 will tend to overestimate the impact of coagulation on viral infectivity. In addition, the coagulation kernel is assumed to be invariant with aggregate size (Kij = K). The coagulation kernel accounts for the frequency with which particles collide and for the efficiency with which collisions result in aggregate formation. The collision frequency between particles should increase with increasing aggregate size, and theories have been proposed for estimating this relationship based on the self-similar nature of aggregate particles (44). The hydrodynamic, electrostatic, and dispersive forces that develop between particles on close approach are also strongly dependent on the size of the colliding particles, and this size dependence can affect coagulation as demonstrated by Monte Carlo simulation techniques (40). Taking these factors into account, the coagulation kernel should increase with increasing aggregate size, and coagulation should occur more rapidly as the size of the particles grows (44). Thus, by assuming a constant coagulation kernel, eq 5 will tend to overestimate the total number of virus particles or, equivalently, underestimate the impact of coagulation on viral infecEnvlron. Sci. Technol., Voi. 28,
No. 5, 1994 831
tivity. However, this is compensated in Figure 2 by assuming a unitary stability ratio (W = 1) which, in practice, will probably lead to an overestimate of the coagulation potential of a given environment, as discussed earlier. It should also be noted that viral aggregates may form during virus assembly in the cytoplasm of host cells, and these “original” aggregates can persist intact after cell lysis or virus release. In an electron microscope study of vaccinia virus, Sharp (6) found that approximately 90% of the virions released from a host cell were present in original aggregates, some of which contained 50 or more virions. Original aggregates have also been reported for herpes virus and other viruses (8), suggesting that intracellular aggregate formation may be a general feature of virus reproduction (18). Thus, to the extent that viral aggregates actually exist in aquatic environments, they are probably formed intracellularly and not by virion-virion coagulation outside the host cell. While the analysis presented in this study assumes that all viruses have the same inactivation rate, D values can also be computed for complex viral assemblages. In this case, upper-limit estimates for D are computed assuming W = 1 and using concentration estimates for the total viral population. The inactivation constant can be estimated from the inactivation rate of the viral population as a whole or from the inactivation rate of the most resistant virus in the population. As with the coagulation of homogeneous viruses, the resulting D values are only useful for determining if coagulation is unlikely (D < 1);if Q > 1,then coagulation is not necessarily implied. The analysis presented here may also prove useful for investigating the effect of coagulation on the survival of airborne viruses and viruses present in hematic fluids. Acknowledgments The author would like to thank the following individuals for their critical review of the manuscript: Drs. C. Chrysikopoulos, L. B. Grant, E. J. List, 0. Ogunseitan, T. 0. Olson, and three anonymous reviewers. This material is based upon work supported by the National Water Research Institute and the U.S. Environmental Protection Agency under Award NWRI/EPA 92-04. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the N.W.R.I. or the U S . EPA. Glossary ni t k Kij N,
932
concentration of ith aggregate size class (l/L3) time elapsed from an initial state in which virions are monodisperse and fully infective ( T ) first-order inactivation rate (UT) coagulation kernel for the collision of aggregates of size class i and j (L3/T) total particle concentration (singlets plus aggregates) (l/L3) initial total particle concentration (l/L3) N,/N,(O), normalized total particle concentration (dimensionless) ni/N,(O), normalized concentration of the ith aggregate size class k t , dimensionless time Boltzmann’s constant (1.38 X 10-l6 erg/K) temperature Environ. Sci. Technol., Vol. 28, No. 5, 1994
7b
dynamic viscosity (M/(L‘I?) (4kT)/3p, rate constant for diffusion-controlled coagulation (L3/T) stability ratio (dimensionless) 2/(KN,(O)), time scale for Brownian coagulation
0
l/(kq-,), ratio of inactivation to coagulation time
P kb
W
(T)
scales (dimensionless) Supplementary Material Available A mathematical proof demonstrating that the closed-form solutions presented in this paper (eqs 4 and 5) solve the modified Smoluchowski equations (eqs 2 and 3) for an initially monodisperse solution of particles (6 pages) will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper or microfiche (105 X 148 mm, 24X reduction, negatives) may be obtained from Microforms Office, American Chemical Society, 1155 16th St. NW, Washington, DC 20036. Full bibliographic citation (journal, title of article, names of authors, inclusive pagination,volume number, and issue number)and prepayment, check or money order for $16.50 for photocopy ($18.50 foreign) or $12.00 for microfiche ($13.00 foreign),are required. Canadian residents should add 7 % GST.
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* Abstract published in Advance ACS Abstracts, March 15,1994.
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