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  • Journal of Virological Methods 135 (2006) 118123

    A statistical method for comparing viral growth curves

    Gary P. Wang, Frederic D. Bushman University of Pennsylvania School of Medicine, Department of Microbiology, 3610 Hamilton Walk, Philadelphia, PA 19104-6076, United States

    Received 16 December 2005; received in revised form 10 February 2006; accepted 20 February 2006Available online 27 March 2006


    Viral replication is often analyzed by growth curves, in which viral multiplication in the presence of host cells is measured as a function of time.Comparing growth curves is one of the most sensitive ways of comparing viral growth under different conditions or for comparing replication ofdifferent viral mutants. However, such experiments are rarely analyzed in a statistically rigorous fashion. Here a statistical method is describedfor comparing curves, using replication of HIV in the presence of an integrase inhibitor as an example. A complication in the analysis arisesdue to the fact that sequential measurements of virus accumulation are not independent, which constrains the choice of statistical method. In therecommended approach, the values for virus accumulation over time are fitted to an exponential equation, then the means of the extracted growthrates compared using a nonparametric test, either the MannWhitney U-test for two samples or the KruskalWallis test for multiple samples. Aw






    eb-based tutorial for implementing this method is available at http://microb230.med.upenn.edu/tutorials/wangTutorial.2006 Elsevier B.V. All rights reserved.

    eywords: HIV; Integrase; Inhibitor; Antiviral; Statistics; Growth curve

    . Introduction

    Studies in virology are often conducted and reported in a fash-on that lacks statistical rigor. As a result, conclusions reachedre sometimes wrong. For most experiments in virology, how-ver, there are statistical procedures that allow a rigorous assess-ent of the possible significance of trends in the data. An excel-

    ent introduction can be found in Richardson and Overbaugh2005).

    Growth curve studies are widely used in virology. Growthurve experiments involve infecting cells with a viral stocknd monitoring viral replication over time (Delbruck, 1940).ften two conditions are compared, for example, a mutant andild-type virus or replication in the presence and absence of an

    ntiviral inhibitor. Some experiments monitor a single cycle ofiral replication, which can vary from 20 min for bacteriophagesp to several days for some animal viruses (Fields and Kinpe,996). Alternatively, growth can be monitored over many cyclesf viral replication, sometimes for many weeks. New virus pro-uction can be measured by removing an aliquot of supernatant

    and analyzing the number of infectious units present (Delbruck,1940), the amount of viral antigen (Michael and Kim, 1999) orthe amount of viral nucleic acid (examples can be found in Butleret al., 2001; Michael and Kim, 1999). Alternatively, the fractionof infected cells can be measured, as with an immunofluores-cence assay for cell-associated viral antigen (e.g. Michael andKim, 1999). For a virus that integrates its DNA into the hostgenome, the number of prophages or proviruses per cell can bemeasured (Butler et al., 2001; ODoherty et al., 2002).

    Viral growth curves are rarely if ever compared in a statis-tically rigorous fashion (for previous work and references; seeRichardson and Overbaugh, 2005; Spouge and Layne, 1999;Weinberg and Lagakos, 2001). To analyze trends in growth curvedata, multiple independent cultures are analyzed over time andthe data for each virus at each time point are typically plot-ted with error bars. Often, if the error bars do not overlap, thedata is judged to be significantly different (in the following,significant is used to imply statistically significant). Thisis unsatisfactory. For one thing, there is wide variation in theformulas used to determine the values for the error bars. Thepopular statistical package GraphPad Prism provides no fewer

    Abbreviations: HIV, human immunodeficiency virus; IN, integrase Corresponding author. Tel.: +1 215 573 8732; fax: +1 215 573 4856.

    E-mail address: bushman@mail.med.upenn.edu (F.D. Bushman).

    than four alternatives and the option selected is rarely specified.Furthermore, there can be cases where error bars overlap but thedifference is significant or where the error bars do not overlap butthe data is not significant (Richardson and Overbaugh, 2005). An

    166-0934/$ see front matter 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.jviromet.2006.02.008


  • G.P. Wang, F.D. Bushman / Journal of Virological Methods 135 (2006) 118123 119

    added difficulty is that experiments carried out on different daysmay have different extents of infection with all viruses tested(including the control) due to slight differences in the experi-mental manipulations. Averaging of experiments from differentdays can thus lead to an unreasonable increase in the apparenterror. For these reasons, researchers often select a representa-tive growth curve they believe represents the true response. Ofcourse, such data selection is very risky and probably accountsfor some cases of irreproducibility between laboratories.

    A method for statistical analysis of viral growth curves ispresented below. The mathematical machinery behind the meth-ods is well worked outreaders seeking more background arereferred to the following reference (Sokal and Rohlf, 1995,pp. 423541). Curves are fit to the exponential phase of viralgrowth and the rate extracted. The collection of rates is thencompared between treatments using a MannWhitney U-testor KruskalWallis test, and significance reported as a P-value.A tutorial describing implementation of this method in detailcan be found at http://microb230.med.upenn.edu/tutorials/wangTutorial.

    2. Methods

    2.1. Measurements of HIV replication

    HIV NL4-3 viral stock was prepared by transient transfectionihccl



    not split over the course of the experiment was also performedin parallel. In this case, culture supernatant was obtained dailyfor p24 analysis as described above; however, three-quarter ofthe supernatant was exchanged for fresh medium every 4 dayswithout removing cells in the culture.

    2.2. Statistical analysis

    The statistical package GraphPad Prism was used for fit-ting exponential growth data, the MannWhitney U-test and theKruskalWallis test. The detection of correlated error structurein the growth curve data was carried out by Dr. Charles Berryas follows. The log-transformed data was fit to linear mixed-effects models using R, and an AR1 model was found to fit thedata better than a repeated measures model.

    3. Results

    3.1. Sample HIV growth curves for statistical analyses

    As a model system for analysis, we have analyzed replica-tion of HIV in a T-cell line over multiple replication cycles(Fig. 1). HIV stocks (from strain NL4-3) were used to infectJurkat T-cells. The cells were incubated with the stock for 3 h,then washed and transferred to fresh medium. Samples werewithdrawn over 20 days and analyzed for viral output by quan-tm(




    n 293T cells using calcium phosphate. Viral supernatant wasarvested at 48 h, then filtered through a 0.45-m filter, con-entrated and stored frozen at 80 C. The amount of p24 viralapsid antigen in the viral stock was determined by enzyme-inked immunosorbent assay.

    Viral titers were determined using the GHOST indicatorell assay as described (Morner et al., 1999). Under conditionsf optimal infection (spinoculation and addition of 20 g/mlEAE dextran), infection of 40,000 GHOST cells with 100 ng24 resulted in 11.9 1.0% infection. Under the conditions usedn the example (no spinoculation or addition of DEAE dextran),nfection of 40,000 cells with 100 ng p24 yielded 0.34 0.043%nfection.

    Jurkat cells at a density of 8 105 ml1 were inoculated with00 ng of HIV NL4-3 p24 capsid antigen, in the presence of00 nM, 2 M or 20 M of integrase inhibitor L-731,988 orithout the inhibitor. Each culture condition was performed inuadruplicate. Cells were cultured for 3 h, washed and trans-erred to 2.5 ml fresh RPMI 1640 medium containing 10%etal calf serum, 50 g/ml gentamicin, 2 mM l-glutamine andntegrase inhibitor at the same concentration, and cultured at7 C with 5% CO2. One hundred microliters of supernatantas removed from each culture daily, and fresh medium con-

    aining the same integrase inhibitor concentration was addedack to the culture. The amount of p24 capsid antigen in theupernatant was measured by ELISA. To maintain similar cellensity for the duration of the experiment, cells were split everydays by diluting one-fourth the culture with four-fold of freshedium containing the same inhibitor concentration. The cell

    ensity prior to splitting reached about 22.5 106 ml1, andas consistent throughout the experiment. A culture that was

    ifying the amount of p24 capsid antigen present in the cultureedium. Three levels of an HIV-1 integrase inhibitor, L-731,988

    Hazuda et al., 2000), were added for comparison.As can be seen in Fig. 1A (no inhibitor), the amount of p24

    n the medium increased over the time of culture. A technicalssue is introduced by the long duration of the experiment. Aypical culture of uninfected Jurkat cells must be split peri-dically to keep the density of cells roughly constant (in thisase, one-quarter the culture was diluted with three volumes ofresh medium). However, splitting dilutes the amount of virus inhe culture medium and reduces the number of virus-producingel