Quantifying relative virulence: When μmax fails and AUC alone just isn’t enough

One of the more challenging aspects in quantitative virology is quantifying relative virulence between two (or more) viruses that have different replication dynamics in a given susceptible host. Host growth curve analysis is often used to detail virus-host interactions and to determine the impact of viral infection on a host. Quantifying relative virulence using canonical parameters such as maximum specific growth rate (μmax) can fail to provide accurate information regarding experimental infection, especially for non-lytic viruses. Although area-under-the-curve (AUC) can be more robust by through calculation of a percent inhibition (PIAUC), this metric can be sensitive to limit selection. In this study, using empirical and extrapolated data from Sulfolobus Spindle-shaped Virus (SSV) infections, we introduce a novel, simple metric that is proven to be more robust and less sensitive than traditional measures for determining relative virulence. This metric (ISC) more accurately aligns biological phenomena with quantified metrics from growth curve analysis to determine trends in relative virulence. It also addresses a major gap in virology by allowing comparisons between non-lytic single-virus/single-host (SVSH) infections and between non-lytic versus lytic virus infection on a given host. How ISC may be applied to polymicrobial infection – both coinfection of a host culture and superinfection of a single cell with more than one virus (or other pathogen type) is a topic of ongoing investigation.

ABSTRACT 10 11 One of the more challenging aspects in quantitative virology is quantifying relative virulence 12 between two (or more) viruses that have different replication dynamics in a given susceptible host. 13 Host growth curve analysis is often used to detail virus-host interactions and to determine the 14 impact of viral infection on a host. Quantifying relative virulence using canonical parameters such 15 as maximum specific growth rate (µ max ) can fail to provide accurate information regarding 16 experimental infection, especially for non-lytic viruses. Although area-under-the-curve (AUC) can 17 be more robust by through calculation of a percent inhibition (PI AUC ), this metric can be sensitive 18 to limit selection. In this study, using empirical and extrapolated data from Sulfolobus Spindle-19 shaped Virus (SSV) infections, we introduce a novel, simple metric that is proven to be more 20 robust and less sensitive than traditional measures for determining relative virulence. This metric 21 (I SC ) more accurately aligns biological phenomena with quantified metrics from growth curve 22 analysis to determine trends in relative virulence. It also addresses a major gap in virology by 23 allowing comparisons between non-lytic single-virus/single-host (SVSH) infections and between 24 non-lytic versus lytic virus infection on a given host. How I SC may be applied to polymicrobial 25 infection -both coinfection of a host culture and superinfection of a single cell with more than one 26 virus (or other pathogen type) is a topic of ongoing investigation .  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45 INTRODUCTION  47  48 Quantifying relative virulence (V R ) is challenging when comparing viruses that exhibit different 49 replication dynamics in a given host. Although host growth curve analysis is often used to detail 50 virus-host interactions and to determine the level of detriment a virus levies on host growth, 51 assessing V R via canonical measures of fitness, such as maximum specific growth rate (µ max ) [1], 52 can fail to accurately describe experimental infection datasets [2], especially for non-lytic viruses. 53 In non-lytic virus systems, progeny virions are released via budding rather than gross cell lysis and 54 growth curves for hosts infected with non-lytic viruses can exhibit non-canonical growth profiles, 55 including absence of a lag phase and very brief exponential growth followed by a prolonged period 56 of non-exponential (but positive) growth prior to stationary phase. 57 58 Using empirical and extrapolated data from Sulfolobus Spindle-shaped Virus (SSV) infections, we 59 introduce a novel, simple metric that overcomes limitations of traditional growth curve analysis 60 when quantifying relative virulence between two viruses independently infecting a common host. Comparison of host growth using maximum growth rate (µ max ) as a metric for relative virulence. 78 Two different intervals were considered. First, an interval from 0 to 36 hours post-infection (HPI), 79 which best represents the archetypal "exponential growth phase" [11] was considered ( Fig. 1C). 80 Using µ max from the Gompertz, SSV1 is less virulent than SSV8 while SSV2 is the most virulent 81 (see Fig. 1C). Given that host growth subject to non-lytic viral infection does not always exhibit a 82 classical Monodian profile, an outer bound at 90 HPI was used to capture more of the growth curve 83 (Fig. 1D). Calculating µ max from the Gompertz for this larger portion of the data changes the results. 84 Specifically, SSV8 appears to exhibit approximately equal virulence to SSV2, while SSV1 remains 85 the least virulent (see Fig. 1D).

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Thus, for non-lytic infections, a significant change in µ max , which drives interpretation of results, 88 can emerge depending on how much of the curve is considered. Depending on culture size and 89 specific virus-host pairing, the truly exponential growth phase may be brief with the majority of 90 positive growth comprising the classically described deceleration, before stationary phase. A widely used and agreed upon alternative is to calculate µ max from a log transformed dataset [12]. 99 Calculating µ max from log transformed data (i.e., ln OD/OD 0 ) using narrow (0-36 HPI) and 100 expanded (0-90 HPI) intervals yields another outcome. Using this method, SSV8 appears to be 101 least virulent while SSV2 and SSV1 are roughly equivalent but greater than SSV8 (Fig. 1E, F). 102 Adding an additional normalization step to compensate for slightly different host cell densities 103 measurements at time of viral inoculation (t 0 ), yields yet another result for the data (Fig. 1G, H). 104 Notably, µ max calculated from the Gompertz fit to the normalized log transformed data suggest that 105 SSV2 is least virulent followed by SSV1 with SSV8 exhibiting highest virulence. Remarkably, 106 none of these analytical adjustments for µ max as the principal parameter for relative virulence 107 captures the known relationship of SSV1, SSV2, and SSV8 virulence on Sulfolobus strain Gq 108 [7,10]. Thus, methods of determining V R , using µ max as the key parameter, are inadequate. 109 110

Comparison of host growth using area-under-the-curve (AUC) as a metric for relative virulence. 111
Given limitations of µ max in determining V R in non-lytic viral infections, an alternative approach is 112 to determine a percent inhibition (PI) based on area-under-the-curve (AUC) [13][14][15][16]. Specifically, 113 determining AUC for infected host growth (AUC infected ) and uninfected control (AUC CTL ) provides 114 a calculated PI AUC on non-transformed data such that: 115 116 This may be alternatively written as: 119 120 Selecting upper and lower bounds of the integration are critical [17]. Yet, approaches for choosing 123 these bounds are varied and often arbitrary [14,15]. In many reports, the upper bound is chosen 124 absent any noted mathematical or biological explanation. For comparing virus infection impacts, 125 the time of inoculation (t 0 ) is a reasonable lower bound; it will capture early changes in host growth. 126 Historically, choice of an upper bound has been subjective. Prior work has relied on a pre-defined 127 end-point after culture initiation (e.g., cancer research). A reasonable upper bound is at the 128 beginning of stationary phase or peak growth (i.e., N asymptote ). However, non-canonical host growth 129 during infection may render this value difficult to determine. 130 131 Using extremes for the outer limit at 36 HPI and 90 HPI for the Sulfolobus strain Gq-SSV dataset 132 (Fig. 2), AUC is calculated. Given that truly "exponential" growth can be brief for non-lytic 133 infections followed by a long non-exponential growth phase, 36HPI represents a conservative 134 lower limit. Alternatively, the 90 HPI bound captures more of the data extending deeply into the 135 positive growth phase and capturing the growth peak of the uninfected control curve (see Fig. 2). 136 Bound at 36 HPI, AUC calculations indicate that SSV1 < SSV2 < SSV8, which is the correct 137 relative virulence between these viruses. However, since this only represents one-third of the 138 dataset, this is only a fortuitous result. 139 140 141 142 To capture a larger component of the virus-host interaction through the peak growth (N asymptote ) of 143 the uninfected control data, the bound was moved to 90 HPI, yielding: SSV1 << SSV8 ≲ SSV2. 144 This is inaccurate and demonstrates that assessing virulence using PI from AUC is unreliable and 145 sensitive to limit selection. What is needed is a metric that captures a significant component of the 146 virus-host interaction (i.e., to peak growth) while yielding the correct V R between viruses.

Comparison of host growth using Stacy-Ceballos Equation as a metric for relative virulence. 149
Although µ max and AUC can be useful parameters for characterizing: drug interactions [18] or 150 attenuated/enhanced growth in mutant versus wild-type cell growth [14,19], when comparing 151 virulence between non-lytic viruses on a host, these parameters are not resilient to changes in 152 integral bounds. An essential component of virus-host interactions is N asymptote (i.e., peak growth), 153 which is a critical but often ignored parameter in growth curve analysis [17]. By considering both 154 percent inhibition of the growth phase as well as the percent inhibition in N asymptote , a more robust 155 representation of V R can be determined. Notably, the square root of the product of PI AUC and PI max , 156 introduced here as Stacy-Ceballos inhibition (I SC ), provides a robust index of V R , where: 157 158 Using I SC , the correct order of increasing virulence emerges for both the 36 HPI and 90 HPI limits 164 ( Fig. 2F) with the latter representing a broad range across the virus-host dynamic (see Fig. 2A-D).

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I SC is a robust index for V R that is resilient to limit selection. Previous cautions against combining 166 parameters into a single metric are acknowledged [20]; however, combining PI AUC and PI max allows 167 for inclusion of variation between growth curves beyond exponential phase. Furthermore, the use 168 of relative measures allows for meaningful comparisons across virus-host systems. Although 169 growth curves with similar growth patterns will result in an I SC value similar to PI AUC , the former 170 more reliably quantifies differences between growth curves exhibiting distinct growth patterns. 171 172 CONCLUSIONS 173 174

Generalizability of Stacy-Ceballos Inhibition (I SC ) as metric for relative virulence. 175
The non-lytic Sulfolobus Spindle-shaped Virus (SSV) system, provides one example of how 176 traditional parameters for assessing relative virulence (i.e., µ max and AUC) between two (or more) 177 viruses on a given host may yield unreliable results and incorrect interpretations of infection data. 178 Using Stacy-Ceballos Inhibition (I SC ) as a metric for calculating relative virulence overcomes the 179 sensitivity of these parameters providing a more robust and reliable approach for determining V R . 180 Importantly, the robustness and reliability of this approach is not constrained to non-lytic viruses.

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It is also useful when comparing non-lytic versus lytic infections. In this case, PI max for the lytic 182 system would be the maximum cell density reached prior to lysis. The ability to accurately assess 183 differences in virulence between lytic viruses and non-lytic viruses or changes in virulence as a 184 virus switches between non-lytic (but productive) and lytic phases offers new opportunities in 185 characterizing single-virus/single-host interactions. In a separate report, reliability and robustness 186 of this approach is demonstrated across multiple areas of application [17]. We are also assessing 187 the applicability of I SC to polymicrobial infections. 188 189 190 191 192 193 194