# Mathematical Modeling Finds Disparate Interferon Production Rates Drive Strain-Specific Immunodynamics during Deadly Influenza Infection

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Development Rationale and Equations

_{V,V}, and a carrying capacity, K

_{V,V}. This form of virus production was selected over target-cell-based modeling approaches because data concerning the number of available target cells in the lung are not available, limiting the viability and accuracy of training a model. The effect of interferon-regulated inhibition of virus replication is modeled using mass action kinetics, where r

_{V,I}is the corresponding rate constant. The inhibition of virus production via macrophage is also modeled with mass-action kinetics, where r

_{V,M}is the rate constant. Virus degrades at a rate, d

_{V}.

_{I,V}, relative to viral load, and decays at rate, d

_{I}. Upregulation of interferon production via macrophages was modeled as a first-order mass-action kinetic with a rate, r

_{I,M}.

_{M,I}, and an apparent dissociation constant, K

_{M,I}. Instead of the classic interpretation of the Hill coefficient, n, as cooperativity in ligand binding [43], it can be interpreted in this context as an activation threshold representing the threshold of interferon needed to induce macrophage production. This is similar to the activation threshold that must be exceeded to induce T cell cytokine production [44,45]. The parameter, K

_{M,I}, is not raised to the Hill-like coefficient, n, to improve parameter fitting. Macrophage decays at a rate of d

_{M}.

#### 2.2. Experimental Data Collected from Literature and Relating the Data to the Model

^{5}PFU. A control group was mock-infected with PBS. At 14 time points spanning the first week of infection, three animals per infection group were sacrificed. Their lungs were harvested and analyzed by a variety of techniques to quantify the viral load and the state of the immune system. The H5N1-infected animals died between days 5 and 7. As such, only the first 13 measurements spanning days 0–5 are included in this work. In all, 234 measurements (78 for each model state) were collected and organized for model parameterization.

_{10}of PFU/mg. To represent the change in interferon concentration over time in Equation (2), log

_{2}fold change of the gene expression of Ifnb1 relative to mock-infected, time-matched samples (unitless) was used. Full details on normalizing the gene expression can be found in the original work [1]. Whole lung macrophage counts were determined at only four time points in the original work, spread across several days [1]. As a result, the concentration of MCP1 (measured using ELISA assay) was selected to act as a surrogate measurement of macrophage cell count (M). Supplementary Figure S1 shows a linear regression of the log

_{10}macrophage cell count and log

_{2}MCP1 concentration (${R}^{2}=0.98$, with a slope of 0.613). The conversion between macrophage and MCP1 is, therefore, given by Equation (4):

_{2}of MCP1 measurements. Equation (4) is then used to transform MCP1 predictions into estimates of macrophage counts in the lung.

#### 2.3. Parameter Training

_{x,t}and O

_{x,t}are the model output and the average of triplicate observed data points, respectively, for each state, x, and time point, t, across all states, X, and time points, T. Each time point was divided by the corresponding data point, O

_{x,t}, to normalize energy values. All MCMC simulations ran across six chains of temperature (0.99, 0.9, 0.8, 0.4, 0.2, and 0.05) to ensure adequate exploration of parameter space. Parameters were unbounded, and priors were defined as uniform between zero and +∞.

#### 2.4. Model and Scenario Prioritization

_{free}is the number of parameters being fit. The number of free parameters in a model depends on the scenario being considered, which is described in the Section 3. The maximum likelihood, MLE, is defined as:

#### 2.5. Sensitivity Analysis

## 3. Results

#### 3.1. In Silico Screenings of Candidate Innate Immune Models Find That H5N1 and H1N1 Viruses Induce Interferon Production at Different Rates In Vivo

_{I,V}, takes on H5N1- and H1N1-specific values. All four models achieve their lowest AIC under this condition (noted in Figure 2), with Model 4 achieving the lowest AIC overall. This suggests that virus-induced interferon production is regulated in a strain-dependent manner, a proposition that is independent of the model, and therefore, macrophage activity, employed. These findings also suggest that Model 4 is the best model for regressing against the H5N1 and H1N1 immunologic data.

#### 3.2. Strain-Specific Interferon Production Is Not an Artifact of Parameter Sensitivity

_{I,V}was the most sensitive model parameter. We conducted a sensitivity analysis of all the models to each of their constituent parameters using the eFAST algorithm [53,54]. The sensitivity of each state is reported in the form of fractional variance that can be explained by the variance of a single parameter, p. These indices are shown in Figure 3.

_{M,I}and r

_{I,M}), whereas the concentration of virus (V) is primarily dependent on the rate of interferon induction by the virus, r

_{I,V}. This trend holds for Models 2 and 3. In Model 4, the concentration of interferon (I) and the macrophage count (M) are most sensitive to the rate of interferon induction by the virus, r

_{I,V}, whereas the concentration of virus (V) is most sensitive to r

_{V,V}. This establishes that the four model structures have unique control schemes, i.e., the most sensitive parameters differ between the different models. This also demonstrates that the minimum AIC values of r

_{I,V}OSSD models during the in silico screen were not simply the result of r

_{I,V}being the most sensitive parameter. Thus, the remainder of this work comprises further analyses using Model 4 to understand the parameter space associated with the model fitting to H5N1- and H1N1-specific data.

#### 3.3. Deep Exploration of Model 4’s Parameter Space Using PT MCMC

_{I,V}is allowed to take on H5N1- and H1N1-specific values. However, further exploration of the parameter space using Parallel Tempering Markov Chain Monte Carlo (PT MCMC) parameterization was needed to determine the breadth of the parameter space that supported Model 4’s best fit to H5N1 and H1N1 data. Using PT MCMC, we re-evaluated all of the scenarios described in Figure 1B for Model 4. For each MCMC optimization, 2 million iterations were run.

#### 3.4. MCMC-Based Parameter Exploration Again Finds That H5N1 and H1N1 Viruses Induce Interferon Production at Different Rates In Vivo

_{I,V}, is allowed to have strain-specific values. Minimum energy values fall between 9 and 13 except in the case where the rate of interferon production, r

_{I,V}, is independently estimated, which yields a minimum energy of 6.65. Though this is closest to the minimum AD energy for Model 4 (3.33), AIC calculations reveal that the resulting value of 35.30 for r

_{I,V}is not only lower than the results of the other nine OSSD parameterizations of Model 4, but is lower than that of the high degree of freedom AD results. Overall, using MCMC instead of basin-hopping for data fitting did not lead to a different conclusion with regards to the optimal solution occurring when r

_{I,V}is independently estimated for H5N1 and H1N1.

_{I,V}OSSD scenario shows a distinct improvement in fit over the NSSD results. When each parameter is allowed to differ between strains, histograms can inform whether the strains’ parameter distributions are unique. Focusing on the r

_{I,V}OSSD scenario histograms, a comparison of the resultant top 1000 parameter distributions across strains yields a significant difference between distribution means (Mann–Whitney test p < 0.001 for r

_{I,V}between H1N1 (blue) and H5N1 (red), Figure 6), indicating that the strains have unique values for this parameter. All other parameter distributions for OSSD models overlap significantly (Supplemental Figure S4), except for d

_{I}. Combined with the AIC results in Table 1, these results highlight that r

_{I,V}OSSD achieves the most statistically defensible fit to the datasets.

#### 3.5. Independent Estimation of Virus Parameters per Strain Does Not Improve Model AIC

_{V,V}, or death rate, d

_{V}. To test this, Model 4 was parameterized such that the viral state parameters, r

_{V,V}, K

_{V,V}, r

_{V,I}, and d

_{V}(denoted {V}), could take on different values when training to the H5N1 and H1N1 data, while all other parameters remained shared between strains. Six additional “Virus-Host” parameterizations were performed with the addition of one of the non-viral state parameters, {V} +OSSD (DoF: 15).

_{I,V}achieving the best fit to data. Corresponding minimum energy and AIC values are found in Table 2. A comparison of the top 1000 parameter distributions per strain yields significant differences between distribution means, except for r

_{V,I}in {V} + K

_{M,I}(Mann–Whitney test p < 0.001 for all independently estimated parameters). This indicates that virus-related kinetic parameters likely vary between strains. Minimum energies associated with the {V} + OSSD parameterizations are lower than that of {V} alone, with a minimum energy of 5.55, associated with the independent fitting of {V} + r

_{I,V}. Compared to the AD and NSSD scenarios, {V} + r

_{I,V}results in a lower AIC value, reiterating the role of interferon production rate in strain-specific infection dynamics. Although strain-specific viral parameters are demonstrably present in the datasets, {V} + r

_{I,V}has a higher AIC than the r

_{I,V}OSSD scenario. This attributes great importance to strain-dependent interferon production rate over simple strain-dependent viral kinetics and implies that increased degrees of freedom are detrimental to model quality. Investigations with higher degrees of freedom were not performed due to the computational time required for each MCMC fit to run 2 million samples per study.

## 4. Discussion

_{I,V}, was allowed to take on different values while training to each infection cohort—regardless of the model employed. The robustness of this finding is further supported by the wide distribution of parameter values which optimally fit the data, quantified by the MCMC analysis, and by the results of the sensitivity analysis. One concern about our in silico screening approach, and indeed in model-based analysis in general, is that the most sensitive parameters are often identified as the most important for maintaining phenotypes, as they are the easiest to use for tuning system dynamics. Across the four models considered here, the top parameters to which the model outputs are sensitive differed (Figure 3). Nonetheless, r

_{I,V}was identified as the most likely candidate across all four models. Finally, in Figure 5, MCMC analysis showed that the best fit for the scenario with strain-specific r

_{I,V}values could be achieved for a wide range of parameter values. It was found that the rate of interferon production, r

_{I,V}, is approximately 2–3 times faster in H5N1-infected lung cells. Additional analyses were performed to consider strain-specific virus replication rates combined with strain-specific immune rates (Table 2; Figure 7). Our work demonstrates that strain-dependent differences arise from host–virus interactions and immunological reactions, rather than strain-specific viral replication behavior.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Parameter | H1N1 | H5N1 | Unit |
---|---|---|---|

${r}_{V,V}$ | 1.22 | 1.21 | $day{s}^{-1}$ |

${K}_{V,V}$ | 3.65 × 10^{1} | 7.80 × 10^{2} | $lo{g}_{10}\left(PFU/mg\right)$ |

${r}_{V,I}$ | 1.20 × 10^{−1} | 1.07 × 10^{−1} | $day{s}^{-1}$ |

${d}_{V}$ | 1.61 × 10^{−1} | 1.10 × 10^{−5} | $day{s}^{-1}$ |

${r}_{I,V}$ | 7.70 × 10^{−1} | 3.06 | ${\left[lo{g}_{10}\left(PFU/mg\right)hours\right]}^{-1}$ |

${d}_{I}$ | 9.59 × 10^{−1} | 3.22 | $day{s}^{-1}$ |

${r}_{M,I}$ | 2.16 × 10^{7} | 9.71 × 10^{3} | $\frac{MacrophageCellCount}{days}$ |

${K}_{M,I}$ | 1.90 × 10^{5} | 1.04 × 10^{9} | unitless |

${d}_{M}$ | 8.80 × 10^{3} | 6.18 × 10^{−1} | $day{s}^{-1}$ |

$n$ | 5.47 | 9.98 | unitless |

${r}_{V,M}$ | N/A | ${\left[MacrophageCellCountdays\right]}^{-1}$ | |

${r}_{I,M}$ | N/A | ${\left[MacrophageCellCountdays\right]}^{-1}$ |

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**Figure 1.**(

**A**) Model schemes of the four models considered in this work. V, I, and M represent virus concentration, interferon concentration, and macrophage cell count in the lungs of infected mice. Arrows represent activating interactions; lines ending in crosses represent inhibiting interactions. The parameters involved in each interaction are indicated in Model 1 (degradation reactions not shown). Model 1 is the fully connected model of the innate immune response model. Models 2–4 are reduced versions of Model 1, wherein select interactions were removed. (

**B**) Each model is analyzed for its goodness of fit to experimental data under three different scenarios. Schemes of the model emphasize the different outcomes that occur under each scenario. Black arrows indicate parameters that retain the same value when fitting the model to H5N1 and H1N1 infection data. Red, broken arrows identify parameters that take on different values when training two copies of a model to the H5N1 and H1N1 infection data.

**Figure 2.**Energy versus AIC values for all four model structures under different parameterization scenarios (All Different (AD), One Strain-Specific Difference (OSSD), and No Strain-Specific Difference (NSSD)). The Model 4 OSSD r

_{I,V}scenario yields the global minima.

**Figure 3.**First-order indices of the eFAST sensitivity analysis for each model as described in Figure 1A. Indices are reported as the normalized change for each model state, for each parameter.

**Figure 4.**The top 1000 fits of Model 4 to the H1N1 (

**top row**) and H5N1 data (

**bottom row**) when using PT MCMC parameterization. The top fits under the AD scenario (all parameters allowed to independently estimate across strains) are shown in black, and NSSD results (all parameters shared between strains) are shown in blue. Intervals represent the standard deviation of the 1000 lowest energy parameter sets. Data from Shoemaker et al. [1] are shown as circles, with bars indicating the standard deviation.

**Figure 5.**Model 4 output for the minimum energy parameter set (lines) for OSSD parameterizations and corresponding training data (markers) for H1N1 (

**top row**) and H5N1 (

**bottom row**). Data from Shoemaker et al. [1] are shown with the standard deviation associated with triplicate data points per time point.

**Figure 6.**Posterior density distributions for all parameters for Model 4, with r

_{I,V}varying between strains. Only r

_{I,V}can have strain-specific values. All other parameters have the same value when fitting the model to H5N1 and H1N1 data. The x axis is given in log

_{10}Parameter Value. Distributions result from the 1000 lowest energy solutions identified using PT MCMC. Narrow posterior distributions indicate that the parameter had a small range of values under which the model optimally fit the data, whereas broad distributions indicate that a range of values would yield fits of the same energy.

**Figure 7.**Model 4 output for minimum energy parameter set (line) for virus-related parameter independent ({V}) and corresponding training data (markers) for H1N1 (

**top row**) and H5N1 (

**bottom row**). {V} is representative of four viral parameters: r

_{V,V}, K

_{V,V}, r

_{V,I,}, and d

_{V}. Data from Shoemaker et al. [1] are shown with the standard deviation associated with triplicate data points per time point.

**Table 1.**The minimum energy, degrees of freedom (DoF), and AIC values achieved by Model 4 for each scenario. The independent parameter column identifies the parameter allowed to take on different values while training two copies of the model to the H5N1 and H1N1 data.

Scenario | Independent Parameter | Energy | DoF | AIC |
---|---|---|---|---|

NSSD | None | 15.04 | 10 | 50.08 |

AD | All | 3.33 | 20 | 46.66 |

OSSD | ${r}_{V,I}$ | 10.83 | 11 | 43.66 |

${r}_{V,V}$ | 9.37 | 11 | 40.74 | |

${K}_{V,V}$ | 9.79 | 11 | 41.59 | |

${d}_{V}$ | 9.65 | 11 | 41.31 | |

${r}_{I,V}$ | 6.65 | 11 | 35.30 | |

${d}_{I}$ | 10.3 | 11 | 42.61 | |

${r}_{M,I}$ | 12.36 | 11 | 46.73 | |

${k}_{M,I}$ | 12.29 | 11 | 46.57 | |

$n$ | 12.28 | 11 | 46.57 | |

${d}_{M}$ | 12.37 | 11 | 46.75 |

**Table 2.**The minimum energy, degrees of freedom (DoF), and AIC values for all seven viral subset ({V}) studies. {V} is representative of four viral parameters: r

_{V,V}, K

_{V,V}, r

_{V,I}and d

_{V}. Model scenarios are given in Figure 1. Independent parameter identifies the parameters allowed to take on different values when training to the H5N1 and H1N1 data.

Scenario | Independent Parameter | Energy | DoF | AIC |
---|---|---|---|---|

{V} | {V} | 9.34 | 14 | 46.68 |

{V} + OSSD | $\left\{V\right\}+{r}_{I,V}$ | 5.55 | 15 | 41.11 |

$\left\{V\right\}+{d}_{I}$ | 8.38 | 15 | 46.75 | |

$\left\{V\right\}+{r}_{M,I}$ | 8.86 | 15 | 47.72 | |

$\left\{V\right\}+{K}_{M,I}$ | 8.89 | 15 | 47.79 | |

$\left\{V\right\}+n$ | 8.92 | 15 | 47.85 | |

$\left\{V\right\}+{d}_{M}$ | 8.89 | 15 | 47.78 |

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## Share and Cite

**MDPI and ACS Style**

Ackerman, E.E.; Weaver, J.J.A.; Shoemaker, J.E.
Mathematical Modeling Finds Disparate Interferon Production Rates Drive Strain-Specific Immunodynamics during Deadly Influenza Infection. *Viruses* **2022**, *14*, 906.
https://doi.org/10.3390/v14050906

**AMA Style**

Ackerman EE, Weaver JJA, Shoemaker JE.
Mathematical Modeling Finds Disparate Interferon Production Rates Drive Strain-Specific Immunodynamics during Deadly Influenza Infection. *Viruses*. 2022; 14(5):906.
https://doi.org/10.3390/v14050906

**Chicago/Turabian Style**

Ackerman, Emily E., Jordan J. A. Weaver, and Jason E. Shoemaker.
2022. "Mathematical Modeling Finds Disparate Interferon Production Rates Drive Strain-Specific Immunodynamics during Deadly Influenza Infection" *Viruses* 14, no. 5: 906.
https://doi.org/10.3390/v14050906