Figures and data in Dynamically linking influenza virus infection kinetics, lung injury, inflammation, and disease severity

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Tables
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11 figures, 1 table and 5 additional files

Figures

Figure 1 with 2 supplements

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Schematic and fit of the CD8⁺ T cell viral kinetic model.

(A) Schematic of the CD8⁺ T cell model in Equation (1)-(6). In the model, target cells ( $T$ ) are infected at rate $β V$ . Infected cells ( $I_{1}$ ) undergo an eclipse phase and transition to become productively-infected cells ( $I_{2}$ ) at rate $k$ . Virus ( $V$ ) is produced by infected cells at rate $p$ and is cleared at rate $c$ . Infected cells are cleared at rate $δ$ by non-specific mechanisms and at rate $δ_{E} (I_{2}, E)$ by effector CD8⁺ T cells ( $E$ ; denoted CD8_E). The dashed lines represent interactions between infected cells and CD8_E. Initial CD8_E influx ( $ξ (E) = ξ / (K_{E} + E)$ ) is proportional to infected cells and is limited by CD8_E with half-saturation constant $K_{E}$ . CD8_E expansion ( $η$ ) occurs proportional to infected cells $τ_{E}$ days ago. Memory CD8⁺ T cell ( $E_{M}$ ; denoted CD8_M) generation occurs at rate $ζ$ and proportional to CD8_E $τ_{M}$ days ago. (B) Fit of the CD8⁺ T cell model (Equation (1)-(6)) to virus and total CD8⁺ T cells from the lungs of mice infected with 75 TCID₅₀ PR8 (10 mice per time point). The total number of CD8⁺ T cells is $\hat{E} = E + E_{M} + {\hat{E}}_{0}$ . (C) Total CD8⁺ T cells in the lung parenchyma (gray circles) and overlay of the model predicted values ( $E + E_{M}$ ). (D) Fit of the model to virus, CD25^–CD43⁺ CD8⁺ T cells (cyan diamonds; $E$ ), and CD43^–CD127⁺CD103⁺CD69⁺ CD8⁺ T cells (blue diamonds; $E_{M}$ ) (five mice per time point). The solid and dashed black lines are the optimal solutions and the gray shading is are the model solutions using parameter sets within the 95% CIs. Parameters are given in Table 1. Data are shown as mean ± standard deviation.

Figure 1—figure supplement 1

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Dynamics of CD8⁺ T cells in the lung parenchyma and vasculature.

Comparison of total CD8⁺ T cells in a non-perfused lung (black) and in a perfused lung (white). Of those from a perfused lung, comparison of CD8⁺ T cells located in the lung parenchyma (CD45^–; gray) and vasculature (CD45⁺; yellow). These data show that perfusion does not alter the number of cells recovered, and that the total number of cells primarily reflects changes in the lung parenchyma with little fluctuation in the lung vasculature. The cells in the lung vasculature may be slightly overestimated late in the infection due to incomplete perfusions during the height CD8-mediated clearance (7–9 d pi) when the lung integrity is low.

Figure 1—figure supplement 2

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Flow cytometry gating strategy for CD8⁺ T cell analysis.

Live cells were first gated on forward scatter (FSC-A) and side scatter (SSC-A) then as singlets. (A) Following neutrophil (Ly6G^hi) and macrophage (F4/80^hi CD11c^hi) exclusion, T cells were gated as CD3e⁺ with total CD8⁺ T cells sub-gated as DX5^–CD4^–CD8⁺. (B) Blood borne CD8⁺ T cells were gated as CD45⁺ cells and those in the lung parenchyma as CD45^– with the total in each sub-population gated as CD3e⁺B220^–DX5^–CD4^–CD8⁺. IAV-specific CD8⁺ T cells were sub-gated as CD25⁺CD43⁺ (recently activated), CD25^–CD43⁺ (effector), and CD25^–CD43^–CD127⁺CD103⁺CD69⁺ (long-lived memory). Expression of CD44, CD69, CD62L, and NKG2D were also assessed to ensure appropriate classification.

Figure 2 with 2 supplements

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Parameter ensembles and histograms.

Parameter ensembles (A) and histograms (B) resulting from fitting the CD8⁺ T cell viral kinetic model (Equation (1)-(6)) to viral titers and total CD8⁺ T cells from mice infected with 75 TCID₅₀ PR8. (A) The rates of infected cell clearance by non-specific mechanisms (δ) and by CD8_E ( $δ_{E}$ ) are slightly negatively correlated. Correlations were also present between the rates of CD8_E clearance ( $d_{E}$ ), CD8_E expansion (η), and CD8_M generation (ζ). The axes limits reflect imposed bounds. Additional ensemble plots are in Figure 2—figure supplements 1–2. (B) The histograms show that the majority of parameters are well-defined with the exception of the eclipse phase transition rate ( $k$ ), one of the half-saturation constants ( $K_{E}$ ), and the CD8_M generation delay ( $τ_{M}$ ).

Figure 2—figure supplement 1

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Parameter ensembles.

Parameter ensembles resulting from fitting the CD8⁺ T cell model (Equation (1)-(6), Main Text) to viral loads and CD8⁺ T cells from mice infected with 75 TCID₅₀ PR8. The axes limits reflect the imposed bounds. Additional ensemble plots are in Figure 2 (Main Text) and Figure 2—figure supplement 2.

Figure 2—figure supplement 2

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Figure 3

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Density-dependent infected cell clearance by CD8⁺ T cells and their impact on recovery time.

(A) The rate of CD8_E-mediated infected cell clearance ( $δ_{E} (I_{2}, E) = δ_{E} E / (K_{δ_{E}} + I_{2})$ ) plotted as a function of infected cells ( $I_{2}$ ) and effector CD8⁺ T cells ( $E$ ; CD8_E). The colored markers (denoted a–g) indicate the infected cell clearance rate that corresponds to different time points during the infection for the best-fit solution. (B) Values of $δ_{E} (I_{2}, E)$ for the indicated time points associated with the markers a–g. (C) Corresponding locations of the various $δ_{E} (I_{2}, E)$ values (markers a–g) on the best-fit solution of the CD8⁺ T cell model for virus ( $V$ ), infected cells ( $I_{2}$ ), and CD8_E ( $E$ ). (D) Solutions of the CD8⁺ T cell model (Equation (1)-(6)) for virus ( $V$ ) and total CD8⁺ T cells ( $\hat{E}$ ) using the best-fit parameters (black line) and when varying the CD8_E expansion rate ( $η$ ; magenta lines) to illustrate how different total CD8⁺ T cell magnitudes alter infection duration. The magenta lines are solutions from when the percent ${\hat{E}}_{\max}$ relative to ${\hat{E}}_{\max}$ from the best-fit solution was 42% (solid line), 39.2% (dash-dotted line), 39.1% (dashed line), or 37% (dotted line). (E) The time at which infected cells reach the half-saturation constant ( $I_{2} = K_{δ_{E}}$ ; gray circles) and the infection duration (time where $\log_{10} V = 0$ ; black diamonds) are shown for the various CD8⁺ T cell magnitudes. The gray line between these points is the time required to eliminate $K_{δ_{E}}$ infected cells and achieve complete resolution of the infection ( $\log_{10} V = 0$ ). (F) Fit of the CD8⁺ T cell model (Equation (1)-(6)) to viral loads and CD8⁺ T cells (magenta diamonds) following depletion at −2, 0, 3, and 7 d pi (magenta arrows). The best model (Supplementary file 2) resulted in fewer target cells ( $T_{0}$ ), a lower CD8_E influx ( $ξ$ ), and a higher CD8_E expansion rate ( $η$ ). All other parameters were fixed to the best-fit value in Table 1. The solid lines are $\hat{E} = E + E_{M} + {\hat{E}}_{0}$ and the dashed lines are $E$ for the cases where CD8⁺ T cells were depleted (magenta) and where they were not depleted (black). (G) Comparison of the log₁₀ ratio of virus to CD69⁺CD8⁺ T cells with and without CD8⁺ T cell depletion (magenta and black, respectively). All data are shown as mean ± standard deviation.

Figure 4 with 2 supplements

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Histomorphometry and inflammation of the IAV-infected lung.

(A) Whole lung sections with histomorphometry showing the areas of influenza NP-positive ‘active’ lesions (red), previously infected ‘inactive’ lesions with minimal antigen-positive debris (green), or mixed active and inactive regions (orange) throughout the infection. Representative images from each group are shown. (B) Percent active lesion (red squares) plotted together with the cumulative area under the curve (CAUC) of the predicted infected cell dynamics ( $I_{2}$ ) obtained from fitting the CD8⁺ T cell model. The linear decline in the active lesion (−28.7%/d; see Figure 4—figure supplement 1) was used to estimate the decline after 6 d pi. (C) Percent inactive lesion (green squares) plotted together with the percent maximum CD8_E ( $E / E_{\max}$ ) obtained from fitting the CD8⁺ T cell model. (D) The total lesion (blue squares) is the addition of the active and inactive lesions. To include all measurements on the same scale, the CAUC of $I_{2}$ was multiplied by a scaling factor of 14.2% per 1 × 10⁷ cells, and the percent maximum CD8_E was multiplied by a scaling factor of 0.46%. (E) Fit of Equation (7) to the alveolar (white triangles) and interstitial (yellow triangles) inflammation scores. The solid black, dashed black, and solid magenta lines are the curves generated using the best-fit parameters obtained from fitting the model to the total CD8⁺ T cells, the CD25^–CD43⁺ and CD43^–CD127⁺CD103⁺CD69⁺ CD8⁺ T cells, and the total CD8⁺ T cells during CD8 depletion, respectively. The gray shading are the curves generated using the 95% CI parameters from fitting the model to the total CD8⁺ T cells. All data are shown as mean ± standard deviation.

Figure 4—figure supplement 1

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Regression analysis of lung injury dynamics and CD8⁺ T cells.

(A) Percent active lesion area decreases by 28.7%/d from 6 -7 d pi. (B) Percent inactive lesion area increases by 15.1%/d from 5 -8 d pi, and decreases by 14.5%/d from 9-10 d pi. (C) Total CD8⁺ T cells increase at a rate of 4.9 × 10⁵ cells/d from 5-8 d pi, and decrease at a rate of 4.1 × 10⁵ cells/d from 9-10 d pi. (D) Lung CD8⁺ T cells increase at a rate of 3.97 × 10⁵ cells/d from 5-8 d pi, and decrease at a rate of 2 × 10⁵ cells/d from 9-12 d pi. (E) CD25⁺CD43⁺CD8⁺ T cells increase at a rate of 1.98 × 10⁵ cells/d from 5-8 d pi, and decrease at a rate of 5.84 × 10⁴ cells/d from 9-11 d pi. (F) Fit of a Hill function ( $L_{I} = c_{m a x} E^{n} / (K_{L}^{n} + E^{n})$ ) to the total CD8⁺ T cells (black; $c_{m a x} = 52.88$ , $K_{L} = 6.12$ , $n = 56.32$ ), lung CD8⁺ T cells (gray; $c_{m a x} = 64.65$ , $K_{L} = 6.02$ , $n = 25.16$ ), and CD25⁺CD43⁺CD8⁺ T cells (cyan; $c_{m a x} = 48.04$ , $K_{L} = 5.23$ , $n = 24.03$ ).

Figure 4—figure supplement 2

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Correlation of inflammation with macrophages and neutrophils.

Regression analysis of alveolar inflammation (white) and interstitial inflammation (yellow) with (A) log₁₀ inflammatory macrophages (F480^hiCD11c^hiCD11b⁺) and (B) log₁₀ neutrophils (Ly6G^hi).

Figure 5

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Weight loss dynamics and its relation to lung injury and inflammation.

(A) The percent total (active and inactive) lesion (blue squares) plotted together with the percent weight loss (black diamonds) to illustrate their similar dynamics. (B) Fit of a saturating function ( $L (W) = l_{\max} W^{n} / (K_{w}^{n} + W^{n})$ ; black line) to the mean percent total lesioned area ( $L$ ) and mean weight loss ( $W$ ) for all time points (black circles). The best-fit parameters were $l_{\max} = 39.7 %$ total lesioned area, $K_{w} = 2.6 %$ weight loss, and $n = 5.2$ . The magenta circles are the predicted percent total lesioned area (Figure 4D) with the corresponding weight loss during CD8 depletion, and the magenta line is the prediction using the best-fit parameters ( $K_{w} = 2.6 %$ weight loss and $n = 5.2$ ) together with the maximum predicted percent total lesion ( $l_{\max} = 17.73 %$ total lesioned area). (C) The alveolar (white triangles) and interstitial (yellow triangles) inflammation plotted together with the percent weight loss (black diamonds). (D) Fit of a saturating function (black line) to the mean alveolar (white circles) and interstitial (yellow circles) inflammation scores and mean weight loss for all time points. The best-fit parameters for alveolar inflammation (white circles; black line) were $l_{a_{m a x}} = 3.63$ score, $K_{a_{w}} = 1.95 %$ weight loss, and $n_{a} = 3.65$ , and for interstitial inflammation were $l_{i_{m a x}} = 3.40$ score, $K_{i_{w}} = 1.96 %$ weight loss, and $n_{i} = 3.15$ . The magenta circles are the predicted percent inflammation score (Figure 4E) with the corresponding weight loss during CD8 depletion, and the magenta line is the prediction using the best-fit parameters ( $K_{a_{w}} = 1.95$ % weight loss, $n_{a} = 3.65$ ; $K_{i_{w}} = 1.96 %$ weight loss, $n_{i} = 3.15$ ) together with the maximum predicted inflammation ( $l_{m a x} = 1.41$ score).

Figure 6

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Summary of the connections between the kinetics of virus, infected cells, CD8⁺ T cells, lung injury, lung inflammation, and disease severity.

Summary of the relations between the dynamics of virus, infected cells, CD8⁺ T cells, lung lesions, lung inflammation, and weight loss established by our analysis. Given that viral loads and weight loss are the most easily measured variables, our mathematical model (Equation (1)-(6)) could be used to estimate the kinetics of infected cells, CD8⁺ T cells, and inflammation. The cumulative area under the curve (CAUC) of the productively infected cell dynamics ( $I_{2}$ ) yields an estimate of the percent lung infected (active lesion) while the predicted relative CD8_E dynamics ( $E / E_{m a x}$ ) yields an estimate of the percent lung resolved (inactive lesion). The total amount of lung involved (% lung infected and % lung resolved) or the inflammation scores can then be used to estimate weight loss through the functions $L_{L} (W)$ and $L_{I} (W)$ . These connections could be reversed and weight loss used to estimate viral load kinetics.

Appendix 1—figure 1

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Fit of alternate CD8⁺ T cell models.

Fit of the CD8⁺ T cell model (Equation (1)-(6); solid black line) compared to the fit of two alternate CD8⁺ T cell models (Equation (A1)-(A6) (solid blue line) and Equation (A7)-(A8) (dashed blue line)) to virus and CD8⁺ T cells (excluding 3–5 d pi; white squares) from the lungs of mice infected with 75 TCID₅₀ PR8 (10 mice per time point). Resulting parameter values were $δ_{E a} = 4.02 \times 10^{- 6} {C D 8}_{E}^{- 1} d^{- 1}$ , $η_{E a} = 3.12 \times 10^{5} {CD8}_{E} / d$ , and $K_{E a} = 9.53 \times 10^{5} i n f e c t e d c e l l s$ , and $κ_{b} = 3.33 d^{- 1}$ , $I_{m a x} = 1.0 \times 10^{6} i n f e c t e d c e l l s$ , $δ_{E b} = 1.53 \times 10^{- 5} {CD8}_{E}^{- 1} d^{- 1}$ , $η_{b} = 1.49 d^{- 1}$ , $K_{E b} = 9.5 i n f e c t e d c e l l s$ , $d_{E b} = 0.95 d^{- 1}$ , and $K_{D b} = 1.90 \times 10^{6} i n f e c t e d c e l l s$ . All other parameters are in Table 1, and the AICs are in Supplementary file 1. Data are shown as mean ± standard deviation.

Appendix 2—figure 1

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Parameter behavior of the density-dependent model and the CD8⁺ T cell model.

(A) Comparison of parameters that were common between the density-dependent model (gray, Equation (A9)-(A12)) and the CD8⁺ T cell model (purple, Equation (1)-(6)). Correlations were evident between parameters relating to the rates of virus infectivity ( $β$ ), virus production ( $p$ ), and virus clearance ( $c$ ). However, the strength of the correlation was significantly reduced in the CD8⁺ T cell model. The eclipse phase parameter ( $k$ ) was not well-defined in either model. (B) In the density-dependent model (gray), the viral kinetics and the infection duration were sensitive to small changes in the infected cell clearance parameter ( $δ_{d}$ ). This parameter was well-defined with a narrow 95% CI. In the CD8⁺ T cell model (purple), changing the CD8_E-mediated infected cell clearance parameter ( $δ_{E}$ ) had little impact on viral kinetics or CD8⁺ T cell kinetics. However, these kinetics were most sensitive to changes in the rate of CD8_E expansion ( $η$ ), which was well-defined with a narrow 95% CI.

Appendix 3—figure 1

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Sensitivity of the CD8⁺ T cell model.

Solutions of the CD8⁺ T cell model (Equation (1)-(6)) with the indicated parameter ( $β$ , $k$ , $p$ , $c$ , $δ$ , $δ_{E}$ , $K_{δ_{E}}$ , $ξ$ , $K_{E}$ , or $η$ ) increased (red) or decreased (blue) 50% from the best-fit value (Table 1). CD8_E denotes effector CD8⁺ T cells.

Appendix 3—figure 2

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Sensitivity of the CD8⁺ T cell model.

Solutions of the CD8⁺ T cell model (Equation (1)-(6)) with the indicated parameter ( $τ_{E}$ , $d_{E}$ , $ζ$ , $τ_{M}$ , or ${\hat{E}}_{0}$ ) increased (red) or decreased (blue) 50% from the best-fit value (Table 1). CD8_E and CD8_M denote effector and memory CD8⁺ T cells, respectively.

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Tables

Table 1

CD8⁺ T cell model parameters.

Parameters and 95% confidence intervals obtained from fitting the CD8⁺ T cell model (Equation (1)-(6)) to viral titers and total CD8⁺ T cells (‘Total CD8’) or viral titers, CD25^–CD43⁺CD8⁺ T cells, and CD43^–CD127⁺CD103⁺CD69⁺CD8⁺ T cells (‘Specific CD8_E,M Phenotypes’) from mice infected with 75 TCID₅₀ PR8. CD8_E and CD8_M denote effector ( $E$ ) and memory ( $E_{M}$ ) CD8⁺ T cells, respectively. The total number of CD8⁺ T cells is $\hat{E} = E + E_{M} + \hat{E_{0}}$ and is denoted by CD8.

Parameter	Description	Units	Total CD8		Specific CD8_E,Mphenotypes
Parameter	Description	Units	Value	95% CI	Value	95% CI
$β$	Virus infectivity	${TCID}_{50}^{- 1} d^{- 1}$	6.2 × 10^–5	[5.3 × 10^–6, 1.0 × 10^–4]	3.7 × 10^–5	[1.1 × 10^–5, 9.4 × 10^–5]
$k$	Eclipse phase transition	$d^{- 1}$	4.0	[4.0, 6.0]	5.1	[4.0, 6.0]
$p$	Virus production	${TCID}_{50} {cell}^{- 1} d^{- 1}$	1.0	[5.8 × 10^–1 × 1.1 × 10²]	1.5	[0.73, 13.6]
$c$	Virus clearance	$d^{- 1}$	9.4	[5.6, 9.5 × 10²]	12.1	[5.8, 17.5]
$δ$	Infected cell clearance	$d^{- 1}$	2.4 × 10^–1	[1.0 × 10^–1, 6.6 × 10^–1]	3.0 × 10^–1	[1.9 × 10^–1, 5.9 × 10^–1]
$δ_{E}$	Infected cell clearance by CD8_E	$cells {CD8}_{E}^{- 1} d^{- 1}$	1.9	[3.3. × 10^–1, 2.0]	5.7	[1.7, 8.5]
$K_{δ_{E}}$	Half-saturation constant	$cells$	4.3 × 10²	[1.0 × 10², 2.9 × 10⁵]	1.3 × 10²	[1.0 × 10¹, 8.6 × 10²]
$ξ$	CD8_E infiltration	$C D 8_{E}^{2} {c e l l}^{- 1} d^{- 1}$	2.6 × 10⁴	[1.3 × 10², 8.7 × 10⁴]	2.9 × 10³	[8.0 × 10², 1.7 × 10⁴]
$K_{E}$	Half-saturation constant	${CD8}_{E}$	8.1 × 10⁵	[1.0 × 10³, 1.0 × 10⁶]	2.2 × 10⁶	[1.1 × 10⁶, 8.3 × 10⁶]
$η$	CD8_E expansion	${cell}^{- 1} d^{- 1}$	2.5 × 10^–7	[1.6 × 10^–8, 6.7 × 10^–7]	3.5 × 10^–7	[2.3 × 10^–7, 5.2 × 10^–7]
$τ_{E}$	Delay in CD8_E expansion	$d$	3.6	[2.1, 5.9]	3.3	[2.6, 3.8]
$d_{E}$	CD8_E clearance	$d^{- 1}$	1.0	[5.1 × 10^–2, 2.0]	1.1	[3.3 × 10^–1, 2.5]
$ζ$	CD8_M generation	$C D 8_{M} C D 8_{E}^{- 1} d^{- 1}$	2.2 × 10^–1	[1.0 × 10^–2, 9.4 × 10^–1]	3.2 × 10^–2	[1.0 × 10^–2, 2.2 × 10^–1]
$τ_{M}$	Delay in CD8_M generation	$d$	3.5	[3.0, 4.0]	3.3	[2.1, 5.3]
${\hat{E}}_{0}$	Baseline CD8 or CD8_E	$CD8$ or ${CD8}_{E}$	4.2 × 10⁵	[3.3. × 10⁵, 5.3 × 10⁵]	6.0 × 10²	[3.1 × 10², 1.8 × 10²]
${\hat{E}}_{M 0}$	Baseline CD8_M	${CD8}_{M}$	-	-	6.5 × 10²	[3.2 × 10², 1.5 × 10³]
$T (0)$	Initial uninfected cells	$cells$	1 × 10⁷	-	1 × 10⁷	-
$I_{1} (0)$	Initial infected cells	$cells$	75	-	75	-
$I_{2} (0)$	Initial infected cells	$cells$	0	-	0	-
$V (0)$	Initial virus	${TCID}_{50}$	0	-	0	-
$E (0)$	Initial CD8_E	${CD8}_{E}$	0	-	0	-
$E_{M} (0)$	Initial CD8_M	${CD8}_{M}$	0	-	0	-