Elucidating the Mechanisms of Dynamic and Robust Control of the Liver Homeostatic Renewal Process: Cell Network Modeling and Analysis

Abstract

Recent experimental investigations of liver homeostatic renewal have identified high replication capacity hepatocyte populations as the primary maintainers of liver mass. However, the molecular and cellular processes controlling liver homeostatic renewal remain unknown. To address this problem, we developed and analyzed a mathematical model describing cellular network interactions underlying liver homeostatic renewal. Model simulation results demonstrate that without feedback control, basic homeostatic renewal is not robust to disruptions, leading to tissue loss under persistent/repetitive insults. Consequently, we extended our basic model to incorporate putative regulatory interactions and investigated how such interactions may confer robustness on the homeostatic renewal process. We utilized a Design of Experiments approach to identify the combination of feedback interactions that yields a cell network model of homeostatic renewal capable of maintaining liver mass robustly during persistent/repetitive injury. Simulations of this robust model indicate that repeated injury destabilizes liver homeostasis within several months, which differs from epidemiological observations of a much slower decay of liver function occurring over several years. To address this discrepancy, we extended the model to include feedback control by liver nonparenchymal cells. Simulations and analysis of the final multicellular feedback control network suggest that achieving robust liver homeostatic renewal requires intrinsic stability in a hepatocellular network combined with feedback control by nonparenchymal cells.

Figure 1

Homeostatic renewal model. (A) Model schematic showing two populations of hepatocytes: SR_high and SR_low. Each population can replicate, or die, at distinct specified rates, and while SR_high cells can transition to SR_low cells, the reverse cannot occur. (B) Phase plane representation of steady-state behavior of the model showing multiple steady states but no stable attractor. Blue arrows represent the sign of the local derivative in the y-direction; the local derivative in the x-direction is zero everywhere. (C) In response to a transient increase in the cell death rate (in this specific simulation an increase of 50% for 30 days), the system shifts to a new steady state. (D) Phase-plane representation of steady-state model behavior in response to a transient 50% increase in the cell death rate lasting 30 days. The initial steady state is 0.95 and 0.05 for SR_high and SR_low cells, respectively. During the transient cell death rate increase, the cell population sizes decrease to a new, lower steady state.

Figure 2

Incorporating feedback mechanisms. (A) Network model schematic modified to include the physiological phenomena of (A) population size capacity constraints, (B) inhibition of proliferation, (C) inhibition of SR_high transformation, and (D) recruitment of SR_high cells from sources external to liver tissue. (B) Robustness of system response to multiple transient disturbances (plotted as MRS) and variance in simulated patient response with altered physiological parameters. The systems that responded with the largest MRS (highest robustness) also showed the smallest normalized variance in response to altered parameters. (C) ANOVA shown as a plot of standardized effects resulted in significant feedback mechanisms A, B, and C as did their interactions. However, feedback D and its interactions did not affect model output significantly. (D) Robust model (Model A+B+C) response to transiently increased and transiently decreased cell death (1.5× and 0.5× nominal value for 30 days). (E) Robust model response to transiently increased and transiently decreased proliferation (1.5× and 0.5× nominal value for 30 days). (F) Robust model response to initial population imbalances shows that multiple starting conditions converge to a steady state. Points represent t = 0, 10, 40, and 60 days followed by every 30 days until 1 year, at which points represent each subsequent year. Gray arrows represent the direction of motion on the phase plane. MRS is the Mean Recovery Score, which is the sum of the recovery volumes for all disturbances.

Figure 3

Homeostatic renewal recovers liver mass when hepatocytes become senescent and are coupled with a large cell death event, shown here using (A) hepatocyte population balances, (B) phase-plane recovery, and (C) relative liver mass recovery. Researchers have shown complete renewal 90 days postinjury, which is consistent in our model. Induced senescence without damage is represented by black lines, and induced senescence with 40% of hepatocytes removed (toxic injury) is represented by blue lines. Our model predicts that stem cells become active even without an additional cell death challenge but that the time frame of recovery is slightly longer (∼120 days to recovery). Points represent t = 0, 10, 40, and 60 days followed by every 30 days until 1 year, at which points represent each subsequent year. Gray arrows represent the direction of motion on the phase plane.

Figure 4

Simulating multiple, pulsatile whole liver cell death events. At low frequencies (f < 2.0 cycles/day), the liver is able to recover between cell death events. At high frequencies (f ∼ 2.0 cycles/day and above), the hepatocyte populations are unable to recover between events, and population sizes become unstable. In this figure a “Cell Death” value of 0 represents no external cell death challenge (only the intrinsic cell death rate).

Figure 5

Nonparenchymal cell control of hepatocyte homeostatic renewal. (A) Model schematic representing the biological process underlying the control system. Dashed lines represent a flow of information, while solid lines represent a flow of mass. (B) A control systems representation of liver homeostatic renewal controlled by nonparenchymal cell networks. In this representation, an increase or decrease in cell death rate disturbs normal homeostatic renewal. The proliferation (k_env^prol) and transformation (k_env^T) rates are modified by the nonparenchymal controllers as shown in eqs 21 and 22. Dashed lines represent a flow of information, while solid lines represent a flow of mass. (C) Ability of the nonparenchymal cell network controller to mitigate the effects of a periodic cell death challenge (increasing extrinsic cell death rate 1 day every other day). Controller action causes an attractor state with minimal deviation around the steady state. In the absence of nonparenchymal cell control, hepatocyte populations fall into an attractor cycle below steady-state population size.