A 3-D agent based model (ABM) (where the agents are dendritic cells (DCs)) is
presented that simulates the stochastic response of DCs to viral infection up to 15
hours post infection. Simulation results are in good agreement with the
experimentally measured DDX58 transcript and IFN-β transcript distributions at 6
and 11 hours post infection.
The results show that both the spatially random distribution of infected DCs (only
infected DCs can express IFNB1) and the broad distribution of times the infected
DCs begin to secrete IFN-β lead to an initially heterogeneous distribution in IFN-β in
the extracellular medium. During the first 6 hours post infection, the spatial
heterogeneity of the IFN-β distribution is amplified in the spatial distribution of the
bound IFNAR receptors across the DC population by extracellular communication
(via autocrine signaling), see Figure 1. By 11 hours post infection the spatial
distribution of IFNAR receptors begins to homogenize due to paracrine signaling
between DCs. Bound IFNAR receptors activate the JAK/STAT pathway that induces
the interferon stimulated genes (ISGs) such as DDX58 (by modulating the rate
DDX58 transitions from a state that results in a basal rate of transcript production to
a state that results in an enhanced rate of transcript production). The results show
that the spatial heterogeneity of the bound IFNAR distribution leads to a broad
distribution of DDX58 transcripts from 7 to 15 hours post infection. The ABM
predicts the other ISGs’ transcript distributions will also be broad during these
times, which may result in the functional consequence of a broad range of antiviral
activity throughout these times. [1]
The stochastic intracellular processes of both infected and uninfected DCs are
simulated using the standard Gillespie algorithm [2]. The reactions are depicted
diagrammatically in Figure 2 (see the supplementary section of [1] for detailed
tables of reactions, rate constants and parameters, and the initial conditions). The
method used to simulate the extracellular diffusion of IFN-β is described in detail in
[3]. The code is written in C++ and requires the user to download the GSL library in
order to compile the code. An annotated version of the source code for the 3-D ABM
can be downloaded below. See Figure 1 for an example of the spatial distributions
that can be produced using this program.
The system is modeled in silico on a three dimensional lattice of 20 x 20 x 20 cubes
that represents the medium. The size of the cubes is chosen to be slightly larger than
the diameter of a DC thus allowing at most one cell per box. The system contained
1050 DCs uniformly randomly distributed with 50% of the DCs infected. The
average diameter of a DC is 30 μm. In the simulation a density of roughly 5 x 106
cells mL-1 is used.
The parameters for the intercellular modeling are based on experimental data and
previous in silico simulations [4]. The experimental value for the diffusion
coefficient for the cytokine IFN-β is approximately 10 μm2 s-1. Given the diffusion
coefficient and the dimensionality of a lattice box we determined the simulation
diffusion time step of 11.25 s for the IFN-β to diffuse to an adjacent square with 50%
probability.
The intracellular part of the simulation is described next. The binding and unbinding
of IFN-β to the IFNAR reactions are included in the intracellular part of the
algorithm. This is justified because only the IFN-β proteins inside a lattice box
containing a DC have a non-zero probability of being bound to a free IFNAR on the
cell surface. The rate constants for the binding and unbinding of IFN-β to the IFNAR
are chosen based on the literature [5].
The induction of IFN-β is described by an extension of an intracellular model of IFNβ mRNA induction developed earlier to explain experimentally observed power-law
behavior in IFN-β mRNA enhanceosome [6]. It has been shown experimentally [7,8]
that the assembly of enhanceosome is promoted by HMGI, an architectural protein;
NFκB is detected initially at the promoter with an IRF and ATF-2 recruited, later
followed by the arrival of IRF-3 or IRF-7. The enhanceosome is modeled with the
sequential cooperative binding of four proteins. As in the earlier model, the cascade
of steps required for the assembly of the pre-initiation complex is represented by a
single step that takes the assembled enhanceosome to a transcribing state. This
gives rise to bursting-type kinetics [9,10,11,12] that leads to good agreement with
the experimentally observed power-law distributions of the IFN-β mRNA copy
number [6].
The stochastic activation of RIG-I is not explicitly modeled and the transcription
factors involved in the enhanceosome assembly to keep the model tractable;
however, the forward rates for the assembly of the enhanceosome are modified to
be a function of RIG-I number. This represents the complex set of reactions that
ensue following viral detection by RIG-I in the cytosol. The transcription rate of IFNβ mRNA is modified to be a function of bound IFNAR. This is done to simulate the
increased production of IFN-β mRNA through signal transduction by means of the
exchange of IRF-3 with the induced IRF-7 [13]. The rate constants for the up
regulation of IFN-β mRNA’s transcription rate via the exchange of IRF-3 with
induced IRF-7 are chosen to match the experimental IFN-β mRNA distribution [4].
The IFN-β secretion is modeled directly as translation of IFN-β mRNA and export
because the transport of the IFN-β is deemed to be rapid with a rate of secretion
chosen based on the [4] paper. We emphasize that the intracellular reactions
involving IFN-β occur only in infected cells.
A more detailed model of DDX58 gene induction than used previously is used: the
gene is assumed to have two states, a basal (low) production state in the absence of
the activated JAK/STAT pathway and an enhanced production state after activation.
The transition rate of the DDX58 gene from a low to a high production state is
modeled in a coarse-grained way as a function of the number of bound IFNARs; this
determines the level of signal transduction of the JAK/STAT pathway that in turn
enhances the production of DDX58. The rate constant for DDX58 degradation is
chosen to be the same as in earlier work [4]. The production rates in the two states
of the gene and the transition rate between them are chosen to match the average
number of DDX58 at six and eleven hours observed experimentally. The RIG-I
production is modeled as translation of DDX58 with the same rate constants as used
earlier [4].
References
[1] Tabbaa OP, et al.: Noise propagation through extracellular signaling leads to
fluctuations in gene expression. BMC Systems Biology 2013 7:94
[2] Gillespie DT: A general method for numerically simulating the stochastic
time evolution of coupled chemical reactions. J Comput Phys 1976, 22: 403-434.
[3] Shimoni Y, Nudelman G, Hayot F, and Sealfon SC: Multi-scale stochastic
simulation of diffusion-coupled agents and its application to cell culture
simulation. PLoS One 2011, 6:e29298-e29298
[4] Hu J, Nudelman G, Shimoni Y, Kumar M, Ding Y, López C, Hayot F, Wetmur JG, and
Sealfon SC: Role of Cell-to-Cell Variability in Activating a Positive Feedback
Antiviral Response in Human Dendritic Cells. PLoS One 2011, 6: 1661-1664.
[5] Coppey M, Berezhkovskii AM, Sealfon SC, and Shvartsman SY: Time and length
scales of autocrine signals in three dimensions. Biophys J 2007, 93: 1917-1922.
[6] Hu J, Iyer-Biswas S, Sealfon SC, Wetmur J, Jayaprakash C, and Hayot F: Powerlaws in interferon-b mRNA distribution in virus-infected dendritic cells.
Biophys J 2009, 97: 1984-1989.
[7] Munshi N, Agalioti T, Lomvardas S, Merika M, Chen G, and Thanos D:
Coordination of a transcriptional switch by HMGI(Y) acetylation. Science 2001,
293:1133-1136.
[8] Panne D, Maniatis T, and Harrison SC: An atomic model of the Interferon-β
enhanceosome. Cell 2007, 129: 1111-1123.
[9] Eldar A and Elowitz MB: Functional roles for noise in genetic circuits. Nature
2010, 467:167-173
[10] Iyer-Biswas S, Hayot F, and Jayaprakash C: Stochasticity of gene products
from transcriptional pulsing. Phys Rev E 2009, 79: 031911
[11] Dar RD, Razooky BS, Singh A, Trimeloni TV, McCollum JM, Cox CD, Simpson ML,
and Weinberger LS: Transcriptional burst frequency and burst size are equally
modulated across the human genome. PNAS 2012, 109: 17454-17459.
[12] Suter DM, Molina N, Gatfield D, Schneider K, Schibler U, Naef F: Mammalian
Genes Are Transcribed with Widely Different Bursting Kinetics. Science 2011,
332: 472-474.
[13] Honda K, Yanai H, Negishi H, Asagiri M, Sato M, Mizutani T, Shimada N, Ohba Y,
Takaoka A, Yoshida N, and Taniguchi T: IRF-7 is the master regulator of type-I
interferon-dependent immune responses. Nature 2005, 434: 772-777.
Figure Caption
Figure 1: Spatial distribution of the fraction of receptors bound on the DCs.
Displayed is the distribution at 8 hours post infection. The colors indicate the range
of the fraction of bound receptors on each DC: green denotes an fraction of bound
less than 0.2, purple a fraction between 0.2 and 0.3, and orange a fraction between
0.3 and 0.4. It is clear that at 8 hours post infection the spatial distribution of bound
receptors is very heterogeneous, which is resulting in noisy induction of the ISGs.
The X- , Y-, and Z-axes are in units of single cell length (~30 μm).
Figure 2: Each dendritic cell has intracellular reactions that are depicted
diagrammatically above. The intracellular reactions are simulated by means of the
Gillespie algorithm.