By Sara Capponi
Defective interfering particles (DIPs) are viral particles that are generated from a genome that lacks structural genes, which are essential for growth and replication. Therefore, DIPs cannot infect cells on their own and require the presence of wild type (WT) viral particles to successfully complete their replication cycle. To replicate and form virions, DIPs compete for cellular resources and viral proteins that are produced by WT particles; such competition can result in an attenuation of the viral infection and/or a reduction of disease intensity. Researchers have recently demonstrated that DIPs activate the immune response both in vivo and in vitro [1]. These findings have reignited interest in the prospect of engineering DIPs for use as therapeutics for many viral infections, including HIV and polio [2, 3].
The combined efforts of experimentalists and theoreticians are crucial to the development of new therapeutics. Experiments can validate the predictions of mathematical models, and these models allow scientists to explore a vast parameter space that describes experimental conditions that are often inaccessible via real experiments. This research area aims to obtain precise design principles of defective particles for use as therapeutics. In this context, our studies focus on building a theoretical framework to guide the engineering of poliovirus-derived DIPs.
Figure 1. Agent-based model (ABM) to describe the dynamics of the wild type (WT) viruses and defective interfering particles (DIPs). 1a. Schematic description of the simulated initial conditions and the intracellular processes that are modeled in the ABM. The two-dimensional square lattice depicts the beginning of the infection, colored proportionally to the \(\textrm{WT/DIP}\) ratio. 1b. The snapshots are taken at \(t=0\), \(30\), and \(60\) hours and represent the dynamics of the WT (blue) and DIP (red) agents (top), and the interferon (IFN) production (gray, bottom) as the infection spreads over the lattice. The gray scale refers to the degree of protection that is provided by the IFN. The plot represents the time evolution of the number of WT and DIP agents when \(\mathit{rf}_{\textrm{WT}} < \mathit{rf}_{\textrm{DIP}}\). Figure courtesy of the author.
Our group previously defined an experimental-based mathematical model to explain and predict the elements that determine the competition between DIPs and viral particles in a single cell [3]. These efforts represented the initial step towards designing DIPs that can limit the spread of infection by interfering with WT viral particles within a host population. We experimentally engineered the DIPs by deleting the poliovirus genome P1 region that encodes structural capsid proteins — the envelope that protects viral RNA and determines cellular entry, among other things. During co-infection, the defective interfering (DI) genome competes with the WT for capsid proteins and can form DI particles at the expense of the WT viral particles. To describe the dynamics of WT and DI particles during replication and encapsidation, we built an ordinary differential equation mathematical model and performed sensitivity analysis to identify the critical parameters that govern the intracellular competition [3]. Our study revealed that the number of WT and DI genomes in co-infected cells is lower than that of the WT genomes in a single infected cell, thus indicating a limiting resource for replication. The DIPs replicate more quickly due to their short genomes, as compared to that of the WT. In addition, we found that the decrease in WT encapsidated genomes from singly to dually infected cells is two folds larger than that of WT naked genomes; this indicates that DI genomes further inhibit WT virion production by trans-encapsidating capsid proteins that are produced by the WT. The sensitivity analysis suggested that the DI relative replication factor and the encapsidation rate are the determinants of the proportion of WT virions at cell lysis in a co-infected cell.
In the current work, we moved our investigation from a single cell to a collection of cells. In other words, we analyze the intercellular competition after characterizing the intracellular competition. We built an agent-based model (ABM) in which cells are represented in a two-dimensional square grid lattice of dimension \(1{,}000 \times 1{,}000\) with periodic boundary conditions; each square represents a cell (see Figure 1). We model the intracellular competition using the reproduction fitness parameter \(\mathit{rf}\), which defines the number of agents that a single parent agent generates at each time step. Consider an example where \(\mathit{rf}=4\): for one particle at a given time \(t\), we obtain four particles at time \(t+1\). In this way, our model describes the competition between the two different agents within a single cell in terms of the total number of virions that are generated at cell lysis according to \(\mathit{rf}\); this parameter can differ among the two agent species. In addition, researchers have observed the existence of a minimum number of WT particles that are necessary for a single DIP to survive in cell. For poliovirus, this quantity is five.
The model’s infection dynamic is as follows: the infection starts at time \(t=0\) with nine co-infected cells that are localized in the center of the grid (see Figure 1). We established the \(\textrm{WT/DIP}\) ratio that infects the cells at the beginning of the simulation based on the multiplicity of infection that our collaborators studied experimentally, and we explored three different initial co-infection conditions—\(\textrm{WT/DIP}=0.01\), \(\textrm{WT/DIP}=0.1\), \(\textrm{WT/DIP}=1\)—to better investigate the dynamics of the competition between DIP and WT particles. For each \(\textrm{WT/DIP}\) ratio initial condition, we analyzed 10 different values of \(\mathit{rf}\) for the DIP and WT particles (\(\mathit{rf}_{\textrm{WT}}\) and \(\mathit{rf}_{\textrm{DIP}}\), varying between four and 40 in steps of four).
The infection starts at time \(t=0\) and continues for 360 time steps, which corresponds to 15 days (one time step \(=\) one hour). We neglected cell proliferation and aging, but we did model the cell death that occurs when a cell undergoes lysis. After death, a cell regenerates in 24 hours; in terms of the ABM, this means that no agents can move into a cell—i.e., a square—for 24 time steps. We fixed the burst time at nine hours, which is appropriate for poliovirus, and the maximum number of occupants per cell at 500. At each time step, every agent replicates according to the corresponding value of the \(\mathit{rf}\); at cell lysis, the agents move to one of the available neighboring cells. This allowed us to describe the spread of a viral infection in the presence of DIPs. A representative example of the system’s time evolution is evident in Figure 1b, which displays three snapshots of the simulations and a plot that represents the DIP and WT dynamics as a function of time. In order to obtain good statistics, we repeated each simulation 10 times. Moreover, we repeated the whole set of simulations that we carried out in presence of DIPs without the DIPs, i.e., with only the WT particle agents. We then used these simulations as points of reference.
We described the native immune response, which we call interferon (IFN), as a mechanism of cell defense (see Figure 1b). We used two parameters: one describes the amount of IFN produced by a cell (IFN generation) and the other describes the degree of protection that is offered to a cell against infection (IFN strength). When a cell is infected by one agent, it begins to produce IFN according to the value of the IFN generation parameter. IFN production propagates, and the neighboring cells that surround the infected one begin to generate IFN. However, propagation occurs in a diffusive way so that the degree of cell protection decreases upon moving far away from the initial infected cell. IFN protects cells from agent infection only if the value of the generated IFN is greater than the IFN strength — i.e., a cell is seldom infected in the presence of low values of IFN strength.
Results
Figure 2 displays representative results from the simulations that are characterized by a \(\textrm{WT/DIP}\) ratio equal to \(0.1\) and by \(\mathit{rf}_{\textrm{WT}}=16\) and \(\mathit{rf}_{\textrm{DIP}}=12\), \(16\), and \(20\) (see the left, central, and right panels, respectively, of Figure 2a and 2b). In the representative snapshots, we observe similar population dynamics but a different spread and distribution of the DIP agents when \(\mathit{rf}_{\textrm{WT}} > \mathit{rf}_{\textrm{DIP}}\) or \(\mathit{rf}_{\textrm{WT}} < \mathit{rf}_{\textrm{DIP}}\). When \(\mathit{rf}_{\textrm{WT}} = \mathit{rf}_{\textrm{DIP}}\), the snapshot reveals the existence of a high-density cluster of DIP agents that are localized in a specific region of the lattice. To better understand the population dynamics at these different conditions, we represented the number of agents as a function of time in Figure 2b. When \(\mathit{rf}_{\textrm{WT}} < \mathit{rf}_{\textrm{DIP}}\) (see left panel of Figure 2b), the number of agents of both species starts to increase as the simulation begins and the number of DIPs is slightly bigger than that of the WT particles. The number of DIPs reaches the maximum approximately 55 hours after the start of the infection and the number of WT particles begins to increase. Because DIP reproduction fitness is lower than that of WT, the number of DIPs goes to zero and the infection spreads without obstacle.
Figure 2. Population dynamics at different \(\mathit{rf}\) conditions. 2a. Left to right: Representative snapshots of the simulations for \(\mathit{rf}_{\textrm{WT}} > \mathit{rf}_{\textrm{DIP}}\), \(rf_{\textrm{WT}} = \mathit{rf}_{\textrm{DIP}}\), and \(\mathit{rf}_{\textrm{WT}} < \mathit{rf}_{\textrm{DIP}}\). 2b. From left to right: Number of wild type (black) and defective interfering (blue) particles represented as a function of time for \(\mathit{rf}_{\textrm{WT}} > \mathit{rf}_{\textrm{DIP}}\), \(\mathit{rf}_{\textrm{WT}} = \mathit{rf}_{\textrm{DIP}}\), and \(\mathit{rf}_{\textrm{WT}} < \mathit{rf}_{\textrm{DIP}}\). Figure courtesy of the author.
Compared to this condition, we observe a remarkable difference in the population dynamics when \(\mathit{rf}_{\textrm{WT}} > \mathit{rf}_{\textrm{DIP}}\) (see right panel of Figure 2b). For time smaller than roughly 80 hours (or 3.5 days), the number of DIPs increases while the number of WT viral particles stays close to zero. Only when the number of DIPs reaches the maximum at 55 hours does the number of WT particles start to increase. This pattern is noteworthy in the context of DIP therapeutic potential. These conditions revealed that the DIPs were able to take control of a viral infection’s spread for about three days — a time window in which one can activate the adaptive immune system or administer a medicine to fight the pathogen. We observe the most interesting case when \(\mathit{rf}_{\textrm{WT}} = \mathit{rf}_{\textrm{DIP}}\) (see middle panel of Figure 2b). In these conditions, the number of DIPs is greater than that of the WT particles for the entire length of the simulations. The viral load’s increase is reduced with respect to the other cases and the number of WT particles starts to increase approximately 200 hours after the infection begins, which allows plenty of time to fight and control the viral spread.
Finally, we analyze the viral infection’s spread and the WT and DIP population dynamics generated by the sets of simulations that are characterized by an initial ratio of \(\textrm{WT/DIP}=1\) or \(\textrm{WT/DIP}=0.01\). In the first case, we observe that when \(\mathit{rf}_{\textrm{WT}} \le \mathit{rf}_{\textrm{DIP}}\), the DIPs can control and reduce the WT viral load increase long enough for a secondary action to stop the infection. However, when the initial \(\textrm{WT/DIP}\) ratio is equal to \(0.01\), the number of DIPs drops to zero within the first hours after the simulation begins — irrespective of the values of the reproduction fitness.
Overall, our results support experimental studies and show that DIPs have therapeutic potential. Our ABM model describes the spread of an infection that is likely to occur in a cell culture or an organ tissue, where cells share functional, structural, and geometrical similarities. Future directions will embrace not only the differentiation of cell characteristics in distinct areas of the lattice so as to mimic the spread of an infection among different type of cells or organs, but also both another type of infection onset that starts in different places of the lattice and a delay between the viral infection onset and DIP action.
Sara Capponi presented this research during a minisymposium at the 2021 SIAM Conference on Applications of Dynamical Systems, which took place virtually in May 2021.
Acknowledgments: This work was supported by DARPA’s INTERfering and Co-Evolving Prevention and Therapy (INTERCEPT) program (grant no. HR0011-17-2-0027). I also thank the following scientists who contributed to this work: Yinghong Xiao, Peter V. Lidsky, Elsa Rousseau, James H. Kaufman, Simone Bianco, and Raul Andino.
References
1. Akpinar, F., Inankur, B., & Yin, J. (2016). Spatial-temporal patterns of viral amplification and interference initiated by a single infected cell. J. Virol., 90(16), 7552-7566.
2. Rast, L.I., Rouzine, I.M., Rozhnova, G., Bishop, L., Weinberger, A.D., & Weinberger, L.S. (2016). Conflicting selection pressure will constrain viral escape from interfering particles: principles for designing resistance-proof antivirals. PLoS Comput. Biol., 12(5), e1004799.
3. Shirogane, Y., Rousseau, E., Voznica, J., Xiao, Y., Su, W., Catching, A., …, Andino, R. (2021). Experimental and mathematical insights on the interactions between poliovirus and a defective interfering genome. PLoS Path., to be published.
Sara Capponi, Ph.D., is a research staff member at IBM’s Almaden Research Center with training in physics, computational biophysics, and biochemistry. At IBM Research, she leverages her expertise to combine molecular modeling with cellular engineering. Capponi holds a Ph.D. in physics from the Donostia International Physics Center in Spain. Prior joining IBM Research at Almaden, she was a postdoctoral fellow at the University of California, Irvine and the University of California, San Francisco. Capponi’s contributions range from studies on macromolecular systems like membrane or viral proteins to works on water dynamics and structure. She has made use of different computational approaches during her career, including molecular dynamics, enhanced sampling techniques, agent-based models, and machine learning methods for data analysis.