Novel mathematical model could predict the evolutionary trend of SARS-CoV-2

In a recent study posted to the bioRxiv* preprint server, researchers used a novel mathematical model to simulate severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) dynamics inside the host. They quantified the relationship between SARS-CoV-2 virulence and transmissibility, any related changes due to mutations affecting viral assembly and tracked its evolutionary trajectory.

Study: More or less deadly? A mathematical model that predicts SARS-CoV-2 evolutionary direction. Image Credit: Adao/Shutterstock
Study: More or less deadly? A mathematical model that predicts SARS-CoV-2 evolutionary direction. Image Credit: Adao/Shutterstock


Historic data indicates that virus-induced pandemics gradually end, like the 2009 H1N1 virus-induced pandemic and the 1918 flu; likewise, the virulence of SARS-CoV-2 appears to be declining; thus, the observed decline in the virulence of its variant of concern (VOC) Omicron compared to Delta. However, amid concerns that a more virulent SARS-CoV-2 strain might emerge, it will be good to base the evolutionary direction of SARS-CoV-2 on a scientific theory rather than relying on historical data that is inherently unreliable.

Moreover, many studies have evaluated the viral assembly capacity; the linkage between viral assembly kinetics and viral evolution has remained relatively unexplored. It will be interesting how a mathematical model could predict SARS-CoV-2's evolutionary trajectory and demonstrate change in its virulence resulting due to mutations in the SARS-CoV-2 nucleocapsid (N) protein or its messenger ribonucleic acid (mRNA).

About the study

In the current study, researchers examined how mutations in SARS-CoV-2 N protein and mRNA affected its assembling capacity and altered its clinical features, including virulence and transmissibility, to track the SARS-CoV-2 evolution. They hypothesized that complete virus particles are infectious and not the single mRNA or the viral N protein.

The first mathematical model, the coarse-grained model, used ordinary differential equations to describe the dynamics of the virus together with antibodies. The equations sequentially explain the SARS-CoV-2 replication process, the translation process of its mRNA, its packaging process, the interaction between antibody and SARS-CoV-2, the degradation process of its different components, the antibody regeneration processes, and finally, the antibody proliferation process.

The proliferation of the antibody correlated with its binding complex since this binding complex further stimulated the regeneration of specific antibodies and did not account for antibody waning in the analysis. The researchers assumed that the immune clearance of antibody binding complex is efficient, and its degradation rate is different between mRNA and proteins. The second study model was a kinetic model that simulated the core part of the virus packing process, and they determined the reaction coefficient of each reaction in the process.

It worked on the assumptions that each capsid protein has a single binding surface toward mRNA or other capsid protein; therefore, the binding energy between An and A is the same as the binding energy between A and A, where A represents the capsid protein, and An is the polymer that contains n structure proteins.

Study findings

The virulence of the virus, i.e., when a virus is generating heterogeneous proteins at its peak, is directly proportional to the overall immune response. The number of antibodies generated in response could help evaluate the virulence as antibodies bind to these heterogeneous proteins. The number of viral particles produced during one infection cycle is the measure of the transmissibility of a virus.

The results of the coarse-grained model revealed that reaction (4) was most sensitive to mutations while the mutational effects on reactions (1) (2) and (3) (5) and (14) were weak.

Intriguingly, they proposed that the enhancement of packaging ability weakens the replication ability of the virus because only the naked mRNA can facilitate replication, while the mRNA wrapped by capsid protein cannot.

Mutations led to an increment in the binding energy between capsid protein and mRNA and capsid monomers. The improved virus packaging ability inevitably led to a decline in its replication and translational ability, followed by a decline in the overall mRNA level and the content of all translation proteins, which corresponded to a relatively lowered antibody production.

The antibody concentrations also reflect the toxicity of a virus, i.e., the sum of mRNA, replicase, and structural protein (heterologous substances) generated by stimulation. Thus, improved SARS-CoV-2 assembling increased viral particles and viral infectivity but reduced toxicity. However, this improvisation in the assembling ability eventually halted at an optimal point and did not continue incessantly. After surpassing this optimal point, further mutations that lead to faster packing will lose their evolution advantages.

When the mutation increased the viral assembly capacity, the reverse reaction constant k-5 decreased. The decrease in k-5 led to a smaller quantity of antibodies, corresponding to milder virulence. The overall particle number is the cumulative virus number over time; a higher value indicated that mutations that promoted the viral packing process displayed an evolutionary advantage over the original strain. However, this trend was not continuous. When the virus packing capacity crossed a certain threshold, a further decrease in k-5 led to a smaller cumulative number.

Overall, the study findings pointed at the underlying mechanism behind the intricate transmission-virulence trade-off, a phenomenon that explains why virulence and transmissibility go hand in hand. Mathematical biologists proposed that the pathogen gets deadly and kills its host such that its own spread halts.


Together, the study model answered several important queries related to SARS-CoV-2 evolution.

First, the study findings indicated that the binding force between capsid protein and mRNA was the driving force behind the evolution of SARS-CoV-2 and its optimal value ensured that maximum viral particles were packed. It showed that mutants with increased assembling capacity selectively evolved, and their transmissibility boomed. Meanwhile, their replication capacity decreased, leading to reduced virulence. This trend was also witnessed in other RNA viruses, rationalizing why epidemics induced by these viruses gradually ended.

Second, a novel virus (not SARS-CoV-2) is more likely to cause a future epidemic because its virulence will decrease to increase its transmissibility. Although SARS-CoV-2 might evolve into a less virulent strain, its virulence will not completely fade away.

Although the model could not accurately compute an optimal point but showed that beyond this optimal point, a SARS-CoV-2 mutant with less virulence would become unfavorable in evolution, and the virulence of this SARS-CoV-2 mutant at this balance point might be the same or weaker than Omicron.

In the future, the current study parameters should be further improved and modified using experimental or ecology-based studies.

*Important notice

bioRxiv publishes preliminary scientific reports that are not peer-reviewed and, therefore, should not be regarded as conclusive, guide clinical practice/health-related behavior, or treated as established information.

Journal reference:
Neha Mathur

Written by

Neha Mathur

Neha is a digital marketing professional based in Gurugram, India. She has a Master’s degree from the University of Rajasthan with a specialization in Biotechnology in 2008. She has experience in pre-clinical research as part of her research project in The Department of Toxicology at the prestigious Central Drug Research Institute (CDRI), Lucknow, India. She also holds a certification in C++ programming.


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