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A collaborative team from the School of Engineering and Applied Science and the Perelman School of Medicine have unraveled the mathematics of a 500-million-year-old protein network that acts like the body's bouncer, "deciding" which foreign materials get degraded by immune cells and which are allowed entry.

"The complement system is perhaps the oldest-known part of our extracellular immune system," says Jacob Brenner, a physician-scientist at the University of Pennsylvania. "It plays a crucial role in identifying foreign materials like microbes, medical devices, or new drugs—particularly the larger ones, like in the COVID vaccine."

The complement system can, however, simultaneously play friend and foe, offering protection with one hand while backhanding the body with the other. In some cases, this ancient network can significantly exacerbate conditions like stroke by targeting the body's own tissues.

As Brenner explains, leaking blood vessels allow complement proteins to target brain tissue, causing the immune system to mistakenly launch an attack on the body's own cells and worsen patient outcomes.

Now, using a combination of wet-lab experimentation, coupled differential equations, and computational-based modeling and simulations, the collaborative team has decrypted the mathematical language behind the complement network's "decision" to attack.

Reporting their in Cell, the team identifies a molecular tipping point known as the critical percolation threshold, which is based on how densely complement-binding sites are spaced on the surfaces of the model invader they engineered.

If spacing between binding sites is too wide—landing above a threshold—complement activation fizzles out; below it, the complement network ignites, a chain reaction of immune agent recruitment which spreads like wildfire.

"This discovery enables us to design therapeutics the way you would design a car or a spaceship—using the principles of physics to guide how the immune system will respond—rather than relying on trial and error," says Brenner, who is co-senior author of the study.

Simplifying complexity

While many researchers attempt to divide complex biological systems into smaller and smaller components—like cells, organelles, and molecules—the team approached the system as a set of simpler mathematical values abiding by parameters like density, distance, and speed.

"Not every aspect of biology can be described that way," says co-senior author Ravi Radhakrishnan, bioengineering chair and professor in Penn Engineering. "The complement pathway is fairly ubiquitous across many species and has been preserved through a very long evolutionary time, so we wanted to describe the process using a theory that's universal."

First, a team from Penn Medicine, led by materials scientist Jacob Myerson and nanomedicine research associate Zhicheng Wang, precisely engineered liposomes—tiny, nanoscale fat particles often used as a drug-delivery platform—by studding them with immune-system binding sites. They generated dozens of liposome batches, each with a precisely tuned density of binding sites, and then observed how complement proteins bound and spread in vitro.

The team then analyzed the experimental data with mathematical tools to assess the binding spread dynamics and immune element recruitment rates and used computational tools to visualize and simulate the reactions to identify when thresholds were being approached.

What they observed in the lab—that closer spacing of proteins ramped up immune activity—became much clearer when viewed through a mathematical lens.

The team's approach drew from complexity science, a field that uses math and physics to study systems with many moving parts. By stripping away the biological specifics, they were able to identify fundamental patterns—like tipping points and phase changes—that explain how the immune system decides when to strike.

"We took that initial observation and then tried to control precisely how closely spaced proteins were on the surface," Myerson says. "We found that there's this threshold spacing that's really the key to understanding how this complement mechanism can turn on or off in response to surface structure."

"If you look only at the molecular details, it's easy to think that every system is unique," adds Radhakrishnan. "But when you model complement mathematically, you see a pattern emerge, not unlike how forest fires spread, or hot water percolates through coffee grounds."

The process of percolation

While much of the research on percolation took place in the 1950s, in the context of petroleum extraction, the physics matched those the researchers observed in complement proteins. "Our system's dynamics map entirely onto the equations of percolation," says Myerson.

Sahil Kulkarni, a doctoral student in Radhakrishnan's lab, not only found that the mathematics of percolation predicted the experimental results that Brenner and Myerson's teams observed, but that complement activation follows a discrete series of steps.

First, an "ignition event" occurs, in which a foreign particle makes contact with the immune system. "It's like an ember falling in a forest," says Kulkarni. "If the trees are spaced too far apart, the fire doesn't spread. But if they're close together, the whole forest burns."

Just like some trees in a forest fire only get singed, percolation theory in the context of biology predicts that not all foreign particles must be fully coated in complement proteins to trigger an immune response. "Some particles are fully engulfed, while others get just a few proteins," Kulkarni explains.

It might seem suboptimal, but that patchiness is likely a feature, not a bug—and one of the chief reasons that evolution selected percolation as the method for activating complement in the first place. It allows the immune system to respond efficiently by coating only "enough" foreign bodies for recognition without overexpending resources or indiscriminately attacking every particle.

Unlike ice formation, which spreads predictably and irreversibly from a single growing crystal, percolation allows for more varied, flexible responses, even ones that can even be reversed. "Because the particles aren't uniformly coated, the immune system can walk it back," adds Kulkarni.

It's also energy efficient. "Producing complement proteins is expensive," says Radhakrishnan. "Percolation ensures you use only what you need."

The next steps along the discovery cascade

Looking ahead, the team is excited to apply their mathematical framework to other complex biological networks such as the clotting cascade and antibody interactions, which rely on similar interactions and dynamics.

"We're particularly interested in applying these methods to the coagulation cascade and antibody interactions," says Brenner. "These systems, like complement, involve dense networks of proteins making split-second decisions, and we suspect they may follow similar mathematical rules."

Additionally, their findings hint at a blueprint for designing safer nanomedicines, Kulkarni notes, explaining how formulation scientists can use this to fine-tune nanoparticles—adjusting protein spacing to avoid triggering complement. This could help reduce immune reactions in lipid-based vaccines, mRNA therapies, and CAR T treatments, where complement activation poses ongoing challenges.

"These kinds of problems live at the intersection of fields," says Myerson.

"You need science and engineering know-how to build precision systems, complexity science to reduce 100s of equations, modeling each protein-protein interaction to an essential three, and medical professionals who can see the clinical relevance. Investing in team science accelerated these outcomes."