For the first time, the ability of proteins within a cell to work together as a complex and strong network has been described mathematically, marking a significant advance in both maths and biology.

The new mathematics – the result of a five-year project undertaken by Robyn Araujo of Australia’s Queensland University of Technology, and Lance Liotta of George Mason University in the US – largely resolves the mystery of how the extremely complex network found within a cell results in a robust, rather than fragile, outcome.

“Proteins form unfathomably complex networks of chemical reactions that allow cells to communicate and to ‘think’ – essentially giving the cell a ‘cognitive’ ability, or a ‘brain’,” says Araujo.

“It has been a longstanding mystery in science how this cellular ‘brain’ works. We could never hope to measure the full complexity of cellular networks – the networks are simply too large and interconnected and their component proteins are too variable.

“But mathematics provides a tool that allows us to explore how these networks might be constructed in order to perform as they do.”

And although the calculations involved in such exploration may be very complicated, the conclusions they yield are unexpectedly elegant.

In a paper published in the journal *Nature Communications*, Araujo and Liotta reveal that evolutionary processes are empowered by simple design principles that scale up to produce robust performance regardless of the size of the underlying networks.

The focus of the mathematicians’ interest is a phenomenon known as robust perfect adaptation (RPA) – which is the ability of a network to reset itself after exposure to a new stimulus.

“An example of perfect adaptation is our sense of smell,” Araujo explains. “When exposed to an odour we will smell it initially, but after a while it seems to us that the odour has disappeared, even though the chemical, the stimulus, is still present.

“Our sense of smell has exhibited perfect adaptation. This process allows it to remain sensitive to further changes in our environment so that we can detect both very faint and very strong odours.”

This type of reset happens in living cells almost constantly. Each cell is subject to a continuous stream of fresh inputs, such as hormones and growth factors. In each case, there is an initial response, after which things settle down again, even though that stimulus is still there.

Science knows a lot about how these stimuli are generated – by gene sequences, for instance – but how the multitude of proteins created by such activity interact with each other remains unclear.

The complexity of these reactions is of a magnitude that defies simple mapping, so Araujo and Liotta wondered whether mathematical analysis could provide some answers.

It turns out that it can.

“I studied all the possible ways a network can be constructed and found that to be capable of this perfect adaptation in a robust way, a network has to satisfy an extremely rigid set of mathematical principles,” Araujo says.

“There are a surprisingly limited number of ways a network could be constructed to perform perfect adaptation.”

To discover what these were, the researchers turned to a branch of mathematics called topology – essentially, the study of how an object’s properties are preserved as it is deformed, twisted or stretched.

Using this approach, they were able to develop an entirely new set of definitions, which were able to describe, interpret and account for all possible flows and controls of biochemical signals through a network. The unifying principles work regardless of the size of the network, and do not require prior knowledge of how its components are interconnected, nor how strong any individual reaction will be.

The result, the pair write in their paper, “suggests a resolution to a baffling paradox in living systems – that while networks of interacting molecules are often unimaginably complex, a property that is generally associated with fragility, such networks are nevertheless characterised by a remarkable robustness”.

The achievement will prove to be of use to a far wider community than simply that of mathematicians.

“Essentially, we are now discovering the needles in the haystack in terms of the network constructions that can actually exist in nature,’ says Araujo.

“It is early days, but this opens the door to being able to modify cell networks with drugs and do it in a more robust and rigorous way. Cancer therapy is a potential area of application, and insights into how proteins work at a cellular level is key.”