# The equation that will help us decode cancer’s secrets

By Trevor Graham - Queen Mary University of London

Lung cancer cell during cel division - Image Credit: United States: NIH/WikimediaCommons

A cancer forms when a cell in the body goes awry, multiplying out of control to form a tumour. A typically-sized cancer tumour is made up of more cells than there are people on the planet, and cells from different areas of a single tumour have different alterations in their genetic code.

This sounds like complete chaos. How can we expect to treat cancer effectively – even with newer “targeted” therapies that hit the products of faulty genes – if every cell is different? In order to find more effective ways to treat the disease, we needed to find some order in the chaos.

## Looking for patterns

When we first began to study the patterns of genetic alterations inside human cancers, chaos was exactly what we expected to see. And the first studies seemed to back this idea up.

Surprisingly though, the more we looked, the less chaotic the patterns became. In fact, we realised that there was often such striking regularity in the patterns that they might be able to be explained by a simple mathematical formula.

In our latest work, published in Nature Genetics, we were able to explain how the patterns we found in the chaos of genetic alterations inside a cancer reveal how that cancer grew.

## When copying goes wrong

Each time a cell divides to form two new daughter cells, the DNA inside it must be duplicated so that each daughter receives a complete set. Unfortunately, although the process of DNA replication is very accurate, it’s not perfect and errors occur during the copying process. These errors are called genetic mutations.

Each time cells divide, they copy their DNA – including any previous errors – meaning that mutations that occur in one division are inherited by all the descendents of that cell. In this way, the DNA from the first cell is progressively distorted as the cancer grows, eventually leading to the genetic chaos we see in tumours.

Our approach was to try and “read” this process in reverse: starting from the end point of a cancer genome with many mutations and attempting to decipher the sequence of cell divisions that would have led to the particular pattern of mutations we observed.

We used data from 14 different types of cancers that had been collected using a technique called next-generation sequencing, which can tell us two things: first, whether a particular mutation is present in a cancer, and second, the fraction of cells within the cancer that has a given mutation in their DNA.

This technology has produced huge amounts of data, stored in publicly available databases. Many researchers have created complex computer programs that search for patterns in it. But we decided to take a different approach. We realised that the pattern of mutations within a cancer might make more sense if we looked at them while thinking about how the cancer had grown.

We realised that mutations that happened early in a cancer’s development would be present in lots of cells within the cancer, because these mutations would be inherited by all the daughter cells as the cancer grew. On the other hand, mutations that occurred later would be present in only a few cells.

The interesting twist was when we realised that there should be many more rare mutations in the cancer – each present in only a relatively small number of cells – than common ones. This is because later on in a cancer’s development, when the tumour is larger, there are many more cells dividing and so much more opportunity for mutations to happen.

The formula we developed perfectly described the pattern of mutations in more than 200 of the cancers we looked at. This meant we knew exactly how these cancers had grown.

And – importantly – it shows that the pattern of mutations in a cancer often follow something known as a “power-law”.

Power-laws are found in all kinds of natural systems. Take earthquakes, for example. There are small seismic events happening almost constantly around the globe, but cataclysmic earthquakes only happen once a decade or so. It turns out that the average time between different types of earthquakes obeys a power law.

Earthquackes also follow the power law equation - Image Credit: Martin Luff/flickr

In cancer, we found that the way the cancer grows means that mutations present in large numbers of cells (large events) were rare, whereas mutations present in only a few cells (small events) were commonplace. In other words, this suggested we could expect to discover a power law underlying cancer’s growth. And that’s exactly what we found.

## What next?

We’re excited to have found a natural law of cancer growth that reveals striking simplicity in the apparent chaos of a cancer genome. We think this work is important as it is a first entry in what we hope will become a “mathematical rulebook for cancer”, that will serve to simplify and improve our understanding of the disease.

There’s a long road ahead, but now we’re starting to find order among the chaos inside a tumour, we are uncovering new clues as to how we can better target the disease and hopefully make a difference to the people who suffer from it.

Source: The Conversation