# Understanding plant growth with the help of mathematics

Most people admire ﬂowers for their beauty and don’t give a second thought as to how a ﬂower has grown into a particular shape.

Researchers at the John Innes Centre are now making breakthroughs in understanding how ﬂowers and plants can form intricate 3D shapes from small initial shapeless masses of cells. They are gaining an amazing insight into the actual mechanism of plant growth and they are doing this with the help of mathematics.

One of the key ideas behind ﬂower formation is to do with what researchers call ‘growth conﬂicts’. A growth conﬂict is where different parts of the ﬂower grow at different rates, causing the ﬂower to form the final 3D shape which we are familiar with.

To understand how growth conﬂicts work, researchers have realised that the ﬂower will display both elastic and plastic behaviour as it grows. If part of a ﬂower is displaying elastic behaviour this means that it is behaving a bit like a spring – stretch is proportional to the force applied. However a ﬂower also needs to display plastic behaviour if the shape is going to be permanent. Plastic behaviour is where a material retains its shape once it has been deformed – a bit like stretching a bit of plasticine – it stays in the shape which you have moulded. This combination of elastic and plastic behaviour allows a ﬂower to stretch and grow, setting into its new shape.

To find out exactly how this growth works, researchers have created mathematical models which simulate the growth of ﬂowers, helping the researchers to understand the underlying pattern of genes which is controlling behaviour. To create these computer simulations, researchers have used the mathematical equations which govern elastic and plastic behaviour.

Elastic behaviour is governed by what is known as Hooke’s Law, whereas plastic behaviour is controlled by more advanced mathematics such as partial differential equations.

### Hooke’s Law

$F = k \times e$

$F$ = Force measured in newtons (N),
$k$ = Spring constant in newtons per metre (N/m) which depends on which material is being used
$e$ = Extension in metres of the spring or material

By modelling the growth of ﬂowers using a mathematical model, researchers are able to test their hypotheses and discover the growth behaviours which produce ﬂowers that are actually observed in nature. This new research has now cast light on a previously mysterious and unknown process, and has been made possible by the use of mathematics. Therefore next time you see a daffodil in spring, take a moment to wonder at the forces which have governed its amazing growth.