# Imperial College/Courses/Spring2008/Synthetic Biology/Computer Modelling Practicals/Exponential Decay Model

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**Exponential Decay Model**

A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. Symbolically, this can be expressed as the following differential equation, where N is the quantity and λ is a positive number called the decay constant.

The equation that describes exponential decay is

- [math]\frac{dN(t)}{dt} = -\lambda N(t)[/math]

- [math]\frac{dN(t)}{N(t)} = -\lambda dt.[/math]

Integrating, we have

- [math]\ln N(t) = -\lambda t + D \,[/math] where D is the constant of integration.

- [math]N(t) = Ce^{-\lambda t} \,[/math]

where [math]C = e^D[/math]. If we evaluate this equation at [math]t=0[/math], we see that [math]e^D = C = N_0[/math].

so, we have :[math]N(t) = N_0e^{-\lambda t} \,[/math]

### Half-Life

A more intuitive characteristic of exponential decay for many people is the time required for the decaying quantity to fall to one half of its initial value. This time is called the **half-life**, and often denoted by the symbol [math]t_{1/2}[/math]. The half-life can be written in terms of the decay constant, or the mean lifetime, as:

- [math]t_{1/2} = \frac{\ln 2}{\lambda} = \tau \ln 2.[/math]

When [math]t = 0[/math], the exponential is equal to 1, and [math]N(t)[/math] is equal to [math]N_0[/math]. As [math]t[/math] approaches infinity, the exponential approaches zero. In particular, there is a time [math]t_{1/2} \,[/math] such that

- [math]N(t_{1/2}) = N_0\cdot\frac{1}{2}. [/math]

Substituting into the formula above, we have

- [math]N_0\cdot\frac{1}{2} = N_0 e^{-\lambda t_{1/2}}, \,[/math]

- [math]e^{-\lambda t_{1/2}} = \frac{1}{2}, \,[/math]

- [math]- \lambda t_{1/2} = \ln \frac{1}{2} = - \ln{2}, \,[/math]

- [math]t_{1/2} = \frac{\ln 2}{\lambda}. \,[/math]