Imperial College/Courses/Spring2008/Synthetic Biology/Computer Modelling Practicals/Exponential Decay Model: Difference between revisions

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'''Exponential Decay Model'''
'''Exponential Decay Model'''
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<font face="trebuchet ms" style="color:#2171B7" size="3">'''Overview:'''</font><br>
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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.
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.
<math> \frac{dN}{dt} = -\lambda.N </math>
<math> N(t) = N_0.e^{-\lambda t}</math>
===Analytical Solution===


The equation that describes exponential decay is
The equation that describes exponential decay is


:<math>-\frac{dN(t)}{dt} = \lambda N(t)</math>
:<math>\frac{dN(t)}{dt} = -\lambda N(t)</math>


:<math>\frac{dN(t)}{N(t)} = -\lambda dt.</math>
:<math>\frac{dN(t)}{N(t)} = -\lambda dt.</math>
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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>.
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===
===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:  
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>
:<math>t_{1/2} = \frac{\ln 2}{\lambda} = \tau \ln 2.</math>


When this expression is inserted for <math>\tau</math> in the exponential equation above, and ln2 is absorbed into the base, this equation becomes:
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) = N_0 2^{-t/t_{1/2}}. \,</math>
 
Thus, the amount of material left is <math>2^{-1} = {1/2}</math> raised to the (whole or fractional) number of half-lives that have passed. Thus, after 3 half-lives there will be <math>{1/2}^3 = 1/8 </math> of the original material left.
 
 
Quantities that are subject to exponential decay are commonly denoted by the symbol <math>N</math>.  (This convention suggests a decaying ''number'' of discrete items.  This interpretation is valid in many, but not all, cases of exponential decay.)  If the quantity is denoted by the symbol <math>N</math>, the value of <math>N</math> at a time <math>t</math> is given by the formula:
 
:<math>N(t) = N_0 e^{-\lambda t} \,</math>
 
where <math>N_0</math> is the initial value of <math>N</math> (at <math>t = 0</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>
:<math>N(t_{1/2}) = N_0\cdot\frac{1}{2}. </math>
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: <math>t_{1/2} = \frac{\ln 2}{\lambda}. \,</math>
: <math>t_{1/2} = \frac{\ln 2}{\lambda}. \,</math>
===Mean-Life===
In exponential decay, the population is governed by the following formula:
:<math>N = N_0 e^{-\lambda t} \,</math>
where ''t'' is the time, ''N'' is the number of elements in the assembly at that time, <math>N_0</math> is the population at the initial reference <math>t=0</math>, and <math>\lambda</math> is a parameter characteristic of the decay called the [[decay constant]].  The mean lifetime <math>\tau</math> is the [[expected value]] of the amount of time before an unstable object undergoes a decay.  First, we let ''c'' be the normalizing factor to convert to a [[probability space]].
:<math>1 = \int_{0}^{\infty}c \cdot N_0 e^{-\lambda t}\, dt = c \cdot \frac{N_0}{\lambda}</math>
:<math>c = \frac{\lambda}{N_0}.</math>
We see that exponential decay is a [[scalar multiplication|scalar multiple]] of the [[exponential distribution]], which has a [[Exponential_distribution#Properties|well-known expected value]].  We can compute it here using [[integration by parts]].
:<math>\tau = \langle t \rangle = \int_{0}^{\infty} t \cdot c \cdot N_0 e^{-\lambda t}\, dt = \int_{0}^{\infty} \lambda t e^{-\lambda t}\, dt = \frac{1}{\lambda}.</math>

Latest revision as of 16:40, 6 February 2008

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

Integrating, we have

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

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

so, we have :[math]\displaystyle{ 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]\displaystyle{ t_{1/2} }[/math]. The half-life can be written in terms of the decay constant, or the mean lifetime, as:

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

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

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

Substituting into the formula above, we have

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