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Team:UNIPV-Pavia/Modeling
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| - | + | = '''Mathematical modeling page''' = | |
In this section we explain two dynamic models that can be used to describe the gene networks in our project. After a brief overview about the motivation of a mathematical model, we will illustrate the general formulas we used, we will show the complete ODE models for Mux and Demux gene networks, and then we will report results of some simulations performed with Matlab and Simulink. | In this section we explain two dynamic models that can be used to describe the gene networks in our project. After a brief overview about the motivation of a mathematical model, we will illustrate the general formulas we used, we will show the complete ODE models for Mux and Demux gene networks, and then we will report results of some simulations performed with Matlab and Simulink. | ||
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| - | + | = '''Why writing a mathematical model?''' = | |
The purposes of writing mathematical models for gene networks can be: | The purposes of writing mathematical models for gene networks can be: | ||
*'''Prediction''': a good and well identificated model can be used in simulations to predict real system behavior. In particular we could be interested in system output in response to never seen inputs. In this way, the system can be tested 'in silico', without performing real experiments 'in vitro' or 'in vivo'. | *'''Prediction''': a good and well identificated model can be used in simulations to predict real system behavior. In particular we could be interested in system output in response to never seen inputs. In this way, the system can be tested 'in silico', without performing real experiments 'in vitro' or 'in vivo'. | ||
| Line 29: | Line 29: | ||
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| - | + | = '''Equations for gene networks''' = | |
In this paragraph, mathematical modeling for gene and protein interactions will be described. | In this paragraph, mathematical modeling for gene and protein interactions will be described. | ||
| + | |||
| + | === '''Binding of a ligand to a molecule: Hill equation''' === | ||
| + | The fraction of a molecule saturated by a ligand ( = the probability that the molecule is bound to a ligand) can be expressed as a function of ligand concentration using Hill equation: | ||
| + | |||
| + | *'''Prob:''' the probability that the molecule is bound to the ligand | ||
| + | *'''T:''' concentration of the ligand | ||
| + | *'''K50:''' dissociation constant. It is the concentration producing a probability of 0.5 | ||
| + | *'''n:''' Hill coefficient. It describes binding cooperativity | ||
= '''Regulated transcription''' = | = '''Regulated transcription''' = | ||
Transcription of a mRNA molecule can be regulated by transcription factors in active form. These factors can be activators or inhibitors: they respectively increase and decrease the probability that RNA polymerase binds the promoter. | Transcription of a mRNA molecule can be regulated by transcription factors in active form. These factors can be activators or inhibitors: they respectively increase and decrease the probability that RNA polymerase binds the promoter. | ||
| - | This probability must be function of active transcription factor concentration and can be modeled using Hill equation: | + | This probability must be function of active transcription factor concentration and can be modeled using Hill equation. |
| + | If factor in active form activates transcription, we can write: | ||
| - | *'''Prob:''' | + | *'''Prob:''' probability that the gene is transcripted |
*'''T:''' concentration of the transcription activator in active form | *'''T:''' concentration of the transcription activator in active form | ||
*'''K50:''' dissociation constant. It is the concentration producing a probability of 0.5 | *'''K50:''' dissociation constant. It is the concentration producing a probability of 0.5 | ||
| - | *'''n:''' | + | *'''n:''' Hill coefficient (n>0) |
| + | |||
| + | while, if factor in active form inhibits transcription, we are interested in unbound promoter and so we can write: | ||
| + | |||
| + | *'''Prob:''' probability that the gene is transcripted | ||
| + | *'''T:''' concentration of the transcription inhibitor in active form | ||
| + | *'''K50:''' dissociation constant. It is the concentration producing a probability of 0.5 | ||
| + | *'''n:''' Hill coefficient (n>0) | ||
The equations show that: | The equations show that: | ||
*In activation formula, if [T]=0 the trascription probability is Prob=0, while the maximum probability, Prob=1, is reached asymptotically for [T]->Inf | *In activation formula, if [T]=0 the trascription probability is Prob=0, while the maximum probability, Prob=1, is reached asymptotically for [T]->Inf | ||
*In inhibition formula, if [T]=0 the trascription probability is Prob=1, while the minimum probability, Prob=0, is reached asymptotically for [T]->Inf | *In inhibition formula, if [T]=0 the trascription probability is Prob=1, while the minimum probability, Prob=0, is reached asymptotically for [T]->Inf | ||
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Revision as of 09:55, 29 September 2008
 Home
|  The Team
|  The Project
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|  Parts Submitted to the Registry
|
|---|---|---|---|---|
 Dry Lab
|  Wet Lab
|  Modeling
|  Protocols
|  Activity Notebook
|
Contents |
Mathematical modeling page
In this section we explain two dynamic models that can be used to describe the gene networks in our project. After a brief overview about the motivation of a mathematical model, we will illustrate the general formulas we used, we will show the complete ODE models for Mux and Demux gene networks, and then we will report results of some simulations performed with Matlab and Simulink.
Why writing a mathematical model?
The purposes of writing mathematical models for gene networks can be:
- Prediction: a good and well identificated model can be used in simulations to predict real system behavior. In particular we could be interested in system output in response to never seen inputs. In this way, the system can be tested 'in silico', without performing real experiments 'in vitro' or 'in vivo'.
- Parameter identification: we already wrote that it is very important to estimate all the parameters involved in the model, in order to perform realistic simulations. Another goal that can be reached with parameter identification is 'network summarization', in fact estimated parameters can be used as 'behavior indexes' for the network (or a part of it). These indexes can be very useful for synthetic biologists to choose and compare BioBrick standard parts for genetic circuits design, just like electronic engineers choose, for example, a Zener diode, knowing its Zener voltage.
Equations for gene networks
In this paragraph, mathematical modeling for gene and protein interactions will be described.
Binding of a ligand to a molecule: Hill equation
The fraction of a molecule saturated by a ligand ( = the probability that the molecule is bound to a ligand) can be expressed as a function of ligand concentration using Hill equation:
- Prob: the probability that the molecule is bound to the ligand
- T: concentration of the ligand
- K50: dissociation constant. It is the concentration producing a probability of 0.5
- n: Hill coefficient. It describes binding cooperativity
Regulated transcription
Transcription of a mRNA molecule can be regulated by transcription factors in active form. These factors can be activators or inhibitors: they respectively increase and decrease the probability that RNA polymerase binds the promoter. This probability must be function of active transcription factor concentration and can be modeled using Hill equation. If factor in active form activates transcription, we can write:
- Prob: probability that the gene is transcripted
- T: concentration of the transcription activator in active form
- K50: dissociation constant. It is the concentration producing a probability of 0.5
- n: Hill coefficient (n>0)
while, if factor in active form inhibits transcription, we are interested in unbound promoter and so we can write:
- Prob: probability that the gene is transcripted
- T: concentration of the transcription inhibitor in active form
- K50: dissociation constant. It is the concentration producing a probability of 0.5
- n: Hill coefficient (n>0)
The equations show that:
- In activation formula, if [T]=0 the trascription probability is Prob=0, while the maximum probability, Prob=1, is reached asymptotically for [T]->Inf
- In inhibition formula, if [T]=0 the trascription probability is Prob=1, while the minimum probability, Prob=0, is reached asymptotically for [T]->Inf
