The idea of the comparator circuit is to provide a modular method of signal integration that can produce a sensor which can specifically and quantitatively measure different chemical species within the cell using non specific promoters. Through mathematical modelling, an equation has been assembled which can predict the expression of each of the reporter proteins such as RFP and CFP.
E = the expression of one of the fluorescent protiens (RFP) when there is transcription of the CFP RNA at any particular level as a proportion of the expression of RFP at the same transcription rate when none of the CFP RNA is present within the cell. So if the promoter (PYEAR) attached to the rfp and construct 1 was expressing at a constant rate with promoter 2 entirely switched off then promoter 2 started transcribing and the amount of rfp in the cell halved then the value of EA would be 0.5 at that transcription rate of CFP (0-1)
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[[File:Equation_7.png| 300px | center]]
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Figure: Theoretical equation to predict the degree of expression of Construct 1 and 2.
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The full equation has been laid out in a way that is relevant only to Construct 1, however, the numbers can be reversed to be relevant to Construct 2. For ease of explanation, everything described will be relevant to Construct 1.
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E = Proportion of expression rate of Construct 1 when both constructs are transcribed (i.e. there is knockdown of one construct) relative to the non-knocked down expression of Construct 1 when only Construct 1 is expressed.
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L = the length of the DNA strand that is transcribed (Leader and protein coding region) .
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A = The rate of transcription of Construct 1 as a proportion of the maximum transcription rate. As a proportion this is measured on a scale of 0 - 1. As an example if the rate of transcription is half of the maximum rate, rate would be 0.5 (arbituary units). It can be assumed the rate of transcription of construct 1 and 2 due to cellular components (e.g. RNA polymerase) is the same, however, the rate of transcription initiation will dictate the transcription rate. The initiation is reliant on the chemical species interacting with the transcription factor which binds to the promoter (i.e. nitric oxide,nitrates,nitrites to PyeaR). The '1' and '2' refer to the Construct 1 or 2 and hence the promoter and the measured fluorescent protein attached (e.g GFP, RFP, CFP, etc).
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L = The length of the Construct 1 in the DNA form that is transcribed (i.e the leader and protein coding region).
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C = the rate of transcription
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Note: Leader refers to the section of RNA at the start of the mRNA that is not translated but has an effect on translation rate.
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C = The rate of transcription. Assuming the rate of transcription of Construct 1 and 2 are the same because the same ribosomes and RBS are involved.
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L/C = the period of time taken for transcription to take place, the time in which translation can be initiated but it is unlikely that the two leaders will bind to one another
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T = Half life of Construct 1 when only Construct 1 is present; the natural half life of Construct 1.
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K = A constant of the biological system. This can only be measured through observation.
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A = the rate of transcription of promoter 1 (the PYEAR) as a proportion of it’s maximum possible transcription rate (0-1)
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The full equation is modelled on the basic equation of:
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A*(L/C) = the number of RFP RNAs that can be translated independently of the presence of other RNA at any one time and so is proportional to translation (and expression) from DNA ascociated RNA (RNA that is still being transcribed). This occurs both when the CFP RNA is and isn’t present so has to be on both the top and bottom of the equation.
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[[File:Equation_2.png| 400px | center]]
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where E is the rate of expression and E(A1) is the same as that explained above.
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H = Half life of the RNA after transcription
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The additional complexity factors in less assumptions, and mimics a biological system, more closely. Below is a breakdown of the full equation.
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L/C + H = the full time for which the RNA would be translated assuming no interactions between leaders for instance when only one of the promoters is inducing transcription.
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[[File:Equation_3.png| 50px | left]]
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This refers to the number of Construct 1 RNA transcripts undergoing transcription at any one time. The length of DNA is particularly important when the chassis is bacterial. In bacteria, as there is no true nucleus, translation occurs simultaneously with transcription. Transcription affects the probability of interaction between construct 1 and 2 and therefore, they are less likely to be translated. As the measurement of fluorescence is the output directly related to the rate of translation, the overall equation measures translation, however, translation rate iis dependent on rate of transcription and degree of knockdown, and hence transcription is factored in here. L/C is the period of time taken for transcription to take place. It is the time in which translation can be initiated but it is unlikely that the two leaders will bind to one another
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B = the transcription of the second promoter within the cell as a proportion of the maximum possible transcription of that promoter (0-1)
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[[File:Equation_5.png| 150px | left]]
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Because there can not be negative expression of A (only positive expression of B to represent negative expression of A) the translation of the RFP RNA(A) = A - AB .
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It also has to be remembered that there will never be full interaction between the two RNA leaders, particularly at low concentrations if only because the two strands never come in to proximity or because of cellular processes; consequently the function B/(D+B) must be used giving the formula; translation of the RFP RNA(A) = A – A (B/(D+B))
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This part of the equation is the deduction of the knockdown of Construct 1 when there is Construct 2 expression and interaction. The biological constant, k, factors in that not all of construct 2 that is expressed will interact with construct 1 and vice versa. Hence, both exist despite construct 2 existing in small quantities. We believe that depending upon the assembly of the orientation of the two constructs within the plasmid, the interaction and hence the binding efficiency can be altered vastly. If the genes have opposite orientations, so that the termination sites are very close then the reduction of distance will increase the chances of interaction and hence make the sensory system more accurate.
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[[File:Equation_6.png| 120px | left]]
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D is a constant of the biological system whose derivation is so complex that it can only really be calculated through observation but can be modelled at various levels.
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This part of the equation encompasses the natural half life of Construct 1 when it alone is expressed (i.e. no expression of or interaction with Construct 2). As described before in the modelling from the basic equation, this is the lower part of the equation and puts it in perspective of Construct 1 and gives expression as a porportion of the maximum transcription. The half life is also Construct 1's half life.
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So to bring it all together; the top half of the equation indicates the degree of translation of the RNA transcribed by the first promoter under any particular transcription rate of the two promoters in arbitrary units. To make this into a meaningful output it is divided by the maximum translation rate at that rate of transcription to equal E(A1); this indicates the degree of attenuation of one RNA from the other.
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So to bring it all together; the top half of the equation indicates the degree of translation of the RNA transcribed by the first promoter under any particular transcription rate of the two promoters in arbitrary units. To make this into a meaningful output it is divided by the maximum translation rate at that rate of transcription to equal EA ; this indicates the degree of attenuation of one RNA from the other.
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To get the degree of translation of the other RNA (EB) just swap A for B throughout the equation.
The idea of the comparator circuit is to provide a modular method of signal integration that can produce a sensor which can specifically and quantitatively measure different chemical species within the cell using non specific promoters. Through mathematical modelling, an equation has been assembled which can predict the expression of each of the reporter proteins such as RFP and CFP.
Figure: Theoretical equation to predict the degree of expression of Construct 1 and 2.
The full equation has been laid out in a way that is relevant only to Construct 1, however, the numbers can be reversed to be relevant to Construct 2. For ease of explanation, everything described will be relevant to Construct 1.
E = Proportion of expression rate of Construct 1 when both constructs are transcribed (i.e. there is knockdown of one construct) relative to the non-knocked down expression of Construct 1 when only Construct 1 is expressed.
A = The rate of transcription of Construct 1 as a proportion of the maximum transcription rate. As a proportion this is measured on a scale of 0 - 1. As an example if the rate of transcription is half of the maximum rate, rate would be 0.5 (arbituary units). It can be assumed the rate of transcription of construct 1 and 2 due to cellular components (e.g. RNA polymerase) is the same, however, the rate of transcription initiation will dictate the transcription rate. The initiation is reliant on the chemical species interacting with the transcription factor which binds to the promoter (i.e. nitric oxide,nitrates,nitrites to PyeaR). The '1' and '2' refer to the Construct 1 or 2 and hence the promoter and the measured fluorescent protein attached (e.g GFP, RFP, CFP, etc).
L = The length of the Construct 1 in the DNA form that is transcribed (i.e the leader and protein coding region).
Note: Leader refers to the section of RNA at the start of the mRNA that is not translated but has an effect on translation rate.
C = The rate of transcription. Assuming the rate of transcription of Construct 1 and 2 are the same because the same ribosomes and RBS are involved.
T = Half life of Construct 1 when only Construct 1 is present; the natural half life of Construct 1.
K = A constant of the biological system. This can only be measured through observation.
The full equation is modelled on the basic equation of:
where E is the rate of expression and E(A1) is the same as that explained above.
The additional complexity factors in less assumptions, and mimics a biological system, more closely. Below is a breakdown of the full equation.
This refers to the number of Construct 1 RNA transcripts undergoing transcription at any one time. The length of DNA is particularly important when the chassis is bacterial. In bacteria, as there is no true nucleus, translation occurs simultaneously with transcription. Transcription affects the probability of interaction between construct 1 and 2 and therefore, they are less likely to be translated. As the measurement of fluorescence is the output directly related to the rate of translation, the overall equation measures translation, however, translation rate iis dependent on rate of transcription and degree of knockdown, and hence transcription is factored in here. L/C is the period of time taken for transcription to take place. It is the time in which translation can be initiated but it is unlikely that the two leaders will bind to one another
This part of the equation is the deduction of the knockdown of Construct 1 when there is Construct 2 expression and interaction. The biological constant, k, factors in that not all of construct 2 that is expressed will interact with construct 1 and vice versa. Hence, both exist despite construct 2 existing in small quantities. We believe that depending upon the assembly of the orientation of the two constructs within the plasmid, the interaction and hence the binding efficiency can be altered vastly. If the genes have opposite orientations, so that the termination sites are very close then the reduction of distance will increase the chances of interaction and hence make the sensory system more accurate.
This part of the equation encompasses the natural half life of Construct 1 when it alone is expressed (i.e. no expression of or interaction with Construct 2). As described before in the modelling from the basic equation, this is the lower part of the equation and puts it in perspective of Construct 1 and gives expression as a porportion of the maximum transcription. The half life is also Construct 1's half life.
So to bring it all together; the top half of the equation indicates the degree of translation of the RNA transcribed by the first promoter under any particular transcription rate of the two promoters in arbitrary units. To make this into a meaningful output it is divided by the maximum translation rate at that rate of transcription to equal E(A1); this indicates the degree of attenuation of one RNA from the other.
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