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	see latex 4 equation		see latex 4 equation
-	The leaky expression of YFP mRNA originates from incomplete repression of mRNA production by the dCas9-sgRNA complex. We considered two possible mechanisms for leaky repression: either RNA polymerase is sometimes able to push past or dislodge the bound complex (which should be represented by an $\alpha_0$ basal expression term) or the complex is unable to bind efficiently (which should be captured by the $K_a$ dissociation constant). Several studies <cite ref="Qi2013"></cite><cite ref="Bikard2013"></cite> have found that almost 100\% repression can be achieved if dCas9 is targeted at the promoter, preventing transcription initiation, while targets downstream of the promoter lead to at most 40\% repression. Since the structure of the DNA at the promoter is not chemically distinct from the DNA in the rest of the gene, these findings support the "`dislodging"' leaky expression hypothesis.	+	The leaky expression of YFP mRNA originates from incomplete repression of mRNA production by the dCas9-sgRNA complex. We considered two possible mechanisms for leaky repression: either RNA polymerase is sometimes able to push past or dislodge the bound complex (which should be represented by an $\alpha_0$ basal expression term) or the complex is unable to bind efficiently (which should be captured by the $K_a$ dissociation constant). Several studies <cite ref="Qi2013"></cite><cite ref="Bikard2013"></cite> have found that almost 100% repression can be achieved if dCas9 is targeted at the promoter, preventing transcription initiation, while targets downstream of the promoter lead to at most 40% repression. Since the structure of the DNA at the promoter is not chemically distinct from the DNA in the rest of the gene, these findings support the "`dislodging"' leaky expression hypothesis.
	From this analysis, we did not tune the dissociation constant $K_a$. In addition, rather than keeping a separately-defined $\alpha_0$ term, we modelled the complex as being able to affect a certain maximum percentage of the production from the promoter. This leads to a new equation for YFP mRNA:		From this analysis, we did not tune the dissociation constant $K_a$. In addition, rather than keeping a separately-defined $\alpha_0$ term, we modelled the complex as being able to affect a certain maximum percentage of the production from the promoter. This leads to a new equation for YFP mRNA:
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	see latex 4 equation		see latex 4 equation
-	When the concentration of the complex is zero, YFP mRNA is produced at the rate expected from the unrepressed sarA promoter. At a large concentration of the complex, the YFP mRNA is produced at only 60% of the possible rate from sarA.	+	When the concentration of the complex is zero, YFP mRNA is produced at the rate expected from the unrepressed <em>sarA</em> promoter. At a large concentration of the complex, the YFP mRNA is produced at only 60% of the possible rate from <em>sarA</em>.
-	CRISPR Model Parameters	+	<h2>Parameter Search and Fitting</h2>
-	We turned to the literature to find parameters for our model. We first looked for exact parameter values in S. aureus. If these could not be found, we next looked for ways to to estimate the parameters using other available data for S. aureus and finally searched for the parameters in other gram-positive bacteria. Aggregating parameters from many experiments across the literature is by nature a somewhat uncertain endeavour, but those parameters about which we are very uncertain are marked with asterisks. A general rationale is given for each parameter, but details on the more circuitously estimated parameters are given after the table.	+	We turned to the literature to find parameters for our model. We first looked for exact parameter values in <em>S. aureus</em>. If these could not be found, we next looked for ways to to estimate the parameters using other available data for <em>S. aureus</em> and finally searched for the parameters in other gram-positive bacteria. Aggregating parameters from many experiments across the literature is by nature a somewhat uncertain endeavour, but those parameters about which we are very uncertain are marked with asterisks. A general rationale is given for each parameter, but details on the more circuitously estimated parameters are given after the table.
	see latex 4 table		see latex 4 table

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Revision as of 00:53, 9 October 2014

Motivation

We decided to create a model of the CRISPR system for two main reasons

• Identifying the parts of the network that could be targeted by our lab team to improve repression efficiency
• To approximate time-series mecA repression data for use in modelling the overall vulnerability of a S. auerus population

Model Construction

After a literature review we were able to construct a network of a CRISPR interference system. The targeted single guide RNA (sgRNA) associates with nuclease-deficient Cas9 protein (dCas9) to form a complex that binds with the DNA complementary to the sgRNA target . The bound complex prevents transcription elongation by RNA polymerase, repressing YFP mRNA expression . The chemical network is shown below:

Definitions

Conventions

Chemical Network

Using standard mass-action kinetics, the network can be translated into the following set of differential equations:

Differential System Here

The choice of a largely first-order model is supported by the findings of several recent studies . To simplify the model further, we made a quasi-steady-state assumption (QSSA) about the formation of of the dCas9-sgRNA complex $b$. That is, we assume that dCas9 and sgRNA associate on a faster timescale than the other reactions (i.e. transcription, translation and the binding of the complex to the DNA). Therefore, we disregard the kinetics of the complex formation reaction and assume that it is always at steady-state relative to the other time-dependent species concentrations. Under the QSSA, the quantity of the complex is given by: see latex 4 equation This is the same assumption made by previous teams . Our model then simplifies to: see latex 4 equation The leaky expression of YFP mRNA originates from incomplete repression of mRNA production by the dCas9-sgRNA complex. We considered two possible mechanisms for leaky repression: either RNA polymerase is sometimes able to push past or dislodge the bound complex (which should be represented by an $\alpha_0$ basal expression term) or the complex is unable to bind efficiently (which should be captured by the $K_a$ dissociation constant). Several studies have found that almost 100% repression can be achieved if dCas9 is targeted at the promoter, preventing transcription initiation, while targets downstream of the promoter lead to at most 40% repression. Since the structure of the DNA at the promoter is not chemically distinct from the DNA in the rest of the gene, these findings support the "`dislodging"' leaky expression hypothesis. From this analysis, we did not tune the dissociation constant $K_a$. In addition, rather than keeping a separately-defined $\alpha_0$ term, we modelled the complex as being able to affect a certain maximum percentage of the production from the promoter. This leads to a new equation for YFP mRNA: see latex 4 equation When the concentration of the complex is zero, YFP mRNA is produced at the rate expected from the unrepressed sarA promoter. At a large concentration of the complex, the YFP mRNA is produced at only 60% of the possible rate from sarA.

Parameter Search and Fitting

We turned to the literature to find parameters for our model. We first looked for exact parameter values in S. aureus. If these could not be found, we next looked for ways to to estimate the parameters using other available data for S. aureus and finally searched for the parameters in other gram-positive bacteria. Aggregating parameters from many experiments across the literature is by nature a somewhat uncertain endeavour, but those parameters about which we are very uncertain are marked with asterisks. A general rationale is given for each parameter, but details on the more circuitously estimated parameters are given after the table. see latex 4 table The only model parameters without some basis in the literature are the association rates for dCas9 and sgRNA. However, since the model is based on the QSSA that those dynamics are much faster than the others in the model, we were able to define a range for those parameters based on the other. Details on the more roundabout estimations are given below: Determining production of dCas9 from dCas9 mRNA We were unable to find a peptide chain elongation rate for \textit{S. aureus}, so instead we used the values reported in BioNumber 107869 which gives a range of 0.59-3.17 amino acids per second per ribosome in \textit{Streptomyces coelicolor}, another gram-positive bacteria. Freiburg's dCas9 part from last year is composed of 1372 amino acids. This translates to a range of 0.0258 to 0.1386 dCas9 molecules per minute per ribosome. We were unable to find ribosome densities in \textit{S. aureus}, but found two different estimates for ribsosome density in Bionumbers: 0.22 ribosomes per 100 codons (i.e. per 3 nt coding sequence) and 3.46 ribosomes per 100 codons . Using our assumption of 3 nt:1 amino acid, we then multiply to get the 0.0057-0.4797 range of dCas9 molecules per minute. Degradation Rate of dCas9 We were unable to find any specific data on dCas9 degradation, so instead we used a protein half-life of \textit{SarA} measured in \textit{S. Aureus} by Michelik et al. . We chose \texit{SarA} rather than a protein more chemically similar to \texit{dCas9} because data on \textit{SarA} was readily available and because \texit{dCas9} is transcribed using the \textit{SarA} promoter, which allows us to at least capture sensitivity of the degradation rate to production. Determining production from the sarA promoter We used the time-series data given by Cheung et al. to estimate the rate of production from the \textit{sarA} P2-P3-P1 promoter in S. aureus. The figure from their paper is reproduced below. After diluting 1:100, the S. aureus strains were serially monitored for $OD_650$. We used data from the SarA+ strain, as that's more like a wild-type S. aureus strain. Using the laboratory-conditions doubling time of 24 minutes given in Using the laboratory-conditions doubling time of 24 minutes given in , we found that the bacteria would re-enter stationary phase after 2.5 hours. For this reason, we considered only timepoints after 3 hours, after the bacteria would re-enter stationary phase and the number of number of \textit{sarA} genes producing fluorescence could be assumed as constant. We then converted from fluorescence units to number of fluorescent molecules using the quantization measurements provided by Wu \& Pollard and, using our assumption of a fixed number of active \textit{sarA} genes, considered the relative change in number of molecules to be representative of the per-promoter rate. We were interested, however, in changes of concentration rather than changes in the raw number of molecules. The diameter of a USA300 \textit{S. aureus} cell has previously been measured as $1.1 {\mu}m$ and Staphylococci are named for their spherical shape, so we assumed the cells to be spherical and found the overall cell volume to be $5.575 \times 10^-15 L$. The number of molecules were thus converted to units of molar concentration in the cell, specifically nanomoles per litre (nM). The exponential fit used to find the rate constant is shown below: This resulted in a exponential model $a e^bt$ with a $b$ rate constant of 0.0011 nM/min.