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	Using the laboratory-conditions doubling time of 24 minutes given in given in <cite ref="Domingue1996"></cite>, we found that the bacteria would re-enter stationary phase after 2.5 hours; for time-points after 3 hours, the number of number of <em>SarA</em> genes producing fluorescence could be assumed as constant. For this reason, we excluded time-points prior to 3 hours. We then converted from fluorescence units to number of fluorescent molecules using the quantization measurements provided by Wu & Pollard <cite ref="Wu2005"></cite> and, using our assumption of a fixed number of active <em>sarA</em> genes, considered the relative change in number of molecules to be representative of the per-promoter rate.		Using the laboratory-conditions doubling time of 24 minutes given in given in <cite ref="Domingue1996"></cite>, we found that the bacteria would re-enter stationary phase after 2.5 hours; for time-points after 3 hours, the number of number of <em>SarA</em> genes producing fluorescence could be assumed as constant. For this reason, we excluded time-points prior to 3 hours. We then converted from fluorescence units to number of fluorescent molecules using the quantization measurements provided by Wu & Pollard <cite ref="Wu2005"></cite> and, using our assumption of a fixed number of active <em>sarA</em> genes, considered the relative change in number of molecules to be representative of the per-promoter rate.
-	We were interested, however, in the changes of concentration rather than the changes in the raw number of molecules. As the name suggests, <em>Staphylococcus Aureus</em> are spherical in shape. Assuming that all <em>S. aureus</em> are spheres, the volume of the cell can be determined. The diameter of a USA300 <em>S. aureus</em> cell was previously measured as $1.1 {\mu}m$ <cite ref="Cheng2014"></cite> resulting in the overall cell volume to be calculated as $5.575 \times 10^-15 L$. 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:	+	We were interested, however, in the changes of concentration rather than the changes in the raw number of molecules. As the name suggests, <em>Staphylococcus aureus</em> are spherical in shape. Assuming that all <em>S. aureus</em> are spheres, the volume of the cell can be determined. The diameter of a USA300 <em>S. aureus</em> cell was previously measured as $1.1 {\mu}m$ <cite ref="Cheng2014"></cite> resulting in the overall cell volume to be calculated as $5.575 \times 10^-15 L$. 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.		This resulted in a exponential model $a e^bt$ with a $b$ rate constant of 0.0011 nM/min.

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Revision as of 01:20, 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. aureus population

Model Construction

After a literature review, we construct the 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 simplifies into the following set of differential equations:

Differential System Here

The choice of a largely first-order model is supported by 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 is represented by an $\alpha_0$ basal expression term) or the complex is unable to bind efficiently (which is 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 modeled the complex such that it limits a 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.

Parameter	Value	Description	Source/Rationale
	0.0011 nM min^-1	mRNA production from sarA P1 Promoter	Determined based on linear fitting to the time-series fluorescence measurements from YFP/P2-P3-P1 fusion, as reported in and fluorescence per molecule from

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:

dCas9 Production Rates from mRNA

We were unable to find a peptide chain elongation rate for 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 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 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 Rates of dCas9

We were unable to find any specific data on dCas9 degradation, so instead we used a protein half-life of sarA measured in S. Aureus by Michelik et al. . We chose sarA rather than a protein more chemically similar to dCas9 because data on sarA was readily available and because dCas9 is transcribed using the sarA promoter, which allows us to at least capture sensitivity of the degradation rate to production.

Production Rates from the sarA Promoter

We used the time-series data given by Cheung et al. to estimate the rate of production from the 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 given in , we found that the bacteria would re-enter stationary phase after 2.5 hours; for time-points after 3 hours, the number of number of SarA genes producing fluorescence could be assumed as constant. For this reason, we excluded time-points prior to 3 hours. 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 sarA genes, considered the relative change in number of molecules to be representative of the per-promoter rate. We were interested, however, in the changes of concentration rather than the changes in the raw number of molecules. As the name suggests, Staphylococcus aureus are spherical in shape. Assuming that all S. aureus are spheres, the volume of the cell can be determined. The diameter of a USA300 S. aureus cell was previously measured as $1.1 {\mu}m$ resulting in the overall cell volume to be calculated as $5.575 \times 10^-15 L$. 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.