Modeling molecular titration approaches to generating ultrasensitivity in dCas9-based gene regulation


Synthetic biologists, in their goal of designing biological systems with novel functions, are particularly interested in design techniques for synthetic genetic networks, given the many emergent properties observed in natural gene circuits. The simplest of these gene circuits is the bistable toggle switch, in which mutually repressing genes create a system that can be tuned to make either gene dominant at will. Previous studies have shown that an essential criterion for the bistability of these systems is the cooperative action of the repressors. Several toggle switch systems, including the first switch pioneered by Collins, have been proposed to introduce cooperativity in the form of the cooperative binding of multiple repressor molecules. In this study, we consider alternative methods for generating cooperativity though molecular titration.

Analogous to chemical titration, molecular titration generates a nonlinear “all-or-nothing” response with a “sink” of buffer molecules that sequester and inactivate repressor molecules at low concentration of repressor. A naturally occurring example of this is protein sequestration, and a previous study has constructed a synthetic circuit with proteins using this principle and demonstrated an ultrasensitive response in yeast. We propose to look at novel forms of molecular titration that involve the widely used dCas9 system in an attempt to construct new building blocks for synthetic circuits. In particular, instead of having a protein and its sequestering protein as a buffer, we wish to use small RNA molecules (anti gRNAs) and decoy binding sites on a plasmid that are complementary to the gRNA component of the dcas9 system to suppress the activity of the cas9 complex and act as a buffer against repression. In this study, we will model the properties of this system by considering how its equilibrium conditions vary with respect to total gRNA concentration and we hope to demonstrate this system’s ability to exhibit an ultrasensitive response under biologically reasonable conditions and to characterize its behavior in general over a wide range of parameters.


The approach we took in modeling our systems is based on Buchler’s approach in modeling molecular titration in a system consisting of a generic repressor and its buffering molecule (1). Their model considers the equilibrium concentration of the active repressor. Taking the total concentration of the other molecules in the system as constants, it is possible to calculate this concentration as a function of the total concentration of repressor. This function is obtained as a solution to a system of equations involving the equilibrium expression for the sequestering reaction in addition to the mass balance equations for each component of the system. The resulting system of equations can be solved as a quadratic polynomial in terms of the concentration of active repressor, giving an explicit expression for this equilibrium concentration in terms of the dissociation constants and the total concentrations of each component in the system.

a) b)

Figure 1: Biochemical reactions involved in the (a) gRNA:anti-gRNA system, and the (b) decoy binding sites system

The system that we want to model using Cas9 is more complex in that neither Cas9 nor the gRNA will actively repress transcription on its own. Because we’re interested in the concentration of the active Cas9:gRNA complex, the equilibrium involving the formation of the active complex needs to be considered, in addition to the equilibrium of the sequestering reaction. Please see the appendix for details of how the resulting polynomial was solved for. This additional equation in the system of equations has the effect of increasing the degree of this polynomial so that it is no longer practical to explicitly solve for the concentration of Cas9:gRNA. To obtain solutions to these polynomials, we used the MATLAB vpasolve function to calculate the numerical solutions for the value of Cas9:gRNA concentration over a range of values for total gRNA concentration. Manual inspection to eliminate the other physically impossible solutions to the polynomial was necessary to obtain the physically relevant graph of active complex versus total gRNA concentration.

A challenge in modeling these systems was finding the equilibrium constants that describe these reactions, as the biochemistry of the Cas9/gRNA system is still a fairly unexplored field of research. Where available, these values were taken directly from published sources; but for some of the parameters, it was necessary to make approximations based on the closest available information.

As can be seen from Figure 1, there are 3 distinct dissociation constants that are relevant to our systems.

Table 1: Values of dissociation constants used in modeling

The first, the dissociation constant of the gRNA:Cas9 complex, was estimated using a dissociation constant obtained through a gel-mobility shift assay of a RNA binding protein that targets small 28bp RNA, which is roughly the same length as the gRNAs we are interested in (2). For the dissociation constant of the gRNA:Cas9:decoyDNA complex, we used the dissociation constant for the complex as reported by Jennifer Doudna (3). Lastly, for the dissociation constants for the formation of the gRNA:anti-gRNA duplex, they were computed from the Gibbs free energy change for the binding of gRNA:anti-gRNA duplexes of similar length. Unfortunately, this approach yielded a very large range in values for the dissociation constant, so we decided to run the model for both extremes reported (4),(5).

Modeling Molecular Titration with anti-gRNAs

a) b) c)

Figure 2: Molecular Titration model with anti-gRNA. (a) Final polynomial in terms of AC, representing the concentration of the gRNA:cas9 complex. k1 and k2 are the gRNA:anti-gRNA and the gRNA:Cas9 dissociation constants respectively; AT is the total gRNA concentration, BT is the total anti-gRNA concentration, and CT is the total Cas9 concentration. (b) gRNA:cas9 concentration versus gRNA concentration for multiple concentrations of Cas9 at k1 = 10-10M (c) gRNA:cas9 concentration versus gRNA concentration for multiple concentrations of Cas9 at k1= 10-18M.

We see that for a dissociation constant of 10-18M, in figure 2(b). we get a strong ultrasensitive response for all concentrations of Cas9. This is not very surprising, as a dissociation constant of that magnitude for gRNA:antigRNA indicates very strong binding and so it makes sense then if the anti-gRNA used has such a powerful sequestering effect that the resulting response curve is quite sharp. However, even for the weaker binding anti-gRNA, with a dissociation constant of 10-10M, we see that we can still get a sharp and nonlinear response under certain conditions. In particular, for Cas9 concentrations of less than 1uM, there is a weaker, but still nonlinear increase in the active complex concentration at the threshold concentration where (gRNA= 100nM. The fact that the response becomes less nonlinear for Cas9 concentrations of 1uM and higher implies that there is a tradeoff that needs to be considered when trying to maximize the concentration of active Cas9:gRNA. While more Cas9 clearly increases the final concentration of the complex at equilibrium, it can also reduce the strength of the ultrasensitivity if the anti-gRNA does not bind tightly enough to the gRNA.

Modeling Molecular Titration with Decoy Binding Sites

a) b) c) d)

Figure 3: Molecular Titration model with decoyDNA (a) Final polynomial in terms of AB, representing the concentration of the gRNA:cas9 complex. k1 and k2 are the gRNA:Cas9 and the gRNA:Cas9:decoyDNA dissociation constants respectively; AT is the total gRNA concentration, BT is the total Cas9 concentration, and CT is the total decoyDNA concentration. (b) gRNA:Cas9 concentrations vs. total gRNA concentration at 100nM Cas9 (c) gRNA:Cas9 concentrations vs. total gRNA concentration at 1uM Cas9 (d) gRNA:Cas9 concentrations vs. total gRNA concentration at 100nM Cas9

For the molecular titration modeling with decoyDNA, there is no need to vary the dissociation constant for the sequestering molecule (decoyDNA) as that value is already well documented in (3). As a result, we attempted to model what would happen as a result of simultaneous changes in total Cas9 concentration and decoyDNA concentration. Note that “decoyDNA concentration” is a function of the plasmid concentration, with a certain number of decoy binding sites per plasmid.

On first glance, the curves for the decoyDNA model appear to be more linear and not as sharp as the ones for gRNA. However, we have much more confidence in the value for the dissociation constant for the gRNA:cas9:decoyDNA complex, and an ultrasensitive response appears to be possible, particularly for higher concentrations of Cas9. An interesting effect of increasing the concentration of the sequestering molecule is that the system appears to be exhausting its supply of Cas9 for the 10uM decoyDNA curves. While we did not show this explicitly for the antigRNA model, it is likely that if we increased the concentrations of antigRNA, we would’ve seen a similar effect there too.

A clear limitation of our current molecular titration models is that they’re only 2D snapshots of what is essentially a four dimensional space (gRNA, Cas9, buffering molecules vs. gRNA:Cas9). Nevertheless, these snapshots do serve the purpose of illustrating the principle of tradeoffs in increasing the concentration of Cas9 and the buffering molecules to create a system that gives a both strong and sharp response to small changes in concentration. Furthermore, we have shown that this ultrasensitive response is indeed achievable through molecular titration with both anti-gRNA and decoy DNA binding site under a range of typical biological conditions. This principle can thus be used to guide the design of genetic constructs that utilize molecular titration to achieve ultrasensitive control of gene expression in vivo.


  1. Cross FR, Buchler NE, Skotheim JM, Philos Trans R Soc Lond B Biol Sci. 2011 Dec 27;366(1584):3532-44.
  2. Ryder SP, Recht MI, Williamson JR Methods Mol Biol. 2008; 488:99-115
  3. Sternberg SH, Redding S, Jinek M, Greene EC, Doudna JA, Nature 6 Mar 2014;507:62-67
  4. Rauzan B, McMichael E, Cave R, Sevcik LR, Ostrosky K, Whitman E, Stegemann R, Sinclair AL, Serra MJ, Deckert AA, Biochemistry 2013; 53:765-772
  5. Lesnik EA, Freier SM, Biochemistry 1995; 34:10807-10815
The appendix can be viewed by clicking here. For more information about the model or the MATLAB scripts, please contact Janan Zhu.

Model 2: Bistable Gene Expression Using CRISPR

Mirroring the use of gRNA-Cas9 repression in lab, this paper attempts to model the transcriptional repression used in lab. It attempts to map the equilibrium transcription rate of a gene as a function of both gRNA concentration and number of gRNA target sites along the gene. Then it explores the effect of "anti-gRNA" nucleus acids which provide molecular titration to the gRNA element of the Cas9 protein. From these findings, a bistable system is created that shows a thousand-fold difference in gene expression. Click here to view the Modeling Paper. For further information or the MATLAB scripts, please contact TJ Ciesla.