Mathematical Model
Background
Synthetic Notch, or synNotch, is an engineered version of the Notch receptor that can function in various mammalian cell lines (Morsut et al., 2016). SynNotch pathways implement user-defined functional responses, and since these pathways are functionally orthogonal, both Boolean response regimes and multi-cellular signaling cascades can be engineered (Morsut et al., 2016). In order to effectively utilize these receptors, their molecular dynamics need to be further understood; since receptors are a recent development, an exhaustive mathematical model of their processes is lacking. We aim to modify Notch-Delta systems-level models, which have already been extensively studied, into a working mathematical model of synNotch. Figure 1 shows the relationships between different components of Notch and synNotch.
Figure 1: Comparison of Notch and components. In synNotch, both extracellular and intracellular domains are replaced. The central regulatory region is present in both systems. Taken from Figure 1 of Morsut et al., 2016.
The Notch mechanism begins with the binding of ligand presented by a neighboring cell (Kopan and Ilagan, 2009). Endocytosis of ligand-receptor complexes induces the unfolding of a juxtamembrane negative regulatory region (Kopan et al., 2012). This is thought to be accomplished by a mechanical pulling force (Gordon et al., 2007; Sprinzak and Blacklow et al., 2021). This leads to subsequent cleavage of the Notch extracellular domain (Bray et al., 2006; Henrique and Schweisguth, 2019; Lovendahl et al., 2018; Sprinzak and Blacklow et al., 2021), since the access is gained by the ADAM10 protease, which cleaves at site 2 (Kopan et al., 2012; Sprinzak and Blacklow et al., 2021; Steinbuck and Winandy, 2018). Following this cleavage, Notch is cleaved again within its transmembrane domain at site 3, which releases the Notch intracellular domain (Gordon et al., 2008; Jarriault et al., 1995; Kopan et al., 2012; Sprinzak and Blacklow et al., 2021). In our system, the intracellular domain is the Gal4 transcription factor released by synNotch (Morsut et al., 2016). It is known that site 2 cleavage is the rate-limiting step (Steinbuck and Winandy, 2018), so we condense the series of cleavages into a single kinetic parameter. However, since exposure of the negative regulatory region is required for the cleavages to occur, we assume that a conformational change is necessary for activation of the receptor. In the synNotch system, a PEST sequence is exposed upon release of transcription factor (Morsut et al., 2016), which is responsible for degradation of the receptor (Sarfraz et al., 2020).
To model molecular synNotch pathways, we introduce the cubic ternary complex approach, previously implemented in G-protein coupled receptor modeling in pharmacology (Weiss et al., 1996). In total, there are three possible dimensions for a state: bound to ligand or not bound to ligand, bound to transcription factor or not bound to transcription factor, and active or inactive conformation of the receptor itself. This allows for eight mathematically possible states, each of which representing a corner of a cube. These assumptions are consistent with applications of previous cubic ternary complex models (Weiss et al., 1996), although we replace the standard G-protein component with transcription factor. The three allowed transformations correspond to the three sets of parallel faces of the cube (Weiss et al., 1996); for our purposes, this includes ligand binding, transcription factor release, and activation of the synNotch receptor.
This cubic structure allows us to consider all possible states and transformations of the synNotch system. While we make assumptions to narrow this system down to fewer equations and parameters, we can always return to the higher level perspective to include new experimental data or observations. Furthermore, this narrowing can be made concisely by interpreting biological phenomena from the literature to make assumptions. In the following section, we explore these assumptions and systematically produce a biologically relevant mathematical system of ordinary differential equations.
Assumptions and Derivations
For our system, the full cubic ternary structure is represented in Figure 2 below.
Figure 2: Cubic ternary complex model representation of synNotch; blue parameters represent ligand binding; green parameters represent transcription factor release; pink parameters represent receptor activation; purple parameters represent degradation rates. The red one-way arrows represent the ligand-dependent activation pathway of synNotch, which is known to be the predominant pathway (Morsut et al., 2016). Also see Figure 3 for the ligand-dependent activation pathway. The yellow arrow represents the expression of on the surface of the cell.
The descriptions of the states are in Table 1 below, as well as the differential equations derived from Figure 2. In this system, all reactions shown are reversible and all species shown are supposedly relevant. Species with the # symbol have a PEST tag shown, which signals degradation of the receptor (Sarfraz et al., 2020). This PEST tag is exposed once the transcription factor leaves the receptor (Morsut et al., 2016). Figure 3 shows the main pathway of interest.
Figure 3: Ligand-dependent activation pathway of synNotch. The blue circle represents the ligand or antigen of interest; the purple curve represents the binding site of synNotch; the dark purple line represents the receptor; the yellow line represents the receptor after conformational change (activation); the pink square represents transcription factor. From left to right, the ligand binds, the receptor activates, the transcription factor releases and is sent to the nucleus, and the ligand dissociates.
| State | Description |
|---|---|
| SynTF | Inactive with transcription factor |
| Syn# | Inactive without transcription factor |
| Syn*TF | Active with transcription factor |
| Syn*# | Active without transcription factor |
| LSynTF | Inactive with transcription factor and ligand |
| LSyn# | Inactive without transcription factor and with ligand |
| LSyn*TF | Active with transcription factor and ligand |
| LSyn*# | Active without transcription factor and with ligand |
Table 1: Variables used in the model and their descriptions
\[\begin{aligned} \frac{d[SynTF]}{dt} = \text{production} + k_{-L,TF} [LSynTF] + k_{-act,TF} [Syn^*TF] \\ + k_{TF} [Syn\#] - (k_{L,TF} + k_{act,TF} + k_{-TF} + \gamma_{SynTF})[SynTF] \end{aligned} \tag{1} \] \[\begin{aligned} \frac {d[LSynTF]}{dt} = k_{L,TF}[SynTF] + k_{-act,L,TF} [LSyn^*TF] + k_{TF,L} [LSyn\#] \\ - (k_{-L,TF} + k_{act,L,TF} + k_{-TF,L} + \gamma_{LSynTF})[LSynTF] \end{aligned} \tag{2} \] \[\begin{aligned} \frac{d[LSyn^*TF]}{dt} = k_{act,L,TF} [LSynTF] + k_{TF,L}^*[LSyn^*\#] + k_{L,TF}^* [Syn^*TF] \\ - (k_{-act,L,TF} + k_{-TF,L}^* + k_{-L,TF}^* + \gamma_{LSyn^*TF})[LSyn^*TF] \end{aligned} \tag{3}\] \[\begin{aligned} \frac{d[LSyn^*\#]}{dt} = k_{-TF,L}^* [LSyn^*TF] + k_{act,L} [LSyn\#] + k_{L}^* [Syn^*\#] \\ - (k_{TF,L}^* + k_{-act,L} + k_{-L}^* + \gamma_{LSyn^*\#})[LSyn^*\#] \end{aligned} \tag{4}\] \[\begin{aligned} \frac{d[Syn^*\#]}{dt} = k_{-L}^* [LSyn^*\#] + k_{act}[Syn\#] + k_{-TF}^*[Syn^*TF] \\ - (k_{L}^* + k_{-act} + k_{TF}^* + \gamma_{Syn^*\#})[Syn^*\#] \end{aligned} \tag{5}\] \[\begin{aligned} \frac{d[Syn\#]}{dt} = k_{-L}[LSyn\#] + k_{-TF}[SynTF] + k_{-act}[Syn^*\#] \\ - (k_{L} + k_{TF} + k_{act} + \gamma_{Syn\#})[Syn\#] \end{aligned} \tag{6}\] \[\begin{aligned} \frac{d[LSyn\#]}{dt} = k_{-TF,L}[LSynTF] + k_{-act,L}[LSyn^*\#] + k_{L}[Syn\#] \\ - (k_{TF,L} + k_{act,L} + k_{-L} + \gamma_{LSyn\#})[LSyn\#] \end{aligned} \tag{7}\] \[\begin{aligned} \frac{d[Syn^*TF]}{dt} = k_{-L,TF}^*[LSyn^*TF] + k_{act,TF}[SynTF] + k_{TF}^*[Syn^*\#] \\ - (k_{L,TF}^* + k_{-act,TF} + k_{-TF}^* + \gamma_{Syn^*TF})[Syn^*TF] \end{aligned} \tag{8}\]
Table 2: All parameters defined in the cubic ternary complex model and their descriptions
We then present several assumptions to simplify these equations, most of which concerning the feasibility and irreversibility of many of the transitions shown in Figure 2. First, we assume that transcription factor cannot bind back to synNotch once it leaves, since a PEST sequence that signals degradation is exposed once the transcription factor is cleaved (Morsut et al., 2016). This gives
\[k_{TF} = k_{TF,L} = k_{TF}^* = k_{TF,L}^* = 0. \tag{9}\]
Second, we assume that activation of synNotch is irreversible, as is the case in the Notch-Delta canonical pathway (Kopan and Ilagan, 2009). Notch activation relies on sequential cleavages as discussed in the background section, so this assumption is reasonable for the Notch-Delta system. This gives
\[k_{-act} = k_{-act,TF} = k_{-act,L} = k_{-act,L,TF} = 0. \tag{10}\]
Since this is extrapolation from Notch-Delta, retroactive sensitivity analysis could be conducted on the above parameters. However, we assume these reactions are all irreversible in order to minimize parameters used in our model.
Third, we employ a quasi-steady state approximation; we treat activated synNotch species with transcription factor as transition states, since transcription factor is released instantaneously upon completion of activation. Recall that we define activation as the conformational change that takes place as a result of a series of cleavages. This eliminates two variables from the model: LSyn*TF and Syn*TF. This also eliminates their degradation rates and several associated pathways, since we assume these only appear for an infinitesimal length of time.
Fourth, we assume that transcription factor cannot release without activation. Thus,
\[k_{-TF,L} = k_{-TF} = 0. \tag{11}\]
By incorporating all of these assumptions, we can then reason that Syn# and LSyn# will always be zero, since their initial values in the system are zero and the parameters in equation 11 are their only input values. From a biophysical perspective, Syn# and LSyn# seem unlikely due to the dependence of transcription factor release on the conformational change of synNotch (Gordon et al., 2008; Jarriault et al., 1995; Kopan et al., 2012; Sprinzak and Blacklow et al., 2021). We can thus ignore these variables; their degradation rates and the equilibrium between them are also irrelevant. Since Syn# and LSyn# always take a value of 0, we also obtain
\[k_{act} = k_{act,L} = 0, \tag{12}\]
which can be reasoned from Figure 2. We have four remaining states, namely SynTF, Syn*#, LSynTF, and LSyn*#. Since we are predominantly concerned with a time-dependent function of released transcription factor, we can combine the LSyn*# and Syn*# variables and ignore the equilibrium between them, since the association and dissociation of ligand does not affect the transcription factor function after the transcription factor has already been released. The combination represents transcription factor released by both ligand-dependent and ligand-independent activation pathways. The ligand-independent pathway parameter is known to be nonzero (Morsut et al., 2016; Roybal et al., 2016; Yang et al., 2020; ).
Finally, we arrive at the three-state model, which we utilize in our simulations. This model describes the interactions between SynTF, LSynTF, and released transcription factor, and it allows us to obtain an output transcription factor function of time. To derive the equation for TF, we take
\[\frac {d[TF]}{dt} = \frac {d[LSyn^*\#]}{dt} + \frac {d[Syn^*\#]}{dt} \tag{13}\]
since the total amount of TF will be the sum of these two states. The initial condition of this differential equation is TF(0) = 0, and this also renders the equilibrium between the two activated states irrelevant, since a decrease by one in a given state is an increase by one in the other state. Expanding this equation from the model in Figure 2 with the applied assumptions, we obtain
\[\begin{aligned} \frac {d[TF]}{dt} & = k_{act,L,TF}[LSynTF] + k_{L}^*[Syn^*\#] - (k_{-L}^* + \gamma_{LSyn^*\#})[LSyn^*\#] \\ &+ k_{-L}^*[LSyn^*\#] + k_{act,TF}[SynTF] - (k_{L}^* + \gamma_{Syn^*\#})[Syn^*\#]. \end{aligned} \tag{14}\]
The degradation rates in equation 14 are for the receptors, so these become zero for transcription factor. While Gal4 does have a nonzero transcription factor (Sprinzak et al., 2010), this is applied after it binds to the nuclear DNA, so we neglect this term for now. Simplifying equation 14, we obtain
\[\frac {d[TF]}{dt} = k_{act,L,TF}[LSynTF] + k_{act,TF}[SynTF] \tag{15}\]
Our final diagram is shown below in Figure 4, and equation 15 is exactly the differential equation we would obtain from the diagram itself.
Figure 4: Three-state system of used in computational work. Simplified from the full cubic ternary complex model using biological assumptions
While ligand-independent activation has proven much more difficult to quantify for the synNotch model, all other parameter values have been fit from Notch-Delta literature values, and these are shown in Table 3.
| Parameter | Value | Units | Notes | Reference |
|---|---|---|---|---|
| \[k_{L,TF}\] | 0.167 | \[\mu m^2 / s\] | Notch-Delta association | Khait et al., 2016 |
| \[k_{-L,TF}\] | 0.034 | \[s^{-1}\] | Notch-Delta dissociation | Khait et al., 2016 |
| \[k_act,{L,TF}\] | 0.34 | \[s^{-1}\] | Notch-Delta activation | Khait et al., 2016 |
| \[\gamma_{SynTF}\] | 0.1 | \[hr^{-1}\] | Notch degradation | Sprinzak et al., 2010 |
| \[\gamma_{LSynTF}\] | 0.1 | \[hr^{-1}\] | Notch degradation | Sprinzak et al., 2010 |
Table 3: Parameter values used in the model simulations
The final ODE system is shown below, as deduced from the schematic in Figure 2 and the derivations above:
\[\frac{d[SynTF]}{dt} = production + k_{-L,TF}[LSynTF] - (k_{L,TF} + k_{act,TF} + \gamma_{SynTF})[SynTF] \tag{16}\] \[\frac{d[LSynTF]}{dt} = k_{L,TF}[SynTF] - (k_{-L,TF} + k_{act,L,TF} + \gamma_{LSynTF})[LSynTF] \tag{17}\] \[\frac {d[TF]}{dt} = k_{act,L,TF}[LSynTF] + k_{act,TF}[SynTF] \tag{18}\]
Note that we take production to be zero, and we simply give SynTF a nonzero initial value to represent the total number of synNotch receptors expressed on the surface of the cell.
Sensitivity Analysis
As stated above, there do not exist specific synNotch parameter values for ligand-independent activation. As such, we want to calculate the sensitivity of this parameter on related model variables. According to the schematic in Figure 4, ligand-independent activation affects both TF and SynTF, and it does not affect LSynTF. We derive the sensitivity equations as follows.
We start with the sensitivity of TF to ligand-indendent activation:
\[[TF]_{k_{act,TF}} = \frac {\partial [TF]}{\partial k_{act,TF}} \tag{19}\]
However,
\[\begin{aligned}\frac {\partial}{\partial t} [TF]_{k_{act,TF}} & = \frac {\partial}{\partial t} \frac{\partial [TF]}{\partial k_{act,TF}}\\ & = \frac {\partial}{\partial k_{act,TF}} \frac {\partial [TF]}{\partial t}\\ & = \frac {\partial}{\partial k_{act,TF}}(k_{act,L,TF}[LSynTF] + k_{act,TF}[SynTF]) \end{aligned} \tag{20}\]
Also,
\[\begin{aligned} \frac {\partial}{\partial t} [TF]_{k_{act,TF}} & = \frac {\partial}{\partial k_{act,TF}} f(k_{act,TF},k_{act,L,TF},[SynTF],[LSynTF])\\ & = \frac{\partial f}{\partial k_{act,TF}} + \frac{\partial f}{\partial k_{act,L,TF}} \frac{\partial k_{act,L,TF}}{\partial k_{act,TF}} + \frac{\partial f}{\partial [SynTF]} \frac{\partial [SynTF]}{\partial k_{act,TF}} + \frac{\partial f}{\partial [LSynTF]} \frac{\partial [LSynTF]}{\partial k_{act,TF}} \end{aligned} \tag{21}\]
Simplifying, we get
\[\frac {\partial}{\partial t} [TF]_{k_{act,TF}} = [SynTF] + k_{act,TF}[SynTF]_{k_{act,TF}} \tag{22}\]
We also need an equation for the sensitivity of SynTF to ligand-independent activation. That is, we want
\[[SynTF]_{k_{act,TF}} = \frac{\partial [SynTF]}{\partial k_{act,TF}} \tag{23}\]
We use the same methodology as above.
\[\begin{aligned} \frac {\partial}{\partial t} [SynTF]_{k_{act,TF}} & = \frac {\partial}{\partial t} \frac{\partial [SynTF]}{\partial k_{act,TF}}\\ & = \frac {\partial}{\partial k_{act,TF}} \frac {\partial [SynTF]}{\partial t}\\ & = \frac {\partial}{\partial k_{act,TF}}(production + k_{-L,TF}[LSynTF] - (k_{L,TF} + k_{act,TF} + \gamma_{SynTF})[SynTF]) \end{aligned} \tag{24}\]
Thus,
\[\begin{aligned} \frac {\partial}{\partial t} [SynTF]_{k_{act,TF}} & = \frac {\partial}{\partial k_{act,TF}} f(k_{act,TF},k_{L,TF}, k_{-L,TF}, \gamma_{SynTF}, [SynTF],[LSynTF]) \\ & = \frac{\partial f}{\partial k_{act,TF}} + \frac{\partial f}{\partial k_{L,TF}} \frac{\partial k_{L,TF}}{\partial k_{act,TF}} + \frac{\partial f}{\partial k_{-L,TF}} \frac{\partial k_{-L,TF}}{\partial k_{act,TF}} + \frac{\partial f}{\partial \gamma_{SynTF}} \frac{\partial \gamma_{SynTF}}{\partial k_{act,TF}} \\ & + \frac{\partial f}{\partial [SynTF]} \frac{\partial [SynTF]}{\partial k_{act,TF}} + \frac{\partial f}{\partial [LSynTF]} \frac{\partial [LSynTF]}{\partial k_{act,TF}} \end{aligned} \tag{25}\]
Simplifying, we get
\[\frac {\partial}{\partial t} [SynTF]_{k_{act,TF}} = -[SynTF] - (k_{L,TF} + k_{act,TF} + \gamma_{SynTF})[SynTF]_{k_{act,TF}} + k_{-L,TF}[LSynTF]_{k_{act,TF}} \tag{26}\]
However,
\[[LSynTF]_{k_{act,TF}} = 0, \tag{27}\]
since LSynTF does not depend on \(k_{act,TF}\). Therefore, it is not sensitive to \(k_{act,TF}\). Our sensitivity equation for \(k_{act,TF}\) on SynTF becomes
\[\frac {\partial}{\partial t} [SynTF]_{k_{act,TF}} = -[SynTF] - (k_{L,TF} + k_{act,TF} + \gamma_{SynTF})[SynTF]_{k_{act,TF}} \tag{28}\]
Thus, our final sensitivity equations are 22 and 28, and these can be solved with equations 16, 17, and 18 to express the entire solved model. The results are shown in Figure 5.
Figure 5: Solved ODE system with equations for sensitivity on ligand-independent activation. Ligand-independent activation was initially estimated as \(\frac{k_{act,L,TF}}{10}\). Sensitivity functions are multiplied by 0.01 to indicate the change in a given variable for a 0.01 unit change in ligand-independent activation. This more accurately captures the scale of this parameter, as it was estimated to be 0.034 \(s^{-1}\).
Success Rates
With its current construction, the model implies that the process goes to completion; that is, both ligand-dependent and ligand-independent pathways successfully yield a released transcription factor for every synNotch receptor. The current parameters that represent these processes only influence the time-scales on which they occur. However, we know this not to be true. There exist experimental data that show a certain proportion of antibody ectodomain expression in the absence of ligand (Morsut et al., 2016; Roybal et al., 2016). Thus, there must be success terms implemented somewhere in this process, since the activation pathways do not release all transcription factor. We introduce success rates for both ligand-dependent and ligand-independent activation in our model. To accomplish this, we fit to experimental data found in Figure 2SA of Morsut et al., 2016, which shows mCherry response against ligand concentration for . Since the fitting should not depend on kinetic parameters, we look at end-state concentration values of released transcription factor for this fitting.
We define (1 - \(\alpha\)) as the success rate for ligand-dependent activation, and we define (1 - \(\beta\)) as the success rate for ligand-independent activation.
We use the following table to outline our procedure:
| "ligand concentration" | mCherry response (relative to total) | Linear relationship between \(\alpha\) and \(\beta\) | Parameter space of \(\alpha\) and \(\beta\) |
|---|---|---|---|
| \[10^{-0.5} = 0.316\] | 20% | \[\beta = -1.5 \alpha + 2\] | \[(\alpha, \beta) \in (0.67, 1) \times (0.50, 1)\] |
| \[10^{0} = 1\] | 50% | \[\beta = -4.54 \alpha + 2.77\] | \[(\alpha, \beta) \in (0.39, 0.61) \times (0, 1)\] |
| \[10^{0.5} = 3.162\] | 75% | \[\beta = -14.18 \alpha + 3.79\] | \[(\alpha, \beta) \in (0.20, 0.27) \times (0, 1)\] |
| \[10^1 = 10\] | 87% | \[\beta = -45 \alpha + 5.95\] | \[(\alpha, \beta) \in (0.11, 0.13) \times (0, 1)\] |
Table 4: Fitting \(\alpha\) and \(\beta\) to experimental data presented in Morsut et al., 2016
For the four experimental data points chosen, the linear relationships between \(\alpha\) and \(\beta\) vary significantly, and the possible values for \(\alpha\) and \(\beta\) vary as well. This implies that the success rate framework implemented into the model is not robust, since an accurate model should predict how changes in ligand concentration affect results without having to change a set of parameters entirely.
It should be noted well that this model only encapsulates the processes from ligand binding to transcription factor release; when validating the model, we use protein expression as a barometer for the entire process. Since we have not yet included transcription factor binding to DNA and subsequent transcription and translation, we may be missing opportunities to introduce success terms that may be more biologically relevant. Furthermore, we are still lacking experimental verification of the ligand-independent activation parameter; it is a guess extrapolated from the ligand-dependent activation parameter, and adjusting the former may yield more concise results for success rates. Overall, this methodology is important, and we have laid the foundation for this aspect of the model. This is ultimately a work in progress, as we still lack data necessary for accurately modeling these processes.
As an example, the solved ODE model with incorporated \(\alpha\) and \(\beta\) parameters is shown in Figure 6 below. We chose the (\(\alpha\),\(\beta\)) pair corresponding to ligand concentration = \(10^0.5\) to display.
Figure 6: Solved ODE system with implemented success rates for ligand-dependent and ligand-independent activation. Transcription factor output only reaches 75% of its synNotch-TF input. Parameters used: \(\alpha\) = 0.25, \(\beta\) = 0.245, from the fit in Table 4.
Future Direction
The ultimate goal is to develop an exhaustive model for synNotch; while we've adopted Notch-Delta parameters to fit the cubic ternary complex structure, we hope to implement specific parameters in order to more accurately represent the system that is the subject of study. We hope to find a Michaelis-Menten production term to describe the expression of on the surface of the cell. We know that expression is not an instantaneous process, which we assumed for our model. This would account for transcription and translation of the receptor.
Additionally, we hope to develop parameters for ligand binding and unbinding to the synNotch receptor; since synNotch can be engineered to bind to chosen antigens and respond with functions defined by the user, we hope to gain parameters related to antigen identity. This would allow us to systematically test antigen-synNotch pairs in silico and determine which ligands may have the best binding affinities to optimize ICARUS response.
To do this, we hope to utilize molecular dynamics simulations in order to determine kinetic parameters. The GlideScore results determined in the Simulation section are only useful for relative thermodynamics, and they cannot be implemented as kinetic parameters to our model.
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