Introduction
In order to improve our design in a clearer and more detailed way, we constructed some mathematical modeling to supplement it.
Model1: In order to analyze the behavior of the protein pathway, I developed a minimal ordinary differential equation model to represent the key components and interactions in the pathway. The model involves three types of interactions: protein production, primary degradation, and protease cleavage.
Model2: We simulated the inhibition of one protease on another protease.
Model3: Simulates our design for fast and slow transcription regulation mRNA degradation rates.
Model4: The efficiency of exosome directed transport to the pancreas and the possibility of injury to other organs were simulated.
Model 1
To analyze the behavior of the protein loop, we constructed a minimal ordinary differential equation model representing the key components and interactions within the circuit. The model incorporated three types of interactions: protein production, first-order degradation, and cleavage by proteases. In the model, protease regulation of substrates is described by differential equations of the following form:
Here, π΄ represents the production rate of a proteolytic substrate, K_cat^Protease represents the catalytic coefficient, assuming that proteolysis can be described as a Michaelis-Menten reaction far from saturation, and the first-order degradation rates K_dAand K_dBrepresent degradation through basal cellular degradation pathways. These rate constants can take higher or lower values depending on whether the substrate protein and its cleaved form are unstable or stable, respectively.
To simplify the analysis without loss of generality, we set π΄ = 1 in the equations for fluorescent reporters, effectively using arbitrary normalized units for the fluorescent protein concentrations. [Substrate] in the normalized version thus corresponds to [Substrate]/A in the original version.
We first considered a CitCFP reporter and CitYFP reporter, who can reflect different colors depending on how the TEVP reacts. In its initial form, the CitYFP reporter intensities at rate πd1 (Equation 3), while its cleaved product, Cit, intensities at a rate ππ2 (Equation 4).
The steady-state solutions for Eqs. 3, 4 are:
Experimentally measured reporter fluorescence corresponds to the sum γCitγ_CFP+ πΆππ‘. The absolute value of the independent variable [TEVP] is not known. However, based on experiments in which protein expression levels correlated linearly with the amount of transfected plasmid, we substituted the concentration of transfected plasmid, πππΈ, for [ππΈππ] in all equations, effectively absorbing the constant of proportionality relating [TEVP] and πππΈ into the πππΈ values. With these simplifications, measured fluorescence can be written:
Using Matlabβs curve fitting toolbox, we determined best fit values of the parameters K_cat^TE , ππ1 and ππ2 by fitting Eq. 7 to the experimentally measured πππΈ- πΆππ‘π‘ππ‘ππ curve.οΌThe energy transfer occurred according to the fluorescence intensity of the opsin excited by the energy. The fluorescence intensity of CFP excitation is 1, and that of YFP excitation is 0.οΌ
Model 2
We modeled repression of one protease by another through direct cleavage, based on the scheme in the figure below. We assume the concentration of the input protease, denoted π0 , is maintained at a constant level, with its activity controlled by a small molecule input, as in the scheme of the figure below. The output protease, denoted π, is produced at a constant rate π΄, and undergoes first-order degradation with rate πΎp . The input protease cleaves the output protease at a single cleavage site, converting it to a cleaved form, whose concentration is denoted π, with a cleavage rate constant π. The cleaved protease irreversibly dissociates at rate πd, and undergoes first-order degradation with rate πΎp for a total rate of elimination of πΎp + πd. We assume a single cleavage for simplicity, but the same conclusions hold true for two independent cleavage sites, cleavage of either of which is sufficient to inactivate the output protease.
The reactions in the protease-protease model are as follows, where π denotes βnothingβ:
Assuming protease cleavage functions in a linear regime far from saturation, consistent with published Km values and our bandpass modeling, the reaction can be expressed as a set of ordinary differential equations :
Model 3
As a comparison to protease regulation, we modeled a logically equivalent transcriptional repression step. The input transcription factor was maintained at a constant concentration of π0, with its activity assumed to be controlled by a small molecule, as with the protease. The input transcription factor regulates the output mRNA, πm, whose production follows a standard rate law:
Β·πm undergoes first-order degradation with rate πΎm. The output protein πp is translated from the mRNA at rate π΄p, and degraded with rate πΎp.
The reactions are as follows:
These reactions can be converted to ODEs for each of the components:
Without loss of generality we set the production rate π΄π = 1 πhβ1 and π΄π = 1 hβ1. We used the same protein degradation rate as in the protease regulation case above: πΎp = 0.1hβ1. For mRNA degradation, we simulate two values at opposite extremes of the biological range for mammalian mRNA: πΎm = 0.1hβ1 (more stable), and 5hβ1 (less stable). As above, we also assumed that the small-molecule-controlled input ON-OFF switch is much faster than the other reactions. To match the protease conditions, we assumed π also undergoes a 20-fold regulation, from π0 = 0.5πΎ (ππππ’π‘ ππΉπΉ) to 10πΎ (ππππ’π‘ ππ), although we note that the exact dynamic range of π or the exact choice of the Hill function does not affect output dynamics.
We simulated this simple model of transcriptional regulation with fast and slow mRNA degradation rates, following the same ONβ>OFFβ>ON input temporal profile used in the protease regulation case. To focus on the timescale of regulation, we normalized each curve to its maximal value. For transcriptional regulation, π‘0.5 = 7.2 hβ1 and 17 hβ1 for fast and slow mRNA decay, respectively, regardless whether the input undergoes ONβ>OFF or OFFβ>ON switch. When input switches from ON to OFF, protease and transcriptional regulation occurs on comparable timescales, although their difference is more apparent in the slower mRNA degradation case. When input switches from OFF to ON, however, protease regulation generates a much faster response time compared to transcriptional regulation and the ON to OFF switch in the protease regulation case. Intuitively, the dynamics of each process is limited by the slowest rate at which a species decays , which is the relatively slow protein degradation rate for transcriptional control (or both protein and mRNA degradation rates when mRNA is more stable); in contrast, the output protease decays at a much faster rate because, in addition to regular protein degradation, it is also cleaved by input protease, and the rate is even higher when the input is switched to its active state.
Model 4
We directed the modified TIL cells into the pancreas. TIL cells secrete exosomes that the pancreas transports to other parts of the body by means of blood circulation, which may result in insufficient exosomes remaining in pancreatic cancer. At the same time, too little exosomes in other sites may affect the efficacy of distal metastatic lesions. In order to ensure the accurate transport of exosomes, a pharmacokinetic model was established to simulate the dynamics of exosome movement after exosome injection into the pancreas.
Based on our model, it is theoretically possible to further more accurately calculate the amount of TIL cells injected to help us better treat cancer.
The modified exosomes can be fully injected into the pancreas and then spread outwards from the pancreas.
The preparation is related to three pharmacokinetic ideas. First, after exosomes follow the blood from pancreatic cancer, they first bind reversibly to components in the blood, which prevents them from entering other parts of the body. Second, we determine the proportion and flow of blood to the body through circulation. Finally, we consider that the pancreas is connected to the stomach, liver, bile, spleen, duodenum, and kidney, and has the greatest effect on these areas. After considering the above three basic ideas, the following formula is obtained:
The preparation is related to three pharmacokinetic ideas. First, after exosomes follow the blood from pancreatic cancer, they first bind reversibly to components in the blood, which prevents them from entering other parts of the body. Second, we determine the proportion and flow of blood to the body through circulation. Finally, we consider that the pancreas is connected to the stomach, liver, bile, spleen, duodenum, and kidney, and has the greatest effect on these areas. After considering the above three basic ideas, the following formula is obtained:
According to our model, the number of exosomes injected into human pancreas can be further more accurately calculated theoretically to help us better treat cancer.
