Model

Perfusion of cells

Computer tomography is a noninvasive medical imaging procedure that uses specialized X-ray equipment and computer processing to create detailed cross-sectional images, or "slices," of the body's internal structures. Measuring the attenuation of X-ray beam allow for imaging soft and dense tissues of the organisms in various shades of gray. As the data is gathered form multitude of angles, it is possible to reconstruct all the individual "snapshots" into two-dimensional cross-sectional images (slices) of the internal organs, bone, and blood vessels. These slices can also be digitally "stacked" to form three-dimensional (3D) images. Difference of contrast allows to recognise organs, and normal tissue, and in extension allows to diagnose disease, trauma, or abnormalities (like tumors, infections, or blood clots), and to plan or guide medical procedures (like surgery or radiation therapy). They provide much greater clarity and detail on soft tissues and internal organs than conventional X-rays, while being significantly less invasive compared to biopsy.

While CT scans can distinguish between tissues of different densities, many soft tissues and organs have very similar densities, making them hard to tell apart. To solve this, contrast agents (or contrast media) are used to temporarily enhance the visibility of specific areas. The most common contrast agents for CT are iodine-based. Iodine has a high atomic number, which makes it very effective at attenuating X-rays. When an iodine-based contrast agent is injected into the bloodstream, it travels throughout the body, accumulating in different tissues and organs. This causes the areas where it is present—such as blood vessels, tumors, or inflamed tissues—to appear much brighter on the CT scan, creating a sharp contrast against their surroundings.

To ensure the contrast agent reaches the target area quickly and at a high concentration, diffusing into tissues of interest, and not just contributing to the background noise due to diffusion, it is often administered as a bolus injection. This means a large volume of the contrast medium is injected rapidly into a vein, typically in the arm. The timing of the CT scan relative to the injection is critical. By carefully controlling this timing, radiologists can capture images during specific circulatory phases. For example:

• Arterial Phase: Scanning shortly after injection highlights the arteries.
• Venous Phase: Scanning slightly later shows the organs as they are perfused with contrast-rich venous blood.
• Delayed Phase: Scanning several minutes later can reveal how organs like the kidneys and liver are processing and excreting the agent, which can be important for diagnosing certain conditions

The Hounsfield Unit (HU) is the standard quantitative scale used to describe radiodensity in computed tomography (CT) images. This scale precisely measures the degree to which different materials attenuate X-rays, translating these physical properties into the shades of gray you see on a scan. Pure water is the neutral baseline, defined as 0 HU, while air is defined as -1000 HU. Using this scale, doctors can identify different tissues. For instance, soft tissues like muscle and organs typically fall within a +40 to +80 HU range, while fat has a negative value of around -100. This precise numerical system allows for the clear differentiation of tissues, even those with very similar densities, and is fundamental to the diagnostic power of CT imaging. Contrast agent however increase radiodensity of body parts they reside in, allowing for analysis of changes of blood flow or permeability of cells or intracellular spaces.

Modeling the perfusion

The goal is to model how the organs and lesions of choice are reacting to changing concentration of contrast in local vicinity. Their perfusion would be then modeled using rates of diffusion of contrast into the area of interest, that is by measuring changes of HU in these areas.

The dye is injected into the vein, and contributes to attenuation of the Blood, that the CT scan is gathering during the arterial phase. While quick, the injection is not instantaneous and may range between injecting 3-5 ml/s of dye, with the total volume being variable as well, and dependent on the patient. The increasing contrast concentration in the vessels around the cells causes it to diffuse into the area of interest, as well as into the Cells themselves. As it is a diffusion reaction, the rate is dependent on the gradient between the two spaces, with a rate corresponding to permeability between the areas. Once the contrast stops being injected, the molecules of dyes are starting to disperse across all of the organism. The attenuation of the blood would slowly start to drop to reach the previous baseline levels, and the cells would then follow.

The equations to describe the system would be the ones bellow

$$ \frac{dB}{dt} = r_b(b+K -B) $$ $$ \frac{dC}{dt} = r_c(B-C) $$

The values of B and C are measured in HU. The attenuation of blood is dependent on the baseline level described by \(b\), rate of diffusion of contrast in blood, called \(r_b\), and then contrast injected in bolus symbolised in the equation as \( K \). As we control parameter of contrast injection in the blood, we may choose how long and how much do we apply the contrast, and without loss of generality, we assume that we start injecting at the time \(t=0\), and and at \(t=t_K\).

Variable \(C\), corresponding to the attenuation of cells inside of the area of interest, is described as the change of its own radiodensity with regard to the blood surrounding it. While the parameter of perfusion of cells \(r_c\) is unknown, it is the goal of the model to estimate this parameter.

The model

Our model attempts to understand the toehold expression system and derivate the parameters for toeholds, given the experimental results. We achive this by creating a complex and robust model which as the end result predicts how much GFP the construct is able to express. By solving the equations for parameters, we may derive the real parameters for effectivnes of our ryboswitches, their specifity and sensitivity.
Our system starts with the transfection of the cells with our plasmids containing sequence, allowing expression of toeholds inside the cell. These plasmids are translated, yielding the free toeholds in process. We take into our considaration degradation of both the plasmids and the toeholds, as well as their respective generation. These plasmids are expressing the toehold sequences, which are folding into their inactive form with hairpin established. Free toeholds have a affinity to a target seqence found in the alfafetoprotin mRNA. Once the toehold recognizes the particular sequence in the mRNA, the complex toehold-mRNA establishes and the toehold unravels into it's active form, designed to be able to express its reporter gene, in this case, GFP. In its active form, it is assumed that the rybosomes may attach to the unfolded toehold, and start translating the sequence that is protected by toeholds hairpin. However, we are predicting that the leackage of expression may occur, and result in translation even without the activated toehold. The inactive toeholds with translation ongoing may by created from hairpin not being of enough obstruction of the ribosome, or by the detachment of toehold from AFP mRNA. The whole system may be described with the following equations:

$$ \frac{dP}{dt}= -\gamma_p $$ $$ \frac{dA}{dt}=-k_a A *T +k_dAT $$ $$ \frac{dT}{dt} = k_p P - \gamma_t T -k_a A * T + k_dAT - k_{off} T + k_tTR $$ $$ \frac{dAT}{dt}= k_a A * T - k_dAT - k_{on} AT + k_t ATR $$ $$ \frac{dATR}{dt}= k_{on} AT - k_t ATR - k_d ATR + k_a TR*A$$ $$ \frac{dTR}{dt}= k_{off} T - k_t TR + k_d ATR - k_a TR*A$$ $$ \frac{dR}{dt}= -k_{on} AT + k_t ATR -k_{off} T + k_t TR $$ $$ \frac{dG}{dt}= k_t ATR + k_t TR - \gamma_G G $$

Let's introduce a notation to track the concentration of all units of molecule, by using "blackboard board" letters, meaning that it tracking the concentration of all the toeholds, regardless of their association. For example $$ \mathbb{T}= T+ AT + ATR+ TR$$ as each unit of this complexes contains one toehold. As the model is quite complex, we may find a ways to reduce it or to find cases where it simplifies.. However before we start with biological explanations of phenomens, we may conduct the properties of the system from the equations themselves.

Toehold concentration stabilization

As the toeholds are directly transcribed from the plasmids transfected into the cells, we may ensure try to predict how much of toeholds will be in the cell itself. Although the ammount, and thus the concentration in the volume of cell is unknown for us, as it is heavily reliant on cell type, transfection success and probabilistic distibution of the plasmids inside of the carrier, we may not assume the exact starting point. However we may know that \( P(0)= P_0 \geq 0\) . Knowing that plasmid DNA lasts and remains expressed after transfection for a significant ammount of time, and its removal is mostly due to dilution when cells divide and plasmid degrades over time, as it is not mantained by cell genome machinery. However in terms of immidiate expression it means that the rate of the dimminishing of plasmid concentration \(\gamma_p\) is small, and may be assumed that it is negligible in terms of immidiate expression, meaning $$ \frac{dP}{dt} \approx 0 $$ This serves another purpouse as well, while trying to find a stationary state, without this assumption, the only stationary state for \(P\) is for the variable to be equal to zero. As the \(k_p P\) is the only factor in differential equation for \(T\), which is not dependent on T it follows that the toehold also has it's stationary state at \(T=0\). Thus to meaningluffy analyse the modelwe must assume at least local stability for \(P\). Once we do the change of toehold concentration and every single of its complexes is then $$ \frac{d\mathbb{T}}{dt} = \frac{dT}{dt} + \frac{dAT}{dt} + \frac{dATR}{dt} + \frac{dTR}{dt} = k_p P_0 - \gamma_t T$$ meaning that the total toehold units, no matter the interactions, and assuming \(\mathbb{T}(0) = 0\), this equal to: $$ \forall_{t>t_{stable}} \mathbb{T}(t) \approx \frac{k_p P_0}{\gamma_t} $$ However, caution has to be taken regarding the time, it would take to achieve such a state. One may ask when such a state would be achieved, and if it is fast enough to not have a significant effect on other parts of equation. As we can see that the rate of approaching the stable toehold concentration is rate of toehold degradation $'\gamma_t'$, and to estimate whether toeholds are transcribed fast enough is to estimate said parameter.

Rybozomes as non-limiting factor

Rybosomes are inherent part of our equation, as they are necessary to express our protein of choice. However, abundance of ribosomal units in cell does, which are present at ~10^7 per cell, suggest that their presence is not a limiting factor in the equation. As the equation involving ribosomes are conserving them, this means that $$ \frac{d\mathbb{R}}{dt} = \frac{dR}{dt} + \frac{dTR}{dt} + \frac{dATR}{dt} = 0$$ This property, alongside with knowledge of overwhelming ribosomal presence in the cell, leads us to assume that the state change to binding rybosomes is dependent on the affinity of binding of rybosomes to the mRNA and not the concentration of rybosomes themselves, as these may not be the limiting factor, which is represented in the equations.

Absence of trigger sequence

AFP is a nucleotide polymer which contains trigger sequence for all of our toeholds. However, it follows that in cells with no target mRNA we would expect no expression, provided the toeholds had perfect specifity. We would like to represent this in our model, as in our experiment we transfected the HEK293T cell line as the negative control. Considering that the AFP content is zero, the model simplifies to the following set of equations. $$ \frac{dP}{dt} = -\gamma_p $$ $$ \frac{dT}{dt} = k_p P - (\gamma_t + k_{off}) T + k_t TR $$ $$ \frac{dTR}{dt} = k_{off} T - k_t TR $$ $$ \frac{dR}{dt} = -k_{off} T + k_t TR $$ $$ \frac{dG}{dt} = k_t TR - \gamma_G G $$ Once again assuming the \(\gamma_p\) is sufficiently small we may calculate the stability of the system. $$ \frac{dT}{dt}=0 \leftrightarrow \frac{k_p P}{\gamma_t} $$ $$ \frac{dTR}{dt}=0 \leftrightarrow \frac{k_{off} T}{k_t} $$ $$ \frac{dG}{dt}=0 \leftrightarrow \frac{k_{t}}{\gamma_G}TR $$

Apllication to experimental results

To recap the experiment, two cell lines HepG2 and HEK293T are transfected with toeholds of our choice. After incubation for two days, the cells are then measured on flow cytometry experiment to gather data on their fluorescence parameters. The results are then normalised, allowing us to compare the ON/OFF ratio. The positive and negative controls allow us te see the maximum possible fluorescence, assuming the toeholds do not obstruct the translation of reporter gene. Normalising it to GFP fluorescence of positive control, allows to simplify the model without the loss of generality. In the table 2 we are presenting the results of the experiments:

From this we may calculate the parameters for the negative control. Assuming \(A = 0\) $$ G^* = \frac{k_{t}}{\gamma_G}TR = \frac{k_{t}}{\gamma_G} \frac{k_{off} }{k_t}T= \frac{k_{t}}{\gamma_G} \frac{k_{off} }{k_t} \frac{k_p }{\gamma_t} P =\frac{k_{off} k_p}{\gamma_G \gamma_t} P $$

General solution

Finally, we would try to solve for the stable state for our experiment with alfafetoprotin. While the calculations are quite long, as the system is not simplified with AFP it is possible to solve for steady state of GFP. $$ G^* = \frac{k_t}{\gamma_G} (ATR^* + TR^*) $$ With the steady states of \(ATR\) and \(TR\) are respectively: $$ATR^*= \frac{k_{on}k_a k_p }{k_t k_d \gamma_t} P A^*$$ $$ TR^*=\frac{k_{off} k_p }{k_t \gamma_t } P $$ Resulting in: $$ G^* = \frac{k_{p} }{\gamma_G \gamma_t }P (\frac{k_{on} k_a}{k_d}A + k_{off} )$$ As the solution of steady stated of \(A\) and \(R\) is arbitrary, the choice of the value of them may depend on the results experiment, yet it is shown in the solutions that concentration of the ribosomes does not matter under previous assumptions. Solving of above equations will yield important knowledge about the nature and effectiveness of the toeholds, with informations regarding their binding efficacy to target sequence, and leakage expression.

While sufficient data was not gathered in time regarding the individual parameters, like GFP half-time or plasmids expression rate, to supply the model estimations may be made from the literature. Limitations of the model may arise in experimental cases, particulary in cells with present AFP, although not abundant. Yet analysis of the model provides important insights into the processes occuring in the cell after transfection with plasmids. The model allows us to grasp which steps of the process are crucial for effective protein expression, highlighting less intuitive parameters such as the plasmid expression rate or the reporter gene degradation. Estimation of these parameters is crucial for future analysis, as computer simulations do not accurately represent the pleathora of interactions happening inside of the cell. Morover the model has a potenitial for as single cell analysis usage, by modeling parameters based on individual cell's fluorescence gathered during cytometry.

Nonetheless, the current analysis and model framework offer immediate, powerful predictive capabilities for understanding the kinetic bottlenecks and determining the optimal expression strategy following transfection. The model's versatility, particularly its capacity for single-cell parameterization using flow cytometry data, suggests a great potential to move beyond population averages and create a truly personalized, quantitative picture of gene expression dynamics, paving the way for significantly optimized and targeted synthetic biology applications in the future.