Model of the cell-free system

We designed a response system to describe the performance of our cell-free biosensor based on literature data for human Androgen Receptor (hAR) biosensor. This sensor responds to the presence of testosterone which, when exogenous, acts as an EDC. Our model was partly based on data from scientific publications on similar receptors or responses [1, 2, 3].

Figure 1 provides a simplified representation of the operation of our biosensor.

Reaction R1 (green) describes the binding of the receptor dimer to the DNA template. Reactions R2a (orange) and R2b (magenta) outline two potential pathways for the formation of the receptor-hormone complex. Reaction R3 (dark blue) represents the transcription process in our system. This is an irreversible reaction. The ribonucleotides are added in excess in the experiments, and therefore are not included in the reaction equations. As a result, their concentration remains approximately constant, and the reaction rate is independent of nucleotide concentration. Reaction R4 (light blue) describes the fluorescence process. Although this reaction is theoretically reversible, under our specific experimental conditions and the composition of the reaction mixture, it is approximated as irreversible [5].

Based on the reactions shown in Figure 1, the reaction rates were derived by subtracting the forward reaction rate from the backward reaction rate. The reaction rates are written as the product of the rate constant and concentrations of reactants.

From this, equations describing the kinetics of these reactions were derived. These consist of differential equations describing the change in concentration of all the involved compounds as a function of reaction rates.

For the reactant R, we use the term -2r1. The negative sign reflects that R is consumed in the forward direction of R1 (hence the terms -k1,on and +k1,off), while the factor of 2 accounts for the stoichiometry as the receptor binds as a dimer. This rationale is applied consistently in the equations that follow:

Although the differential equation approach has significant applications, it is not the primary method we will use throughout the rest of the text.

In addition to assisting wetlab with deciding starting concentrations and reaction times, the main aim of modelling for us was to predict the fluorescence output of our cell-free biosensor for a given testosterone concentration and fixed initial substrate concentrations. Additionally, a second aim was to determine the range of testosterone concentrations that our biosensor can reliably detect by adjusting concentrations and reaction times.

A tertiary, less critical objective could have been to estimate the kinetic parameters of the reaction as accurately as possible. Achieving this would require conducting numerous experiments, which was not feasible within the given time frame.

Our primary focus is therefore on the steady state of the reaction. For reactions involving receptor:ligand interactions (R1 and R2b), we can apply the dissociation constants (K_D) from the literature. The dissociation constant (K_D) is calculated as the product of the receptor and ligand concentrations, divided by the concentration of the receptor-ligand complex.

In reaction R1 the ligand is the DNA template, and in reaction R2(b) the ligand is testosterone.

Kinetic data in first iteration

In the first iteration of our model, the kinetic parameters were derived from a combination of literature data [1, 4] , protocols from the Rosalind system [6] and stochastic modelling. For the human androgen receptor, we selected the dissociation constant K_D = 5 × 10^-8 M, the dissociation rate parameter: k_1,off = 1560 [1/(M·s)], and the association rate parameter: k_1,on = 7.8 × 10^-4 [1/s], all based on established literature sources [1,4].

According to the Rosalind protocol [6], DNA template concentrations in the range of 10–50 nM should be tested. Lower concentrations of DNA template are preferable because they reduce the amount of receptor required. Although it is not recommended to use DNA template concentrations below 10 nM [6], we opted to allow it for the model in case it became necessary to further reduce the required receptor concentration.

Stochastic modelling, conducted using an unpublished tool by PhD student Johan Cassias and Professor Mogens Kilstrup from the Technical University of Denmark, was also employed. This tool models the progression of chemical reactions over time based on a numerical implementation of the Gillespie algorithm [7]. The program and the related scientific article is still in pre-publication and cannot be shared here. However, it was very helpful in estimating the appropriate testosterone concentration range (0–20 μM) and receptor concentration (1.1 μM) for the initial set of wet lab experiments.

Kinetic data in subsequent modelling iterations

The next step in kinetic modelling involves analysing data from wet lab experiments. This is necessary to accurately determine the parameters for each reaction and to discuss which of the two possible mechanisms for the formation of the receptor-hormone complex (R2a or R2b) is correct.

We will focus exclusively on experiments conducted with a DNA template containing a single receptor binding site (EREmin). Having multiple receptor elements on the DNA template would allow more than one receptor dimer to bind simultaneously [8]. This approach is chosen because modelling multiple receptor binding events to DNA would require more experimental data than we could collect due to time constraints.

Analysis of reactions R3 and R4

The first wet lab experiments examined the transcription and fluorescence in the absence of the receptor and hormone. In these experiments, the concentration of the RNA polymerase was 10 ng/uL and the concentration of the DFHBI-1T fluorophore was 0.2 mM. Using data from these experiments, we analysed reactions R3 and R4. The graph in Figure 2 shows the results from these experiments - the fluorescence levels over time for different DNA template concentrations.

As seen in the graph, fluorescence levels increase linearly over time. When the DNA template concentration reaches a critical threshold (somewhere between 10 nM and 20 nM), the slope is no longer influenced by adding more DNA template concentration. Thus, using DNA concentrations higher than 20 nM provides no additional benefit for our system. For instance, using a 40 nM DNA template in receptor experiments may result in no noticeable reduction in fluorescence, despite the progressive binding of the receptor to the DNA. This is because the concentration of free DNA (unbound to the receptor) would still exceed the critical threshold, leading to no significant difference in fluorescence with or without the receptor.

In summary, the obtained result indicates that, in our setup, DNA concentrations should not be higher than 10nM of DNA template. To improve future modelling efforts, it would be beneficial to analyze more DNA template concentrations to identify the crucial threshold more precisely. The conducted experiment is not sufficient to determine individual kinetic parameters of the R3 and R4 reactions. To do this, we would have to analyze the effect of varying the concentration of the fluorophore. Additionally, it would be required to isolate and purify the aptamer to analyse the course of the R4 reaction.

However, the analysis of this experiment allowed us to find the DNA template concentration used in further experiments. Additionally, this analysis allowed us to determine the value of the linear increase in fluorescence depending on the concentration of template DNA when the receptor and testosterone are not present.

Analysis of reaction R1

To investigate the R1 response, we conducted experiments in the absence of EDC/hormone but in the presence of the androgen receptor at concentrations of 0.5-1.25 μM. These experiments revealed that as receptor concentration increases, the fluorescence readout decreases, reaching near-zero fluorescence at a receptor concentration of 1.25 μM. Since no EDC is present, these results indicate that the receptor binds to the template DNA independently of EDC involvement.

As receptor concentration increases, the readout stabilises after around 105 minutes. The intensity of the signal is expressed as Micromolar Equivalent Fluorescein (MEF), more specifically Fluorescein Sodium Salt (FSS). The average MEF values for the last four measurements are; 1.108 for 0.5 μM receptor concentration, 0.331 for 0.75 μM, and 0.100 for 1.0 μM. Therefore, for between 0.5 and 1.0 μM receptor concentration, each 0.25 μM increase in receptor concentration results in an approximately 3.3-fold decrease in readout at equilibrium This suggests an exponential relationship.

We decided to use a 4th degree polynomial trend line for modelling the steady state, as it provided a better fit with a higher coefficient of determination than an exponential function. For other calculations in the model, we needed the coefficient of determination to be as high as possible. We assumed the steady state was reached at the last recorded time point (259.83 minutes), and, based on the readout data, we plotted the relationship in Figure 3 below.

Figure 3: Fluorescence readout by receptor concentration at steady state. Result performed in the absence of EDC/hormone, but in the presence of the androgen receptor.

This relationship allows calculation of the expected value of MEF readout based on the receptor concentration without the influence of EDC.

The steady state of the readout observed after 105 minutes suggests that all the DNA templates are bound to a receptor dimer . This is plausible as the DNA template concentration of 10 nM was much lower than the receptor concentrations (0 - 1.2 uM). The remaining leaking fluorescence can likely be explained by aptamer formation during transcription before the receptor binds to the DNA.

To test this hypothesis, future experiments should gradually decrease the receptor concentration and identify the minimum concentration at which the MEF readout in the steady state ceases to increase and stabilises. If this hypothesis is correct, the dissociation constant (K_D) for the binding reaction could be determined from these data.

Analysis of reactions R2a and R2b

Proposed reactions R2a and R2b describe the action of testosterone/EDC in the model. However, during the experiments it turned out that the MEF readout value changed under presence of the hormone, but in absence of the androgen receptor.

Therefore, the effect of testosterone on the readout was investigated in an experiment without the receptor present. The result of this experiment is presented in Figure 4 below.

Figure 4: MEF readout (μM FSS) without receptor present. The readout with 4 μM testosterone was divided by 0 μM and is shown against time [min]. This result was validated in independent repetition of the experiment.

Figure 4 shows that the presence of testosterone caused a decrease in the MEF readout. The results may indicate a reduction in T7 RNA polymerase or the Broccoli aptamer activity upon testosterone addition. The addition of testosterone leads to signal reduction, compared to assay in absence of testosterone. The percentage of remaining signal gradually decreased from 69% to 59% over the course of 260 minutes.

In the same experiment, we measured samples with different receptor concentrations and a constant hormone concentration. These results show a decrease in readout with increasing receptor concentration, reaching complete inhibition at a receptor concentration of 1.25 μM.

To investigate the testosterone-receptor interaction, the influence of hormone individual effect (IE) was eliminated by standardising the MEF value to the amount without IE-induced decrease. Then it was possible to quantitatively investigate the influence of testosterone-receptor interaction on the value of MEF readout. For this purpose, Figure 5 shows a comparison between the values ​​of MEF readout before and after IE correction for 1μM of testosterone.

Figure 5: Comparison of MEF readout over time before and after IE correction. The concentration of testosterone (H) was 4 μM and the concentration of the androgen receptor was 1 μM. The T7-HREminimal-sB-T template was used. A negative control is shown in purple containing only the template.

The same measurements were performed for other testosterone concentrations. For testosterone concentration 1.25 µM no fluorescence was observed. The presence of testosterone at a concentration of 4 μM causes an increase in MEF by an average of 16.9% in the case of 0.5 μM receptor, 43.7% in the case of 0.75 μM receptor and 216.3% in the case of 1.0 μM receptor. Given that we do not observe sudden percentage changes in MEF readout — changes that would likely result from the release of DNA molecules from inhibition, especially early in the process — Reaction R2b appears to be more plausible than reaction R2a. However, confirmation of this hypothesis requires more experiments.

Formation of a receptor-hormone complex reduces the concentration of free receptors that can bind to the DNA template. Consequently, the inhibitory effect caused by the receptor is reduced in the presence of the hormone.

Assuming steady state is reached after 260 min, we now read the fluorescence values ​​for different receptor concentrations in the presence of 4 μM testosterone after IE-correction. As a reminder, this is the steady state fluorescence. Then, we used the relationship in Figure 3 describing fluorescence against receptor concentration (steady state approximation). From this relationship, we evaluated the concentration of the receptor in the free state that could react with the DNA template. The results from this is shown in Table 1 below:

Now, we were able to calculate the K_D of R2b as our biosensor from this point will operate at a receptor concentration of 1 μM (see engineering section):

The obtained value of K_D is slightly higher than for literature data for other receptors from the same family. It should be noted that this is an estimate based on literature data and initial experimental data.

Knowing the value of the K_D, we were able to calculate the concentration of the receptor-testosterone complex (x below) for a receptor concentration of 1 μM and a given testosterone concentration (input) from the relationship below. This can be solved analytically or numerically.

From this, we derived the concentration of the free receptor that can react with the DNA template (Reaction R1) by subtracting the amount of receptor in the receptor-testosterone complex from the total concentration. Then we used the relationship from Figure 3 (steady state approximation) to calculate predicted fluorescence values. These do not account for the individual hormone effect (IE). Assuming this effect is linear with respect to hormone concentration (e.g., if 4 μM testosterone reduces the steady state MEF readout by 41.36%, then 2 μM testosterone would reduce it by 20.68%), we were able to calculate the expected value of the MEF readout accordingly.

Using this system it is possible to calculate an expected value of the MEF readout for different values ​​of testosterone concentration in a sample.

Figure 6: Expected MEF readouts as a function of the testosterone concentration. Data calculated for 1 µM receptor and 10 nM DNA template.

As shown in Figure 6, the biosensor is viable with the current receptor and DNA template concentrations in the range of 0.1–8.39 μM testosterone concentration (MEF amount ​​higher than for zero testosterone concentration). To extend this detection range, further experiments should explore increasing DNA template concentrations without exceeding the critical threshold, as well as optimising the receptor concentration further.

To determine the exact testosterone concentration in a sample, serial dilutions should be tested. If at least two dilutions show a readout higher than 0.102, the more diluted sample can be used to determine the hormone concentration. If only one dilution exceeds 0.102, additional similar dilutions are necessary to pinpoint the concentration or rule out false positives. If none of the samples exhibit a readout above 0.102, and some show lower values, it indicates the presence of another compound that negatively affects transcription or fluorescence.

Conclusion

The modelling we performed allowed us to predict what fluorescence value will be measured for our biosensor with fixed initial receptor and DNA concentrations for a sample with a given testosterone concentration. Additionally, the limit of detection for testosterone concentrations that the biosensor is able to detect with the current substrate composition (0.1-8.39μM of testosterone concentration) was determined.

In the case of reaction R1, kinetic parameters were not determined directly, but a correlation was established that allows for calculating the effect of receptor concentration on observed fluorescence. For reaction R2a/b, the R2b was concluded to be more probable than R2a and the value of the dissociation constant was calculated. In the case of the reaction sequence R3-> R4, it was determined that the change in MEF readout is constant over time and the course for selected concentrations in the range of 10-50nM DNA template was determined.

This process has supported our wetlab work immensely by supplying plans for experiments and reducing the number of experiments required to achieve a functional biosensor.