Exemplary Results Draft 5

RESULTS

Figure 2

Original Results	Our Results
Figure 2. Shankaran et. al.. EGFR system response to varying downregulation and ligand introduction.	Figure 2. Group Simulation. EGFR system response to varying downregulation and ligand introduction.

In Figure 2, the effect of downregulation on the receptor function was studied for the EGFR system. Downregulation is the reduction of a cellular component caused by some stimulus. In this case, the stimulus is an increased rate of internalization of receptor-ligand complexes, quantified as an increase in k_e. Of the four receptor systems studied in this paper, EGFR was selected for this analysis because it is a signaling receptor which makes its ability to quickly respond to ligand concentrations integral to its function. TfR, LDLR, and VtgR are transport receptors, so internalization rates are important but not quite as key to function as with EGFR. Figure 2a serves primarily as a first test of the model to confirm that it is exhibiting logical responses and to get a first look at the impact of downregulation on receptor function. The curves plotted are dimensionless functions for the number of receptor-ligand complexes, C*[t]. In response to a one time impulse of ligand that has been quantified as an initial condition of L[t], we see the number of complexes rise to a peak then fall once again to zero. Logically this makes sense since a sudden introduction of ligand would cause a bunch of complexes to form immediately after introduction, but as more and more of the ligand has been successfully internalized there soon becomes no ligands for the formation of complexes. Thus, the number of complexes degrades to zero in time. Additionally, as downregulation is increased by increasing k_e, we see the height of the maximum peak is reduced, which also makes sense because a higher rate of internalization of complexes would reduce the ligand concentration more quickly as well as preventing large numbers of complexes from building up on the cell surface. Another key impact of increasing downregulation is that the peak occures at an earlier time, indicating a quicker response to the initial ligand input. With these two trends in mind, the rate of introduction of new ligand was then switched to a pulse function with active and resting periods of 360 minutes and an amplitude small enough to keep the vast majority of the receptors unbound on the cell surface, $\frac {0.05 Q_R}{N_{av} * V * 10^{-9}}$. Also, the functions were normalized by dividing each curve by its respective maximum value. This allowed us to look primarily at the efficiency of the responses without being distracted by response magnitudes. Once again, higher downregulation, led to a closer fit to the ligand introduction curve, and thus indicated a quicker response to the ligand stimulus. Then, in Figure 2c, the pulse curve was randomized using a Gaussian distribution. These results serve only to confirm the positive impact of downregulation with respect to response efficiency to perturbations of the extracellular ligand concentration.

Figure 3

Original Results	Our Results
Figure 3. Shankaran et. al.. .	Figure 3. Group Simulation. .

Remembering the dimensionless system of equations that was attained earlier, we note that $\beta$, the partition coefficient used to quantify a system's dependence on consumption, was defined as the ratio of the rate of complex internalization, k_e, to the rate of dissociation of receptor-ligand complexes, k_off. Pairing this with the study of downregulation from Figure 2, we see that by increasing k_e to increase downregulation we were subsequently increasing the partition coefficient as well, and therefore identifying whether or not the function of the given receptor system, EGFR is this case, was responsive to changes in consumption. Varying downregulation through increasing k_e has now been applied to the remaining systems to determine whether or not they are consumption controlled. One important change has been made in the analysis process however. Because the other receptor systems are transport receptors who's primary function is to internalize their designated ligand, the original authors decided that the percentage of the extracellular ligand that had been internalized at a given time would be a more appropriate response measure than the dimensionless or normalized number of receptor-ligand complexes used in Figure 2. Results of this analysis indicate that EGFR and VtgR systems are consumption controlled while TfR and LDLR systems are robust to changes in consumption. Looking at the parameter values in Table 3, it appears that lower values of $\beta$ indicated sensitivity to its variation. Also, it is interesting to note that, contrary to intuition, EGFR, the sole signaling receptor, did not exhibit the quickest and most efficient responses. In order of increasing efficiency/speed of response the receptors are VtgR, EGFR, TfR, and finally LDLR.

Figure 4

Original Results	Our Results
Figure 4. Shankaran et. al.. .	Figure 4. Group Simulation. .

Now, the authors needed to address the second element of the hypothesis which discussed the receptor systems' responsiveness to changes in avidity. Specific avidity is a measure of the system's efficiency at forming receptor-ligand complexes, and it is quantified in the dimensionless equations as $\gamma = \frac{Q_R * k_{on} * 10^9}{k_t * k_{off} * V * N_{av}}$. After considering this mathematical definition of specific avidity, the authors decided that varying the extracellular volume would be the most biologically significant method of altering avidity. The upper volume limit, 4*10^-10 L, was calculated by observing that "a typical cell culture experiment would have 10 mL of media and a cell count of 2.5x 10^7 cells." Then, the lower limit, 4*10^-13 L, "corresponded to approximately one cell volume per cell." Results from this analysis reveal LDLR and TfR to be highly sensitive, EGFR to be moderately sensitive, and VtgR showed no discernible variation. Once again looking at the parameter values in Table 3, it appears that, similar to the partition coefficient, higher specific avidity values correspond to systems that are more robust to variation in avidity.

Figure 5

Original Results	Our Results
Figure 5. Shankaran et. al.. .	Figure 5. Group Simulation.

As receptor systems who's primary function is to internalize their complementary ligand, the time it takes these systems to do so becomes the primary measurement of the efficiency of the system as a whole. To quantify this response time, Shankaran et. al. used a relaxation time, $\tau$, which they defined as the time it takes the number of complexes, C[t], to decay to a value of 1/e times its maximum value. Therefore, an increase in $\tau$ indicates a decrease in efficiency of the system and vise versa. In order to attain an expression for $\tau$ in terms of $\beta$ and $\gamma$, the linearized functions presented in the Model Description were used as a good approximation due to the small initial concentration of ligand that was being studied. Thinking purely from a physical standpoint, since specific avidity describes the system's ability to form receptor-ligand complexes and the partition coeffficient describes the system's ability to internalize those complexes once formed, we would expect an increase in either component to lead to greater efficiency of the entire system, and in fact we do see this trend in Figure 5a. The highest values of $\tau$ are located in the lower left hand corner of the plot and the lowest values are located in the upper right hand corner. As a quick check of Figure 5a, we can estimate the log[$\tau$] value for the EGFR system to be about 1.1 which yields a dimensionless $\tau$ of 10^1.1 which is then divided by k_off as the reverse of the nondimensionalizing of time seen in Table 5 to yield a relaxation time of 52 min for the default EGFR system. This value can now be compared with the Figure 2a curve for a downregulation of 7.5 which is the default EGFR system. The curve has a maximum of roughly 0.45 which means $\tau$ corresponds to the dimensionless C*[t] value of 0.17. The time at which the plot reaches this value is approximately 50, thus demonstrating an agreement between the two figures.

Having seen that increasing either $\beta$ or $\gamma$ leads to a more efficient system, the logical next step is to find out which increase will cause the greatest gain in efficiency for a given receptor system. To do this, Shankaran et. al. defined r_s, the relative sensitivity of $\tau$, as the change in $\tau$ with respect to a 1% increase in $\gamma$ divided by the change in $\tau$ with respect to a 1% increase in $\beta$. Therefore, high values indicate that increasing the specific avidity, $\gamma$, is the best option while low values indicate that increasing consumption, $\beta$, is the best way to increase efficiency. In Figure 5b, we see that the contour plot of r_s has divided the plotted area into three distinct regions. The red region is most responsive to avidity, the blue to consumption, and the green/yellow region indicates relatively equal sensitivities to both.

Figure 5 Linear Approximation Test

Test Confirmation of the linear approximation used to create figure 5

To create Figure 5 we used a linearization of the dimensionless equations at the point ( R* = 1 , C* = 0, L* = 0 ) that would allow us to solve symbolically. However, we needed to make sure that this approximation was appropriate, so we set up a Manipulate function in Mathematica that would allow us to compare the symbolic equation for C*[t*] attained from the linearization to the interpolating function Mathematica's NDSolve created from the non-linearized equations. Our results show a near perfect approximation for all but the lowest values of specific avidity, and even then the difference is relatively slight. We conclude that the approximation used for the creation of Figure 5 was appropriate.

Figure 6

Original Results	Our Results
Figure 6. Shankaran et. al.. .	Figure 6. Group Simulation.

In Figure 5a, we saw that increasing $\beta$ and $\gamma$ lead to more efficient receptor systems, and in Figure 5b we discovered the regions for which increasing $\beta$ had the greatest impact, increasing $\gamma$ had the greatest impact, and where both increases has relatively the same impact. Now, the next step is to quantify the magnitude of these responses to determine the robustness of a system with given $\beta$ and $\gamma$ values. To quantify the magnitude of the robustness of the system, Shankaran et. al. simply added in quadrature the change in $\beta$ with respect to a 1% increase in $\gamma$ and the change in $\gamma$ with respect to 1% increase in $\beta$. This allows for the creation of the contour plot displayed as our result. Low values indicate robustness while high values indicate sensitivity, or relatively large increases in efficiency for increases in $\beta$ or $\gamma$. This plot shows us that while increasing $\beta$ and $\gamma$ does produce gains in efficiency, it does so at a declining rate such that eventually the increases in efficiency will become negligible. Looking at the schematic provided as Figure 6 in the original paper, we note that it is simply a combination of the results of Figures 5a and 5b along with the robustness plot we have provided. Figure 5a is used as the background, Figure 5b is used to mark the avidity controlled, consumption controlled, and dual sensitivity regions, and then our robustness plot is used to mark the robust and sensitive regions of their schematic.

Hypothesis

Remembering from the introduction...

"Specific avidity describes the system's ability to form receptor ligand complexes while the partition coefficient is a measure of consumption, the system's ability to internalized receptor-ligand complexes. The paper then hypothesizes that, by parameter manipulation, the selected receptor systems may be classified as avidity controlled, consumption controlled, or a combination of the two which will in turn all them to be partially, if not completely, distinguished from one another."

With the figures originally presented within the paper that we have now recreated and confirmed, this hypothesis has been addressed. First, the results in Figure 2 confirmed a working and reasonable model. Next, in Figure 3, we varied the internalization rate of receptor-ligand complexes, k_e, which in turn varied the partition coefficient $\beta = k_e / k_{off}$. The corresponding plots have shown the EGFR and VtgR systems to be particularly sensitive to changes in the partition coefficient, while TfR and LDLR are generally unresponsive. Then, in Figure 4, we varied the extracellular volume which caused variation in the specific avidity, $\gamma = \frac{Q_R * k_{on} * 10^9}{V * k_t * k_{off}}$. This time, TfR and LDLR were both very responsive, EGFR was reasonably responsive, and VtgR showed no change. Now we see that EGFR has been isolated as responsive to both avidity and consumption, Vtgr has been isolated as responsive only to consumption, and TfR and LDLR were unable to be separated from one another, both displaying responsiveness to only changes in avidity. Disappointingly, TfR and LDLR even expressed nearly identical magnitudes of responsiveness to varying $\beta$ and $\gamma$. In any case, we have still managed to identify each system as avidity controlled, consumption controlled, or both as well as using these identifications to at least partially separate the receptor systems from one another based on qualitative differences.

Discrepancies

Correction 1

From the original text:

"For computing these sensitivity indices, we generated a linear grid for the variables (b, g). Logarithmic b and c values at a particular (bi, gj) grid point are given by $\beta_i = log10(bi)$ and $\gamma_j = log10(gj)$. The i and j indices vary from 1 to 100; that is, the size of the grid was 100x100. The linear grid spanned the values from -2 to 2 and -3 to 3 for the b and g variables, respectively."

This description is given in the Materials and Methods section of the original text. It is describing a portion of the methodology used to create Figure 5a. However, a mistake has been made in how bi and gj are defined with respect to $\beta$ and $\gamma$. Looking at Figure 5a, we see that the contour plot is a log log plot of $\beta$ and $\gamma$. Therefore, the arguments of the logarithms in the definitions above should be $\beta$ and $\gamma$ respectively instead of bi and gj. This mistake is made more obvious by the ranges given for bi and gj, "-2 to 2 and -3 to 3," because the log of a negative number is undefined. Our recreations above were made using the correct definitions of $bi = log10(\beta_i)$ and $gj = log10(\gamma_j)$.

Correction 2

Our first attempt at recreating Figure 5b resulted in a contour plot that was scaled appropriately, but was in fact a 180 degree rotation of the correct contour plot shown in the paper.

Incorrect Figure 5b Rotated version of Figure 5b caused by a discrepancy in original text.

After carefully checking our work and still being unable to find the discrepancy between our code and the mathematical definitions given in the paper, Professor Chiel suggested that we contact the authors of the original paper to see if they had any insight as to what we were doing wrong. A special thanks is due to the original authors of the paper for their quick response, within one business day. In his response to our email, Harish Shankaran pointed out that there had been a typo in the original paper.

Email response from Harish Shnkaran describing discrepancy

With this typo corrected, we were then able to successfully recreate Figure 5b as well as the robustness plot we created for Figure 6.

Extension

• To be completed for final paper

MATHEMATICA NOTEBOOKS

Figure 2a

Figure 2b

Figure 2c

Figure 3

Figure 4

Figure 5a

Figure 5b

Figure 5 Approx. Test

Figure 6

Equilibrium Extension

Varying kt Extension

New Receptors Extension