Exemplary Discussion Draft 5

DISCUSSION

Hypothesis Analysis

There are two dimensionless parameters that play a major role in the function of both signaling and transport cell surface receptors. The first is specific avidity which is used to quantify the receptor system's ability to form receptor-ligand complexes. The next is a measure of consumption, the partition coefficient, which quantifies a receptor system's ability to internalize receptor-ligand complexes before they dissociate. The analysis undertaken in the original paper that has been recreated here sought understanding of how these two functional parameters could be used to predict receptor function and sensitivity to alterations in these parameters. Four receptor systems, EGFR, TfR, LDLR, and VtgR, were studied in hopes of gaining insight as to the classification of and distinctions between one another based solely on responsiveness to changes in avidity, changes in consumption, or a dual responsiveness to variation in both parameters. Analysis confirms the possibility of distinguishing receptor systems from one another based on responsiveness to changes in avidity and consumption. Of the systems studied, TfR and LDLR were found to be responsive to avidity, VtgR was responsive to consumption, and EGFR displayed a dual sensitivity to both parameters. Furthermore, it was discovered that increases in either parameter led to higher levels of receptor system function and efficiency. Then, by inspecting the mathematical definitions for these parameters, further insight was gained about optimal strategy to improve receptor system efficiency. Specific avidity is defined as $\gamma = \frac{Q_R*K_{on}*10^9}{k_t*k_{off}*V*N_{av}}$. In the paper, avidity was increased by simply decreasing the extracellular volume, V, but we feel it is unlikely that in practice cells will be able to exert control over extracellular volume. On the other hand, a more reasonable approach for the cell to increasing its avidity would be to increase its balance between the rate of introduction of new surface receptors, Q_R, and the rate of internalization of receptors that haven't yet bonded with a ligand, k_t. This ratio is defined as R_T, the steady state receptor abundance, and other studies have found the LDLR and TfR systems to have receptor levels that can readily be controlled by growth factor signals. The VtgR system was found to be consumption sensitive which was quantified through variation of the partition coefficient, $\beta = k_e/k_{off}$, by increasing k_e. Findings from Opresko et. al. support this method of variation, finding that hormones were able to significantly impact the internalization rate of vitellogenin in frog oocytes. Now, we examine the EGFR system which displayed a dual sensitivity to both avidity and consumption, and will consequently experience the greatest gain in efficiency if both avidity and consumption are simultaneously increased. The original authors present that the optimal method of causing this increase would be a to both increase the rate of internalization of free receptors, k_e, while simultaneously decreasing the the rate of internalization of free receptors, k_t. Research supports this decoupled control over the two internalization rates for the EGFR system. However, we believe that another likely option would be for the receptor system to develop a stronger bonding mechanism with the relatively small signaling ligands which would reduce k_off, one of the factors in the denominator of both $\beta$ and $\gamma$. Perhaps the biological pressure of increased signaling efficiency will eventually lead to this adaption in the EGFR system. At this point, we see that this mathematical modeling study has allowed us to classify and distinguish between the selected receptor systems as well as providing biologically relevant results that agree with other work in the field of receptor mediated endocytosis.

Recreation Discrepancies

My partner and I selected to recreate the differential model for receptor mediated endocytosis as used by Shankaran et. al. to study the sensitivity and categorize four different receptor systems. We have been able to entirely recreate results presented in the original paper and have found no final differences between our figures and those presented in the paper. Our only difficulties in recreating the original work stemmed from two typos we encountered in the Materials and Methods section of the paper. The first was an error in defining a pair of intermediate terms used for the log log scaling in the contour plots. This mistake was mathematically obvious and we were able to notice and correct it easily. Unfortunately, the second typo proved more difficult to recognize. This error led to a 180 degree rotation of the contour plot presented as Figure 5b. After carefully checking our work multiple times, we contacted to the original authors for insight as to what our mistake might be. Thankfully, the authors responded quickly with an answer to our problem. A second typo had been made, again with a pair of intermediate terms used in the creation of the contour plots. After making this simple correction in our Mathematica code, we attained a correct plot for Figure 5b and were subsequently able to create the robustness contour plot. Once again, we are happy with our completely accurate recreation and the timely help provided to us by the original authors.

Limitations

While the receptor mediated endocytosis model studied by Shankaran et. al. has clearly produced meaningful results that are supported by other biological evidence, it is subject to simplifications that may somewhat limit its application. One such limitation is the model's lack of account for receptor recycling after internalization, a process that is know to occur in both LDLR and TfR. Instead, the model merely pushes all recycling effects into a single constant creation rate of new free receptors on the cell surface, an approximation that has not be investigated for appropriateness. Stemming from the disregard for receptor recycling, the model also ignores all processes post internalization such as receptor ligand dissociation, receptor and ligand degradation, and in some cases ligand recycling. As with chemical reactions, its possible that one of these steps could be rate-limiting for the entire endocytosis process and consequently lead to qualitatively different results. Another possible rate limiting process that was not addressed was the formation of clatherin coated pits in the cell wall that eventually close off to complete the internalization of the receptor-ligand complexes. Possible limitation could stem from either a shortage of clatherin or and over-reduction of the cell wall available for internalization. Lund et. al. displays support for such possible rate limitations with their Satin plots which show a saturation of internalization rate with increased ligand concentrations. However, while these possible limitations are clearly relevant to the modeling or receptor mediated endocytosis, they do not necessarily retract from the significance of the results obtained by Shankaran et. al. We remember that the original paper emphasizes low ligand concentrations and likely avoids running into such limitation by doing so. Nonetheless, it is likely to be an important development to this field of study when a model is developed that incorporates the total kinetics of the process of receptor mediated endocytosis.

Comparison to Other Models

After recognizing some of the limitations experienced by the model that we had selected to reproduce, we searched for similar results in models that had incorporated some of the disregarded kinetics in our model. In one paper we found, Radhakrishnan et. al. examined the impact of varying the ligand affinity of receptor systems, k_off. Their model includes elements of recycling and both receptor and ligand degradation absent from our model. We remember the mathematical definitions of specific avidity and the partition coefficient from our model possessed k_off terms as factors of their respective denominators. Therefore, as we suggested for the EGFR system, a reduction in k_off, symbolizing an increase in bonding affinity, would increase both parameters and hence would be expected to increase efficiency of the system. Consequently, results from Radhakrishnan et. al. show reduced k_off correlating to higher initial receptor system activity and greater quantities of internalized ligands which is consistent with our expectations. A similar model used by Krippendorff et. al. that also incorporates post internalization processes looked at the impact of increasing the complex internalization rate, k_e. K_e is the numerator of the partition coefficient in our model, so we would expect an increase to result in at least some degree of increase in efficiency depending on how robust the given system is from the start. Once again, true to our expectation, their results showed increased efficiency with increased k_e. Then Krippendorff et. al. looked at the same changes in k_e but reduced both k_on and k_off by factors of 100. Again remembering k_off is a term in the denominators of both avidity and partition coefficient, we would expect this change to make the system much more robust to changes in k_e, which is in fact what their results show. Having identified these similar results in other papers that examined more complete models for receptor mediated endocytosis, we have come to believe that the assumptions made in our model have not had significant impact on its qualitative results.