Mathematical Modeling Homework
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A1: Modify the lead-in-the-body model by doubling the rate of elimination of lead from the bones. Plot the amount of lead in the three compartments in this case and the amount of lead in the three compartments in the unmodified case over a 40 year time period in each of the three cases (remaining in the high-lead environment, leaving it after 400 days for a medium-lead environment, and leaving it after 400 days for a lead-free environment). How effective is this treatment?
A2: Consider equation (5.1). If the initial dose is changed to 2C0 but the regular doses remain C0, how is equation (5.5) changed? Show that equation (5.6) does not change.
A3: Verify the statements in equations (5.9)-(5.12).
B1: Solve equation (5.37a,b) using the eigenvalue-eigenvector method under the assumption that the eigenvalues are real, negative, and distinct. (You don't need to find formulas for the eigenvectors but do show how they occur in the solution.) Show that your answer corresponds to equation (5.44a,b).
B2: Modify equation (5.37a) by adding a constant V to the right-hand side. Under the assumption that the eigenvalues are real, negative, and distinct, find a particular solution of the system in equation (5.37a,b).
B3: Notice that in equation (5.37a,b) the coefficients Li,j and Mi,j are positive and M21 <L11 and M12 ≤L22 for a physiologically meaningful model. Must the eigenvalues of the system be real? Under what additional conditions, if any, will they be negative?
B4: Use equation (5.46a,b,c,d) to find the coefficients in equation (5.37a,b) if it has been determined that the eigenvalues are λ1=-2, λ2=-3, A11=1, A12=4, A21=5, and A22=6.
B5: Explain the meaning of equation (5.60a,b). What do the variables and coefficients represent and what values may they take?
B6: How would equation (5.60a,b) change if the destruction of the drug was proportional to the square of its concentration rather than to its concentration (as in equation (5.50) and equation (5.58))?
C1: Solve the PPA model numerically in the case where only one pill is taken. Plot your solution over a sufficiently long time period to reveal the long-term result. Repeat for the CPM model.
C2: Solve the PPA model numerically in the case where only two pills are taken exactly 6 hours apart. Plot your solution over a sufficiently long time period to reveal the long-term result. Repeat for the CPM model.
C3: Solve the PPA model numerically in the case of uniform dosing (in which one unit of the drug is dissolved continuously in every 6-hour period). Plot your solution over a sufficiently long time period to reveal the long-term result.
C4: Find the analytical solutions of the PPA and CPM models in the case of uniform dosing.
C5: Explain the Michaelis–Menten kinetics model.
C6: Compare the Widmark, Wagner, and Pieters models. Discuss their strengths and weaknesses.
D1: Consider the Upton anaesthesiology model. Extend the graphs in Fig. 4 to the case of cardiac output of 1 litre/min and the case of 12 litre/min. Describe the trend in these graphs.
D2: Extend the graphs in Fig. 5 to the case of injection time of 1 s and the case of 300 s. Describe the trend in these graphs.
D3: Reproduce the graphs in Fig. 5 for a dose of 10 mg and a dose of 1000 mg. Describe the trend in these graphs.
D4: Reproduce the graph in Fig. 6 for a dose of 50 mg and a dose of 200 mg. Describe the trend in these graphs.
D5: Reproduce the graph in Fig. 6 for a cardiac output of 2, 3, 4, 6, 7, and 8 litre/min. Describe the trend in these graphs.
D6: Reproduce the graph in Fig. 6 but with the variation in cardiac output modeled by a single sine wave of period 120 (that is, with only a single cycle fitting on the graph). Compare your result to that in Fig. 6.
E1: Discuss the limitations of the SRM0 model. What types of phenomena does it fail to take into account? How convenient is it to use, given the number of likely presynaptic neurons?
E2: Suggest a plausible functional form for the EPSP funtion εij(t).
E3: Sketch a graph similar to Fig. 1.3B if neuron 1 fires again when the EPSP from neuron 2 is at its maximum height.
E4: Consider the action potential shown in Fig. 1.2A (inset) and in Fig. 1.4 (dashed curve). The model of Eq. 1.5 captures the shape of the curve only very roughly. What would be a way to get a higher-fidelity approximation to it than η(t)?
E5: How could the absolute refractory period be put into the SRM0 model?
E6: What are the arguments for and against a spike-count/average-over-time rate code as the means of neuronal coding?
F1: Use Kirchoff's voltage and current laws to derive Eqs. (2.3)-(2.5).
F2: Use Eq. (2.2) to compute the reversal potentials ENa and EK.
F3: Explain what happens to the ionic currents when the potential difference across the cell membrane is less than EK, between EK and ENa, and greater than ENa. What is the significance of urest in this regard?
F4: Write the Hodgkin-Huxley model as a system of differential equations (including initial conditions).
F5: Discuss the significance of, and the models for, the activation variables m and n and the inactivation variable h.
F6: How would the Hodgkin-Huxley model change if we were to separate the leak current into three separate currents with variable conductances? Draw a circuit and give a differential equation as part of your answer.
G1: The Hodgkin-Huxley model treats a neuron as a single point and models the potential difference only as a function of time. Discuss why a compartment model might be appropriate for modeling the potential difference as a function of both space and time and why this might be valuable.
G2: Compare the Morris-Lecar equations to the Hodgkin-Huxley model.
G3: Compare the FitzHugh-Nagumo model to the Hodgkin-Huxley model.
G4: Why are the quasi-steady-state approximation for m(t) and the approximation an=b-h reasonable when reducing the Hodgkin-Huxley model to a two-dimensional model?
G5: Assuming I(t) is identically zero, what is the Jacobian matrix for the FitzHugh-Nagumo model? Could it ever be singular?
G6: Assuming I(t) is identically zero, what is the Jacobian matrix for the Morris-Lecar equations? Compare it to the Jacobian matrix for the FitzHugh-Nagumo model.
H1: Discuss the issues of redundancy and unambiguity in the standard DNA genetic code.
H2: Use GenBank to find the number of times that the amino acid glycine occurs in the first 1018 amino acids in contactin in Homo sapiens (sequence CAA79696). Write out the possible nucleotide sequence(s) for the first 6 amino acids in that sequence.
H3: Based on the contactin data, what would be the 20 probabilities pΩ in a mutinomial sequence model for this amino acid sequence, where Ω is any of the 20 possible symbols in the amino acid alphabet?
H4: Based on the contactin data, what would be the 20 probabilities pAΩ in a Markov sequence model for this amino acid sequence, where Ω is any of the 20 possible symbols in the amino acid alphabet? Is pΩA equal to pAΩ, in general?
H5: What is a reasonable alignment of the amino acid sequences MICELIKERICES and MRICELIKEMICE?
H6: Discuss the significance of the A-T and C-G frequencies in a DNA sequence.
I1: Consider the three amino acid sequences VIVALASVEGAS, VIVADAVIS, and VIVADALLAS. (See pg.53). Using the standard scoring function σ(x,y), find the three optimal pairwise global alignments and the optimal multiple global alignment of the three sequences.
I2: Find the optimal global and local alignments of VIVALASVEGAS and VIVADAVIS if we modify the original scoring function σ(x,y) so that a gap now scores -2.
I3: Find the optimal global and local alignments of VIVALASVEGAS and VIVADAVIS if we modify the original scoring function σ(x,y) so that a substitution of L to D or of A to I now scores -2.
I4: Why are bifurcated rooted trees appropriate graphs for representing phylogenetic trees? What types of phenomena are excluded from consideration when using such a representation?
I5: What factors make whole genome comparisons difficult?
I6: Discuss the issues of synonymy and polysemy in the context of Latent Semantic Analysis. What are the corresponding issues in comparing DNA or amino acid sequences?
J1: Consider the sample term-by-document matrix on pg.33. What documents are returned for a simple query consisting of terms T1 and T2? On terms T3 and T7? On terms T1, T2, T6, and T9? Are these results intuitively reasonable?
J2: Repeat Problem J1 based on the rank-reduced QR decomposition of rank 4 (see pg.59).
J3: Repeat Problem J1 based on the rank-reduced SVD decomposition of rank 4 (see pg.59).
J4: How does Latent Semantic Analysis address the problems of synonymy and polysemy in a document?
J5: Explain in detail how Latent Semantic Indexing can be used to create a phylogenetic tree.
J6: Discuss the reasonableness of the evolutionary distance formula di,j=-ln(1+cos(θ)/2).
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