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Illinois Institute of Technology

This lecture explores the concepts of survival models in radiation and molecular biology, focusing on DNA, chromosomes, and mathematical modeling. Topics include target theory, linear-quadratic models, and molecular models.

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Illinois Institute of Technology

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  1. Illinois Institute of Technology RADIATION BIOPHYSICS Fourth Lecture: Radiation and Molecular Biology;Survival Models ANDREW HOWARD Thursday 12 June 2014 Rad Bio: mol bio and survival

  2. Survival • Cells sometimes survive radiation exposure, and sometimes they don’t; we need to develop mathematical models for survival • … but first, I want to say a few things about DNA and chromosomes Rad Bio: mol bio and survival

  3. What we’ll discuss • Discussion of homework and relativity • DNA, chromosomes, and DNA repair • What we mean by survival • Models for survival • Target theory • STSH • MTSH • Molecular models • Linear-quadratic models of various sorts Rad Bio: mol bio and survival

  4. Bookkeeping • Generally I’m happy with your homework performance:most of you have turned in the first assignment • Don’t obsess over getting them in on time, but get caught up by the first midterm • Midterm (Wed-Thu 6/18 – 6/19) • Get a proctor! • Closed book, closed notes, but a help-sheet will be provided • Covers chapters 1-7 • My midterms are long but not hard:budget your time accordingly • I will proctor Wed 11am-12:15pm and Thu 1-2:15pm • Chuck Scott will proctor late Wednesday afternoon Rad Bio: mol bio and survival

  5. Typos in Alpen • Alpen seems to have replaced D with D! at least twice • Several terms have too many factorial signs in them; • Eqn. 7.3 should beP(,h,D) = (DCh)(h)(1-)(D-h)(H(h)) • Eqn. 7.4 should be S(,D) = h=0h=D P(,h,D) • Axis labels are faulty sometimes too:Pp. 136-137: the lowest number on the Y axis should be 0.01, not 0.001 Rad Bio: mol bio and survival

  6. The APS problem • Remember we said we have 7 GeV electrons,and we asked you to calculate c-v. • Keep the calculation in algebra until the very end! • So: E = m0c2 = (1-v2/c2)-1/2m0c2 • Therefore (1-v2/c2)-1/2 = E/m0c2 • Squaring both sides, (1-v2/c2)-1 = (E/m0c2)2 • Inverting, 1-v2/c2 = (m0c2/E)2, so1 - (m0c2/E)2 = v2/c2 • Hence v = c(1 - (m0c2/E)2 )1/2 • Therefore c - v = c - c(1 - (m0c2/E)2 )1/2, i.e.c - v = c[1 - (1 - (m0c2/E)2 )1/2] Rad Bio: mol bio and survival

  7. APS problem, continued • Use our preloaded knowledge that the rest energy of the electron, m0c2, is 0.511004 MeV, so m0c2/E = 0.511004 MeV/7000 MeV = 7.300*10-5 (unitless!) • Therefore (m0c2/E)2 = 5.3291 * 10-9, so1 - (m0c2/E)2 = 0.99999999467,(1 - (m0c2/E)2)1/2 = 0.99999999734,1- (1 - (m0c2/E)2)1/2 = 2.6645418 * 10-9. • Hence c-v = 2.99797*108 ms-1 * 2.6645418*10-9i.e. c - v = 0.7988 ms-1, a moderate walking pace. Rad Bio: mol bio and survival

  8. APS acceleration • This part doesn’t need to be viewed relativistically: • Acceleration a = v2/r • We’ve just convinced ourselves that v is pretty close to c, so we can ignore that tiny difference. • Circumference of the APS is 1100m,so radius = C/2 = 1100m/2 • a = c2/r = (2.99797*108 ms-1)2 / (1100m /2)= 2 * 8.171 * 1013 ms-2 = 5.11 * 1014 ms-2 = 5.239 * 1013 * g • The force exerted on the electron isF = ma = m0a = (1-v2/c2)-1/2m0a Rad Bio: mol bio and survival

  9. Relativistic kinetic energy • I’ve mentioned in passing that, according to special relativity, the formula for energy is E = m0c2 = (1-2)-1/2m0c2 = (1-v2/c2)-1/2m0c2 • Where  = (1-2)-1/2 = (1-v2/c2)-1/2m0c2 and  = v/c • Our intuition should tell us that this total energy consists of mass energy, m0c2, and something else that is related to movement, namely, kinetic energy. • The kinetic energy, then, should beKE = E - m0c2 = m0c2 - m0c2 = (-1)m0c2 Rad Bio: mol bio and survival

  10. So what? • Our nonrelativistic formula for kinetic energy, which you all learned in the sixth week of freshman physics, isKE = (1/2)mv2, or using the notation we’re using here, KE = (1/2)m0v2. • Does our relativistic formula reduce to that in the low-speed limit, i.e. when v << c? • Let’s convince ourselves of that, both because it’s interesting and to give us some practice using Taylor (or Maclaurin) expansions Rad Bio: mol bio and survival

  11. Maclaurin expansion • Remember that the Taylor expansion of a continuous function f(x) about a point x0 isf(x) = f(x0) + [(df/dx)|x0](x-x0) + (1/2)[d2f/dx2|x0](x-x0)2 + (1/6)[d3f/dx3|x0](x-x0)3 + higher-order terms, • Where z|y means “z evaluated at x=y”. • Or, more precisely,f(x) = f(x0) + ∑k=1∞(1/k!)[dkf/dxk|x0](x-x0)k • In the specific case x0 = 0, it’s a Maclaurin expansionf(x) = f(0) + ∑k=1∞(1/k!)[dkf/dxk|0]xk Rad Bio: mol bio and survival

  12. Using this for kinetic energy • To apply this to the kinetic energy problem, we define a variable u = v2/c2 = 2, so that • KE = (-1)m0c2 = [(1-u)-1/2 - 1]m0c2 • And we will expand that in u around u = 0: • KE = KE(0) + d/du { [(1-u)-1/2 - 1]m0c2}|0}*u + higher-order terms. • But KE(0) = 0, clearly, since if u=0,  = 1. • So KE = d/du { [(1-u)-1/2 - 1]m0c2|0}*u + higher-order terms. Rad Bio: mol bio and survival

  13. So does this work? • d/du[(1-u)-1/2] = (-1/2)(1-u)-3/2(-1) = (1/2)(1-u)-3/2 • Evaluating that at u=0, • d/du[(1-u)-1/2]|0 = (1/2)(1) = 1/2. Therefore KE = d/du { [(1-u)-1/2 - 1]m0c2}|0}*u + h.o.t.= (1/2)m0c2 * u = (1/2) m0c2 * (v2/c2) = (1/2) m0v2 • … which is what we were hoping for. • You can do this directly as an expansion in v, but it’s a little bit harder. Rad Bio: mol bio and survival

  14. APS problem reconsidered • Remember that we wound up with a number very close to zero for (m0c2/E)2, and we were at the mercy of our calculator’s ability to keep a whole lot of zeros. • Is there a way to do this that doesn’t depend on 10-digit calculators? Yes: • For small x, (1-x)n = 1 - nx • This is a binomial expansion, (W.S. Gilbert:About binomial theorem I’m teeming with a lot of news) • … which is just a special case of a Taylor series. Rad Bio: mol bio and survival

  15. How do we use this? • Well, remember that our equation isc - v = c[1 - (1 - (m0c2/E)2 )1/2] • But we’re saying x = (m0c2/E)2 << 1, so c - v = c[1 - (1 - x)1/2] = c[1 - (1- x/2)] = cx/2 • Remember that x = 5.3291 * 10-9 • Therefore c - v = 2.99797 * 108 ms-1 * 5.3291 * 10-9 / 2 • Thus c - v = 0.7988 ms-1 • That’s identical to what we got with all those 9’s on our calculator (0.7988 ms-1) — differences are in the next decimal place or two Rad Bio: mol bio and survival

  16. Force on the electron • We said F = ma = m0a = (1-v2/c2)-1/2m0a • But (1-v2/c2)-1/2 = E / m0c2= 7000 MeV/0.511004 MeV = 13698, so • F = (1-v2/c2)-1/2m0a= 13698 * 9.11*10-31 kg * 5.11 * 1014 ms-2= 637695 * 10-17 N = 6.38 * 10-12 N • Comparable to the forces measured in some atomic force microscopy experiments • That’s a small force, but it’s acting on a very small object! Rad Bio: mol bio and survival

  17. Some definitions from biochemistry • A catalyst is a species that increases the rate of a reaction without ultimately being changed in the net reaction • An enzyme is a biological macromolecule capable of catalysis • Most enzymes are proteins • In the last two decades we have begun to study RNA molecules that function as enzymes(may do damage surveillance in RNA?) Rad Bio: mol bio and survival

  18. Catalytic vs. noncatalytic mechanisms • Uncatalyzed reaction: A  B (slow) • Catalyzed reaction, with E as the catalyst:A + E A-E B-E B + EAll three of these reactions are likely to be much faster than the uncatalyzed reaction (~ 107 times faster) Hermann Emil Fischer Rad Bio: mol bio and survival

  19. DNA nucleotides W. H. Brown & J. A. McClarin, Introduction to Organic and Biochemistry, 3rd Ed., 1981 Rad Bio: mol bio and survival

  20. Protein Backbones and Nucleic Acid Backbones R1, R2 = one of 20 amino acid sidechains(H, CH3, CH2OH, … ) Image courtesy McGraw-Hill Education.The circled “P” indicates a phosphate linkage, i.e.. Rad Bio: mol bio and survival

  21. Realities of DNA damage • Covalent damage to one strand doesn’t always result in failure to replicate correctly • … but it increases the rate of copying errors • Base-pairing can be destroyed by covalent damage • A-T pairs(2 hydrogen bonds per base pair)are more fragile than C-G pairs(3 H-bonds / pair) H.J. Muller:characterized DNA damage from X-radiation Rad Bio: mol bio and survival

  22. Reactions of Radiation with DNA • Single-strand breaks and double-strand breaks SSB High-reliability enzymatic repair DSB caused by a single event Lower-reliability enzymatic repair Rad Bio: mol bio and survival

  23. DSBs from two events • If the breaks are not at the same position, they can be more readily repaired:5’-C-G-A-T-C-C-G-A-3’ 5’-C-G-A--C-C-G-A-3’3-’G-C-T-A-G-G-C-T-5’ 3’-G-C-T-A--G-C-T-5’ DSB caused by two neighboring events Moderate-reliability enzymatic repair Rad Bio: mol bio and survival

  24. Chemistry of DNA damage • Damage to sugars and bases(not removed but covalently damaged) • Loss of base(apurination or apyrimidation) • Strand scission due to radical chemistry on a nucleotide • Single-strand breaks on the backbone • Double-strand breaks (see above) on the backbone Rad Bio: mol bio and survival

  25. Chromatin • This refers to DNA in a cell. • In between cell divisions, DNA is spread out in the nucleus. • At a particular stage in the cell cycle, the DNA becomes highly coiled and organized in preparation for replication. Thomas Hunt Morgan, pioneer in the understanding of the role of chromosomes in heredity Rad Bio: mol bio and survival

  26. How does chromatin become organized? • At the lowest level of organization, ~200 base-pairs of DNA wrap themselves around a group of nitrogen-rich proteins called histones which have been organized into the nucleosome core particle • That interaction is stabilized by charge-charge interactions between the negatively-charged phosphate groups in the DNA and positively charged amino acids in the histone Rad Bio: mol bio and survival

  27. Higher levels of organization • Neighboring nucleosomes group together to form even higher levels of coiling through an interaction with another histone, H1.This forms a solenoid-like structure. Rad Bio: mol bio and survival

  28. Why does this matter? • DNA tends to be more radiation-sensitive when it is more organized • More tightly packed—harder for the repair enzymes to get access to the lesions • Closer to the time of replication • So we need to be conscious of these levels of organization Rad Bio: mol bio and survival

  29. DNA repair • All organisms have some DNA repair mechanisms. • Eukaryotes have more and nimbler repair systems than prokaryotes • DNA repair enzymes can recognize and repair • Single-strand breaks (SSBs) • Double-strand breaks (DSBs) • Chemically altered bases • Chemically altered sugars • Damage to DNA-related proteins (e.g. histones) Rad Bio: mol bio and survival

  30. DNA repair, continued • Some mechanisms are more error-prone than others • Certain kinds of damage are effectively irreparable • Repair in eukaryotes is much more effective than in prokaryotes Rad Bio: mol bio and survival

  31. Excision repair • The least error-prone type of repair • It relies on the complementary strand’s base to define which base (dA, dC, dG, or dT) to insert • Characteristic of repair of certain kinds of damage, like pyrimidine dimers; but it can come into use whenever the damage involves a single base on one strand Missing Base Template Strand Rad Bio: mol bio and survival

  32. Other forms of repair • “Error-prone repair”: recA and similar mechanisms • Underlying notion: any base is better than no base. • This is particularly true if the replaced base happens to be the wobble (3rd) base: changes at the 3rd base often are harmless, or nearly so • Recombination repair: see figure 6.5. Recombination repair: image from answers.com Rad Bio: mol bio and survival

  33. How do we define survival? • It’s harder at the cellular level than you might think. • It takes a lot of radiation to destroy metabolism. • It takes a lot less to compromise DNA replicationbadly enough to either: • Prevent replication after 1-4 generations • Produce large changes in morphology or function, again after 1-4 generations • Therefore: we concentrate on clonogenic survival as a definition for cell survival Rad Bio: mol bio and survival

  34. What kind of experiments do we envision here? • A few cells placed on a growth medium • Cells are exposed to a toxicant or to radiation • Cells allowed to divide for several generations • Compare number of progeny in the treated cell group to number in untreated group • Damage is said to be significant if the treated group produces fewer progeny than the control group Rad Bio: mol bio and survival

  35. What do we mean by clonogenic survival? • Clean definition: clonogenic survival is the ability to produce six generations of viable offspring • This works well for prokaryotic cells and cultured eukaryarotic cells, particularly immortalized ones Rad Bio: mol bio and survival

  36. Clonogenic survival in differentiated eukaryotic cells • The definition works less well for differentiated eukaryotic cells: • A respectable eukaryotic cell has a chromosomal component called a telomere that regulates the number of cell divisions before the cell undergoes programmed cell death (apoptosis) • If the cell you’re studying is close to its natural cutoff point for cell divisions, it’s unfair to blame the treatment for its inability to produce 5 generations of progeny! Rad Bio: mol bio and survival

  37. Six generations… • Roughly corresponds to 50 surviving progeny 6 5 0 4 1 3 2 Rad Bio: mol bio and survival

  38. Contact inhibition • Many cells change behavior when they come into contact with neighbors • Often the change involves inhibition of replication • That complicates the definition of clonogenic survival: • If the cells stop dividing because they’re getting too crowded, it’s unfair to blame that on the treatment! Changes Rad Bio: mol bio and survival

  39. What’s an immortalized cell line? • Certain transformed cell lines lose their responsiveness to cell-cell communication and to the apoptotic count • These cells can replicate without limit • Often this kind of transformation is associated with cancer • It’s always questionable whether experiments on transformed cell lines are telling us anything useful about the behavior of untransformed cells • But we’re somewhat stuck with this kind of system Rad Bio: mol bio and survival

  40. Mechanisms of Reproductive Cell Survival and Death • Up until around 1970 there were two highly disparate lines of research surrounding these issues: • Modelers, who carried out mathematical studies of dose-response; • Biologists, who sought understanding of the mechanisms of the cellular response • Enzymatic • Molecular-biological • Since 1970 there has been better communication between these two communities ln(Survival fraction) Dose Rad Bio: mol bio and survival

  41. Sorting out multiple causes • … can be tricky. • Ancient study of uranium mine workers:Status Smoking Non-smokingMiner 1 2Non-miner 3 4 • Result:cancer(1) > cancer(3) >> cancer(2) ~ cancer(4) • So the effect of mining is potentiated by smoking • We’d like to know why! Rad Bio: mol bio and survival

  42. Lea’s model for cellular damage • Four basic propositions (1955): • Clonogenic killing is multi-step • Absorption of energy in some critical volume is step 1 • Deposition of energy as ionization or excitation in the critical volume will give rise to molecular damage • This molecular damage will prevent normal DNA replication and cell division • More details about his assumptions on next slide. • Alpen argues that this predates Watson & Crick.That’s not really true, but it probably began independent of Watson & Crick Rad Bio: mol bio and survival

  43. Lea’s assumptions • There exists a specific target for the action of radiation • There may be more than one target in the cell, and the inactivation of n of these targets will lead to loss of clonogenic survival • Deposition of energy is discrete and random in time & space • Inactivation of multiple targets does not involve any conditional probabilities, i.e., P(2nd hit) is unrelated to P(1st hit) Rad Bio: mol bio and survival

  44. The role of DSBs • We will eventually want to emphasize unrepairable DNA damage as the true bad actor in all of this • We saw at the end of last class that double strand breaks are harder to repair with high fidelity • So DSBs are likely to be the real issue here • You can begin to see the utility of an interaction between the modelers and the biologists! Rad Bio: mol bio and survival

  45. Log-linear response • With cells that are distinctly deficient in DSB repair (e.g., bacterial cells): • Log-linear dose-response to radiation over several logsln(N/N0) = -D/D0 • N0 is the number of cells in the absence of treatment 0 ln(Survival fraction) -12 Dose, Gy 0 40 Rad Bio: mol bio and survival

  46. The cellular damage model • Cell has volume V; target volume is v << V • Mechanistically we view v as the volume surrounding the DNA molecule such that absorption of energy within v will cause DNA damage. Cell, volume V Nucleus Sensitive volume v 5 µm Rad Bio: mol bio and survival

  47. Single-target, single-hit model • In this instance, each hit within the volume v is sufficient to incapacitate the cell • Define S(D) as the survival fraction upon suffering the dose D. Define S0 = survival fraction with no dose. • Note that S0 may not actually be 1:some cells may lack clonogenic capacity even in the absence of insult • Then: S/S0 = exp(-D/D0) • D0 = dose required to reduce survival by 1/e. Rad Bio: mol bio and survival

  48. STSH model: graphical behavior • Slope of curve = -1/D0 • Y intercept = 0(corresponds to S/S0 = 1) 0 -1 Slope = -1/D0 ln(S/S0) Dose, Gy D0 Rad Bio: mol bio and survival

  49. When is STSH insufficient? • Any situation in which more than one hit might be necessary, or in which the repair capacity of the cell includes the ability to eliminate certain kinds of damage, may result in a plot of ln(S/S0) vs. D that isn’t linear. • We seek plausible mechanistic mathematical models that can account for this behavior • In practice the Multitarget Single-hit (MTSH) and Linear-Quadratic (LQ) models have been used most Rad Bio: mol bio and survival

  50. Multi-target, single-hit model • Posits that n separate targets must be hit • Probabilistic algebra given in Alpen • Outcome: S/S0 = 1 - (1 - exp(-qD))n,or for D0=1/q,S/S0 = 1 - (1 - exp(-D/D0))n Rad Bio: mol bio and survival

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