Top 10 Reasons to Understand Macromolecular Dimensions**

1. Top 10 Reasons to Understand Macromolecular Dimensions** 10. Proteins—the darn things change size before acting on a substrate. 9. Fast way to follow polymerization of monomers, aggregation of polymers, binding of proteins or small molecules to vesicles, etc. 8. $1,000,000,000/year rests on getting size vs. mass relations to prove branching of polyolefins. 7. Multi-viscosity motor oils use polymers that form structures whose size depends on temperature. 6. Contractile polymers—simulating muscle. 5. How do I buy the best GPC column? 4. Assessment of early stages of crystal formation for protein crystallography. 3. Do dendrimers change size when they bind metals? 2. Is it Alzheimer’s yet? And the number one reason to measure and understand polymer dimensions…. 1. Will that condom stop HIV virus?

2. Skipping Many Pages of Written Lecture Now In F2003 basic info on scaling and freely jointed was done through the device of a quiz that asked students to compute the # of configurations and estimate the size through Stockmayer’s mnemonic about 106 PS = 1000A. We also know fractal dimension and that it’s ½ for FJ, that it can differ from this for real chains. The remaining slides in this lecture are to introduce long and short range nonideality.

3. The yardstick from the freely jointed chain model Characteristic ratio chemical term short-range structure Expansion factor Physics long-range nonideality

4. Characteristic Ratio Define: Cn = characteristic ratio. It allows for expansion of the chain (compared to FJ) due to chemical, short-term effects such as a fixed bond angle or “crashes”. It does not actually depend strongly on n. In the high-n (large chain) limit, it’s called C and it is just a constant. Actual values: C= 6.7 for polyethylene C= 10 for polystyrene It should make sense that C is bigger for PS than for PE: PS has those bulky sidechains that will require a preponderance of extended trans configurations.

5. q Why we need C • Three things missed in the FJ model: • Bond angle is fixed (here, the bond angle supplement is shown). • The monomer units have a finite size. • Correlations between bond choices can lead to crashes. Slightly more realistic with bond angles FJ model

6. Ouch! Why we need a2 It is hard to predict from chemical bond considerations, but eventually the chain can come back and crash into itself. Since two beads (monomer units) cannot occupy the same space (as in the freely jointed chain of thin bonds) this leads to expansion. This expansion effect gets worse as the chain gets longer. Gets better or worse if polymer is restricted to two dimensions? This expansion effect can be opposed by attractions between chain segments (in a bad solvent).

7. Theories for the short-range, chemical effect, C

8. FR model Holy cow! Now all those off-diagonal terms don’t fade away anymore. In computing the term First you project l3 onto l2 and then l2 onto l1. This introduces cos(q)…twice! The rest is just math and tricks…see if you can do it by following the notes on the website.

9. The answer is…. <r2> = where: The lead term is most important (it contains the large number, n, so…

10. So….. Not even close to experimental values (from light scattering measurements of M and Rg) but nevertheless this freely rotating model is a useful pedagogical tool. The full equation (atop previous slide) is used to get a theory for weakly bending free rotators—e.g., semiflexible rods. Also, we learned in this about the importance of projection! One bond onto another onto another. What if we could do this better, with proper statistics? That is what Flory did.

11. f Cl Cl Cl H Cl H H ClL H H H H H H H H Cl H Hindered rotation Energy -180 -120 -60 0 60 120 180 f Assuming a realistic potential. Bottom line: a nice example of a statistical mechanical average of some quantity, the cosine of f, but….this does not raise C by nearly enough.

12. Pentane Effect

13. What’s missing? Correlation! When bond configurations repeat, it leads to a crash. We need a properly weighted average of the bond configurations, along with the resultant geometrical effects. In other words, weighted projections. This can only be done for a limited subset of the f angles.

14. Rotational Isomeric State Model Energy -180 -120 -60 0 60 120 180

15. The RIS model is less accurate for simple things like <cosf> but the simplicity lets us perform the projections….and actually enumerate all those zillions of conformations. That is, we can tame that W~3n problem. For this, and other amazing achievements, one gets a Nobel prize.

16. To see how it works, consider some simple average… Note that a statistical weight (e.g., W120 is the probability of that angle) is always multiplying into the thing whose average we want, cos(120), and these terms are added. Then you have to sum, and “normalize” by the sum of weights. A kind of math operation that multiplies as it adds is matrix algebra.

17. Matrix Multiplication If you need to sum together all these terms, other matrix multiplications can do that.

18. i t g+ g_ i-1 t g_ g+ The computational miracle Each bond gets a matrix, whose elements represent the probabilities that it and a nearest neighbor are in a set of states. This matrix is ui. Each element represents the probabilitiy that bond i is in a given state and that the preceding bond is in a given state.

19. This weight matrix will prevent crashes. In trans, out trans =1 In g-, out trans = 1 In g+ , out trans = 1 In trans, out g+=s In g-, out g+ = s In g+, out g+ = 0

20. Miracle, continued The probability of the whole chain – its total weighted number of states—is related to: u1 x u2 x u3 x u4…. This thing, a partition function in statistical thermodynamics, functions like the denominator when we were computing the average age of students: you divide it into the numerator to get whatever average you need.

21. So what? Suppose we have n = 1024 bonds, producing W~31024 conformations. It just multiplies right up, thanks to efficient exponent laws! If you can do that, you can also do weighted bond projections and get <r2>. Instead of 3 x 3 matrix multiplication, it becomes 15 x 15, but this is still easy and fast. Bottom line: RIS predicts C accurately.

22. Long range effects. Switch back to notes on web.

23. The probability of all the chains is not necessarily normalized