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Structural Tools, Concluded

This article explores the structure of nucleic acids, including their polynucleotide chains, torsion angles, and helical forms. It also discusses the mathematics involved in biochemistry, such as exponentials, logarithms, and complex numbers.

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Structural Tools, Concluded

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  1. Structural Tools, Concluded Andy HowardBiology 555, Fall 2018 30 August 2018

  2. Structure of Nucleic Acids • polynucleotide chains highly flexible • range of allowed conformations much greater than for polypeptide chains • because number of torsion angles in back-bone greater • some restrictions, e.g. sugar pucker • The  angle around the glycosidic bond is restricted to be between -180º and -90º for the anti comformation and -90º and 180º for the syn conformation Biology 555: Environments and Components

  3. Torsion Angles in Nucleic Acids Biology 555: Environments and Components

  4. Helical structures composed of nucleic acids • Secondary structures DNA, RNA all helical • minimum of two strands • Watson-Crick base pairing genomic DNA • RNA base pairing intra-molecular • tRNA, rRNA complicated range of possible structures Biology 555: Environments and Components

  5. Watson Crick Base Pairing • There are circumstances where other base-pairing arrangements occur (e.g. Hoogsteen pairs) Biology 555: Environments and Components

  6. Helical Forms of DNA • B-DNA, A-DNA and Z-DNA • Discovered by X-ray diffraction • Roles in DNA: • B-DNA physiological • A is found only in dehydrated DNA • Z is found with specific sequences and can be interconverted with B even in those • A-form typical for RNA and for RNA-DNA hybrids Biology 555: Environments and Components

  7. Different Forms of DNA Biology 555: Environments and Components

  8. B-DNA • First X-ray patterns by Rosalind Franklin • 10 base pairs/turn helical rise of 0.34 nm per base pair • Nucleotides in anti conformation • sugars 2'-endo • Watson -Crick model not as rigid in solution ~10.5 pairs/turn Biology 555: Environments and Components

  9. Other helical forms may be involved in transcription • TATA box sequence similar to A-form • RNA/DNA hybrids during transcription probably A-form • A-form has 0.26 nm rise/base pair • much wider helix radius • sugar pucker 3’-endo • Lots of variants H-DNA , G-quartets etc. Biology 555: Environments and Components

  10. Base Pair Parametersfor Nucleic acids Biology 555: Environments and Components

  11. RNA-folding • unusual bases, no automatic Watson-Crick pairing - highly variable teritary structure • tRNA - flat cloverleaf structure, two loops, two stems with regions of base-pairing • ~10 bp each stem • Base pairs within and between helices stacked as much as possible • Bases will pair with other bases as much as possible Biology 555: Environments and Components

  12. Transfer RNA Biology 555: Environments and Components

  13. Ribosomal RNA • Several types & sizes of RNA make up the ribosome itself • Bacterial rRNA: large subunit has 5S and 23S rRNA; small subunit has 16S rRNA • Like tRNA, there are some nonstandard bases Three-dimensional views of the ribosome, showing rRNA in dark blue (small subunit) and dark red (large subunit). Lighter colors represent ribosomal proteins. Biology 555: Environments and Components

  14. Small RNAs • Many varieties • Most operate in the nucleus and have roles that aren’t directly related to translation • Many are at least partially base-paired • Some are catalytic, e.g. sRNAs responsible for modifying A,C,G,U into nonstandard bases Biology 555: Environments and Components

  15. Mathematics in biochemistry • Ooo: I went into biology rather than physics because I don’t like math • Too bad. You need some here:but not much. • Biggest problems in past years: • exponentials and logarithms • complex numbers • Fourier series • statistics Biology 555: Mathematics

  16. Exponentials • Many important biochemical equations are expressed in the formY = ef(x) • … which can also be writtenY = exp(f(x)) • The number e is the base of the natural logarithm system and is, very roughly, 2.718281828459045 • I.e., it’s 2.7 1828 1828 45 90 45 Biology 555: Mathematics

  17. Logarithms • First developed as computational tools because they convert multiplication problems into addition problems • They have a fundamental connection with raising a value to a power: • Y = xa logx(Y) = a • In particular, Y = exp(a) = ealnY = loge(Y) = a Biology 555: Mathematics

  18. Algebra of logarithms • logv(A) = logu(A) / logu(v) • logu(A/B) = logu(A) - logu(B) • logu(AB) = Blogu(A) • log10(A) = ln(A) / ln(10)= ln(A) / 2.30258509299= 0.4342944819 * ln(A) • ln(A) = log10(A) / log10e= log10(A) / 0.4342944819= 2.30258509299 * log10(A) Biology 555: Mathematics

  19. Complex numbers Im(r) • Someone probably introduced you to the idea of complex numbers: • r = a + ib, where i = √-1 • a = real part of r • b = imaginary part of r • We can represent this graphically r b Re(r) a Biology 555: Mathematics

  20. Alternate description Im(r) • What your grandmother may not have told you about complex numbers is that they can also be represented as a length and an angle:r = |r|ei = |r|(cos a + i sin a) • Where  represents an angleand |r| = sqrt(a2+b2) represents a length r b  Re(r) a Biology 555: Mathematics

  21. Why bother? • Pythagorean theorem says|r| = amplitude of complex number = sqrt(a2+b2) • So we can pick two parameters—either • a andb, or • |r| and — • to describe our complex number. Biology 555: Mathematics

  22. Gauss’s formula • This is a simple relationship describing a complex exponential:exp(ix) = cosx + isinx • This actually works even for complex x, but we often think in terms of real values of x. • Therefore for our angle :r = a + ib = |r|ei = |r|(cos + isin) Biology 555: Mathematics

  23. Why do we care? • You were probably first introduced to complex numbers as roots of a quadratic equation: • ax2 + bx + c = 0 has rootsx = {-b ± sqrt(b2-4ac)}/2a • For b2 < 4ac, i.e. arg(sqrt) < 0, the roots are complex, and we can write this asx = (-b/2a) ± i{sqrt(4ac-b2)}/2a Biology 555: Mathematics

  24. But this appears a lot! • Complex numbers appear in a lot of other contexts besides roots of quadratics. • Many physical phenomena can be compactly and effectively described using complex numbers. • We’ll see some examples in a few minutes. Biology 555: Mathematics

  25. A specific point • We’ve seen from Euler’s formula that exp(ix) = cosx + i sinx • Note that if x is real, then the norm, i.e., the absolute value of exp(ix), is|exp(ix)| = |cosx + i sinx|= sqrt(cos2x + sin2x) = sqrt(1) = 1. Biology 555: Mathematics

  26. What if x itself is complex? • Let’s say x = a + ib. • Then exp(ix) = exp(i(a+ib)) = exp{ia+(i2)b} • But i2 = -1, so • exp(ix) = exp(ia - b) = exp(ia)e-b • This will have a norm not equal to 1; • In fact, if b > 0 the norm will be < 1. • This comes up naturally in spectroscopy Biology 555: Mathematics

  27. Differentiation • I hope that all of you has been exposed to calculus at least once in your lives • Let’s remind you of some fundamentals • Differentiation is the process of calculating the derivative of a function • A derivative is (among other things) the slope of a function at a particular point

  28. Some basic derivatives • d/dx f(u(x)) = df/du du/dx • d/dx{xn} = nxn-1 • d/dx{cosx} = -sinx • d/dx{sinx} = cosx • d/dx{ex} = ex • d/dx{f(x)g(x)} = (df/dx)g(x) + f(x)dg/dx

  29. Indefinite Integrals • These express a relationship that is the inverse of differentiation; in fact, they’re sometimes called anti-derivatives • The result of every indefinite integral has a constant associated with it becaused/dx(C) = 0

  30. Some indefinite integrals • ∫xndx = {1/(n+1)}xn+1 + C for n ≠ -1 • ∫ cosxdx = sinx + C • ∫ sinxdx = -cosx + C • ∫ x-1dx = ln(x) + C • ∫ exdx = ex + C • ∫ u dv = uv - ∫ vdu (“integration by parts”)

  31. Definite integrals • Provide a way of determining the area under a curve from a specific starting point to a specific end point • Related to indefinite integrals in that they are calculated by determining the value of the indefinite integral’s function form at the two limits

  32. Fourier series • In its most general form, this is a discrete summation of terms that enables us to approximate any continuous function as a set of sines and cosines or complex exponentials • This is especially appropriate for periodic functions, i.e., functions that repeat with a constant periodicity Biology 555: Mathematics

  33. Specific definition: 1-dimensional Fourier series • For a continuous function f(x) we writef(x) = a0/2 + ∑n=1∞ ancosnx + bnsinnx • The terms in front of the cosines and sines are called Fourier coefficients • With many periodic functions we find that only a few of the Fourier coefficients an and bn are required to provide a very reasonable approximation to our original function. Biology 555: Mathematics

  34. Finding the coefficients • Unlike a Taylor expansion, where we have to do derivatives of f(x) to figure out what the coefficients in the expansion are, we do these by integration: • an = (1/)∫- f(t) cosnt dt, n ≥ 0 • bn = (1/)∫- f(t) sinnt dt, n > 0 Biology 555: Mathematics

  35. Example (from Wikipedia!) • Sawtooth function:f(x) = x for - ≤ x ≤ f(x) = f(x - 2) for all x. • That doesn’t look much like sines and cosines. But it’s an easy one to think about because the integrals are easy. Biology 555: Mathematics

  36. Finding the coefficients • All the cosine terms vanish: • for n=0, a0 = (1/)∫-t cos(0*t) dt, i.e.a0 = (1/) ∫-t cos(0) dt = ∫-t dt = t2/2 | -a0 = (1/) {2/2 - 2/2} = 0 • For n>0 we say • an = (1/)∫-t cosnt dt ={1/(n2)}∫-nnw cosw dwfor w=nt, t = w/n, dw = dw/n Biology 555: Mathematics

  37. Integrating by parts • In the dim recesses of your mind you may recall that ∫udv = uv - ∫vdu • Here we set u=w, dv=coswdw, so • du = dw, v = sinw, and∫w cosw dw = wsinw - ∫sinw dw= wsinw + cosw + C = nt sinnt + cosnt + C • Therefore an = (1/) ∫-t cosnt dt ={1/(n2 )}(nt sinnt + cosnt)|-= 0  n. Biology 555: Mathematics

  38. Now, the sine terms • Similar change of variables (n>0):bn = (1/) ∫-t sinnt dt= (1/n2) ∫-nnw sinw dw • As before, we integrate by parts withu = w, du = dw, dv = sinwdw, v = -cosw • So ∫w sinw dw = - wcosw - ∫ (-coswdw)= -wcosw + sin w + C • bn = {1/(n2)}(-wcosw)|-nn = (2/n)(-1)n+1 Biology 555: Mathematics

  39. Putting this together • Our sawtooth function isf(x) = 2∑{(-1)n+1 /n}sinnx • And we find that even 5 terms gives us a pretty clean approximation • We will use these functions often to approximate periodic functions of either time or distance Biology 555: Mathematics

  40. Formulation with complex exponentials • This is a bit less intuitive but easier to work with: • f(x) = ∑-∞∞cneinx • With the coefficients cn given by • cn = (1/2)∫-f(x)e-inxdx • These integrals are sometimes easier to do analytically even though they (!) involve complex numbers Biology 555: Mathematics

  41. Time and frequency • If our independent variable is time, then the Fourier domain values n have dimensions of inverse time, i.e. frequency • A lot of spectroscopy, including NMR, can be analyzed by time-domain Fourier analysis! Biology 555: Mathematics

  42. Multi-dimensional Fourier series • There’s no reason these notions can’t be extended to 3 spatial dimensions: • (x,y,z) = ∑h=-∞∞ ∑k=-∞∞ ∑l=-∞∞ Fhklei(hx+ky+lz) • This is a natural way to formulate the relationship between atomic positions and structure factors in crystallography • Definable triple integral for Fhkl • Remember: h,k,l are integers! Biology 555: Mathematics

  43. Why so natural? • Because waves look like cosines and sines • Or… waves obey the differential equation known as the wave equation, and the wave equation has solutions that look like cosines and sines (or like complex exponentials) Biology 555: Mathematics

  44. Okay. That explains it for light… • Remember that X-rays are just light at a shorter wavelength (~1Å), as compared with visible light (~5000Å) • But this even helps us recognize the relevance to electron or neutron diffraction: matter can behave in a wavy fashion too! Biology 555: Mathematics

  45. Is that all the math you need? • Probably not, but I hope this will help prepare you for the journeys ahead. Biology 555: Mathematics

  46. How do we determine structures? • Big picture:We need to perturb these molecules with some source of energy whose characteristic wavelength is comparable to the distances we’re trying to find • The energy could be coming from X-rays, neutrons, or electrons • The molecules are often arranged in some regular pattern so that we can take advantage of aggregate effects Biology 555: Environments and Components

  47. Specific tools • Scattering of X-rays, visible light, or neutrons by solutions (SAXS, DLS, SANS) • Diffraction of X-rays by 2-D ordered fibrous arrays (fiber diffraction) • Diffraction of X-rays, neutrons, or electrons by 3-D ordered crystalline arrays (crystallography) • Scattering and absorption of electrons from samples with molecules laid out on a grid (cryoEM) • Excitation of unpaired nucleons by interaction with electromagnetic radiation (NMR) Biology 555: Environments and Components

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