Structural Tools, Concluded
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Structural Tools, Concluded Andy HowardBiology 555, Fall 2018 30 August 2018
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
Torsion Angles in Nucleic Acids Biology 555: Environments and Components
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
Watson Crick Base Pairing • There are circumstances where other base-pairing arrangements occur (e.g. Hoogsteen pairs) Biology 555: Environments and Components
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
Different Forms of DNA Biology 555: Environments and Components
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
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
Base Pair Parametersfor Nucleic acids Biology 555: Environments and Components
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
Transfer RNA Biology 555: Environments and Components
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
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
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
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
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) = ealnY = loge(Y) = a Biology 555: Mathematics
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
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
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
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
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
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
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
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
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
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
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
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
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”)
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
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
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
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
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
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)}∫-nnw cosw dwfor w=nt, t = w/n, dw = dw/n Biology 555: Mathematics
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
Now, the sine terms • Similar change of variables (n>0):bn = (1/) ∫-t sinnt dt= (1/n2) ∫-nnw 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)|-nn = (2/n)(-1)n+1 Biology 555: Mathematics
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
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
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
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
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
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
Is that all the math you need? • Probably not, but I hope this will help prepare you for the journeys ahead. Biology 555: Mathematics
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
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
