Falla por variación brusca de la rigidez a lo largo de la altura del edificio.

Small angle X-ray and neutron methods exam ine the scattered waves produced from the diffraction o f an incident beam by protein m olecules in solution. It is a

diffraction technique that can be used to study the overall structure o f proteins.

2.3.1. X-rav scattering theory

A n X-ray scattering experim ent is perform ed by irradiating a sample w ith a

highly collim ated beam o f monochromatic X-rays, and measuring the intensity o f

scattering / as a function o f Q, w here Q corresponds to the scattering angle 20, from

which the scattering curve I{Q) is obtained (Figure 2.1). Scattering is due to interaction

o f X-rays w ith electrons in the sample: upon irradiation each electron oscillates and

emits electrom agnetic waves o f the same w avelength X in all directions, but phase-

shifted by 71 w ith respect to the incident X-ray beam. This type o f scattering is known

as coherent scattering. The intensity o f scattering by an electron is proportional to the

X-ray scattering length is the atomic num ber (the num ber o f electrons it contains)

m ultiplied by the scattering length o f an electron. Accordingly, the hydrogen isotopes,

’H and ^H, both have the scattering length o f a single electron o f 2.81 fm, while the

atoms '^C, and which are also relevant to proteins, have/ values o f 16.9 fm,

19.7 fm, 22.5 fm and 45.0 fm respectively.

2.3.1.1. The Debve equation

The form o f the scattering curve I{Q) is described by the Debye equation. An

incident planar wave o f wavelength X is scattered by a m acrom olecule in the form o f a

spherical wave (Figure 2.2a). The intensity o f the radiation is m easured w ith a planar

detector as a function o f the scattered angle, 20 relative to the direction o f the incident

wave. D iffraction phenom ena arise from interference betw een scattered waves and

consequently the scattering curve 7 ( 0 is determined by the spatial arrangem ents o f

electrons in the protein. X-ray scattering from two points in a protein is depicted in

Figure 2.2b. The incident X-ray beam is defined by the unit vector Sq and is scattered

from an origin point O in a direction denoted by the unit vector s. Since scattering is

elastic, Sq and s have the same amplitude, and for convenience this is set as InlX . Q is

the scattering vector (j - s ^ , and its amplitude is 4?! sin Q/X (Figure 2.2b). W hen the

scattering angle is zero, waves scattered from all points in the protein will be in phase

and the intensity o f scattering is the sum o f all scatterers. W hen the scattering angle is

non-zero, interference is produced by phase differences betw een scattered waves, and this can be considered using the path difference betw een scattering points. In Figure

2.2b, the incident X-ray beam is scattered by a second point P, and the path difference

betw een waves scattered by points O and P is A O + OB. This path difference

corresponds to a phase difference o f 2 n (A 0 + OB)/X. If the vector betw een O and P is

r, then A O = -vsq and OB = rsj and the phase difference is r{Sj - or more simply rQ.

The phase difference rQ betw een each individual scatterer in a macromolecule

determ ines its scattering curve 7 ( 0 , and this relationship is contained w ithin the Debye

o

O) o

7

Guinier region: gives molecular weights and radii of gyration

6

5 4

3 _{W ide angle region: gives more shape}

/ Information

2

1

0

0.2

0.6

0.8

Angle from central beam [Ü (n m “^)]

Figure 2.1. G eneral features o f a solution scattering curve I{Q) measured over a Q

range. The scattering curve is analysed in two regions, that at low Q giving the Guinier

plot from w hich the overall radius o f gyration Rq and the forward scattering intensity

Kfi) values are calculated, and that at larger Q from which m ore structural information

is obtained. A t low Q, the scattering curve is truncated for reason o f the beamstop. (Adapted from Perkins, 1994).

a)

O) O)

In l(Q )

In cident p lan e w a v e S cattered spherical w a v e D ete cto r

b) S cattered radiation Incident radiation ;-0

F igure 2.2. a) Diffraction o f electromagnetic radiation. The incident planar wave, denoted as T(z) =e'''^, where k = InlX, X is the wavelength and r is the distance o f the scattering particle from the observer, excites the scattering particle to radiate a spherical wave:-

¥ (^ ) -

r

where b is the scattering length o f the particle. The intensity o f scattering as a function

o f scattering angle g , is measured at the detector. The plot o f In I{Q) against Q gives a scattering curve characteristic o f the macromolecule.

b) Schematic representation o f X-ray scattering from two points in a protein

molecule. Diffraction is by two points O and P separated by a distance r within a single

particle in a solution scattering experiment. A and B correspond to the perpendiculars

to the incident and scattered beams. The unit vectors Sg and 5 define the incident and

scattered radiation, and Q defines the scattering vector {s-s,^. (Adapted from Perkins,

1988).