0README

driven-protein-motions
|-- 0NOTE                        obvious
|-- 0README                      obvious
|-- exam1.data                   data: driven motion chi-2 of BPTI Tyr-35
|-- exam1.data.ps                plot of data
|-- fit.s                        R commands for analysis of data
|-- fit-output.s                 same, but to generate ps and txt files
|-- fit-output.ps                plots generated by analysis
`-- fit-output.txt               text generated by analysis



The data in "exam1.data" and shown in the PostScript file "exam1.data.ps"
are from a molecular dynamics simulation of bovine pancreatic trypsin
inhibitor (BPTI).  There are 1024 data points, taken at 2 fs
intervals, for the dihedral angle chi-2 of tyrosine residue 35 of
BPTI.  Details of the simulation are summarized in the file "0-NOTE".

The angle chi-2 describes rotation about the C-beta--C-gamma bond,
which links the tyrosine ring to the beta-carbon of tbe tyrosine
residue.  It describes the rotation of the tyrosine ring relative
to the C-alpha--C-beta bond and in effect also to the protein main
chain.

To generate the data, a sinusoidal torque was applied at chi-2, to
produce oscillation of the tyrosine ring around the axis of the
C-beta--C-gamma bond of Tyr-35.  The data (listed in "exam1.data"
and plotted in "exam1.data.ps") are values in degrees of the dihedral
angle chi-2, given as a function of the time in femtoseconds.
Because the variation in chi-2 in response to the sinusoidal driving
force is conditioned by the structure and potential function for
BPTI, the response can be expected to lag the sinusoidal driving
force.  This lag is seen as a phase shift of the response relative
to the driving force.  We may model the time dependence of the
dihedral angle chi-2 as

    chi-2   =   A ~ sin (2 pi nu t ~+~ B)

where A is the amplitude in degrees of the driven motion of chi-2
, B is the phase, t is the time in femtoseconds, and nu is the
frequency of the driving force in fs^-1.  For these data, nu = 1/128
fs^-1; there are 16 cycles of forced rotation of the ring over 2.048
ps of the experiment.

In order to fit the above model to the data, we use the R language
and system (freely available as part of the GNU project).  We assume
that the values of the dependent variable, chi-2 , have equal weights.
The commands are in the R-language script "fit.s".

	the data are read from the file "exam1.data";

	they are fit to the model by a nonlinear least squares
	algorithm;

	a summary of the fit is written to the text window (or to
	"fit-output.txt");

	an X11 window (or a PostScript file "fit-output.ps") is
	opened for display of the three plots:

		data and fitted values; 
		residuals; 
		a quantile-quantile plot for analysis of the residuals.


The Fourier transform approximates a real function by a set of
sin/cosine terms (or exponentials) of various frequencies.  If the
function is a pure sine function, then there is only one component
in the Fourier transform, corresponding to the frequency of the
pure sine function.  The amplitude and the phase are obtained
immediately from this component.  One can do this easily with the
R functions "fft()", "Mod()", and "Arg()".


RESULTS:

fit-output.txt:

	Formula: chi ~ A * sin((2 * pi)/128 * t + B)

	Parameters:
	   Estimate Std. Error t value Pr(>|t|)    
	A 34.370745   0.240202  143.09   <2e-16 ***
	B -0.196677   0.006989  -28.14   <2e-16 ***
	---
	Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 

	Residual standard error: 5.435 on 1022 degrees of freedom

	Correlation of Parameter Estimates:
		   A
	B -4.682e-09


	parameters from fft of signal
	A =  34.37076  ;   B =  -0.0985027 



good fit, good model:

	std error A and B << values A and B ( |t| >> 1);

	residuals normally distributed (quantile plot, plot 3 of
	file "fit-output.ps") and no periodicity correlated with
	that of the sinusoidal signal (compare plots 1 and 2 of
	file "fit-output.ps");

	values of A and B reasonable - cf plot of data with fitted
	values;

	parameters A and B uncorrelated;

	change in starting estimates have no effect (except phase B
	undefined by a multiple of 2*pi) [data not recorded here,
	but easy to generate in R by varying the "start" parameter
	of the "nls" fitting function].


amplitude, A: 
	34 degrees is a hefty maximum displacement, ca 4x rms fluctuation.
	The amplitude of the driving force is greater, 40 degrees. 
	The supression of the amplitude, like the phase lag, reflects the
	viscosity of the protein matrix and the (high) frequency of the
	driven oscillation.

phase, B:
	phase near zero implies torque large enough to force ring to
	follow closely the driving force, i.e., |torque| >= |dV/dchi|;

	the negative value of the phase of the response means it lags
	the driving force, as is reasonable for motion of the ring in a
	viscous medium.


John Rupley
 rupley@u.arizona.edu -or- jar@rupley.com
 30 Calle Belleza, Tucson AZ 85716 - (520) 325-4533; fax - (520) 325-4991
 Dept. Biochemistry & Molecular Biophysics, Univ. Arizona, Tucson AZ 85721