def fit_std_curve(i_vals, i_sds, dpx_concs):
# Plot the intensity ratios as a function of [DPX]:
plt.figure()
plt.errorbar(dpx_concs, i_vals, yerr=i_sds, linestyle='', marker='o',
color='k', linewidth=2)
plt.xlabel('[DPX] (M)')
plt.ylabel('$I / I_0$')
plt.title('Quenching vs. [DPX]')
# Fit the curve using scipy least squares
kd = fitting.Parameter(0.05)
ka = fitting.Parameter(0.490)
def eq2(dpx):
return 1. / ((1 + kd()*dpx) * (1 + ka()*dpx))
fitting.fit(eq2, [kd, ka], i_vals, dpx_concs)
# Plot curve fit
dpx_interp = np.linspace(dpx_concs[0], dpx_concs[-1], 100)
plt.plot(dpx_interp, eq2(dpx_interp), color='r', linewidth=2)
plt.xlabel('[DPX] (mM)')
plt.ylabel('$I / I_0$')
plt.title('DPX quenching, standard curve')
print "kd: %f" % kd()
print "ka: %f" % ka()
return (ka(), kd())
def fit_stoichiometric_comp_ols(bid_concs, fret_means, fret_ses):
"""Obtained the equation for analyzing competitive binding assays from
Motulsky's article published here:
http://www2.uah.es/farmamol/Public/GraphPad/radiolig.htm
"""
f = fitting.Parameter(0.40) # FRET efficiency
ic50 = fitting.Parameter(6.) # kd for binding lipo sites
nonspec = fitting.Parameter(0.1) # baseline quenching
total = fitting.Parameter(0.80)
def fit_func(bid_conc):
frac_bound = nonspec() + ((total() - nonspec()) /
(1 + 10 ** (np.log10(bid_conc) - np.log10(ic50()))))
return f() * frac_bound
fitting.fit(fit_func, [f, ic50, nonspec, total], fret_means, bid_concs)
plt.figure('Bid FRET, competitive')
plt.errorbar(np.log10(bid_concs), fret_means, yerr=fret_ses, linestyle='',
color='r')
bid_pred = np.logspace(-1, 3, 100)
plt.plot(np.log10(bid_pred), fit_func(bid_pred), color='k')
plt.xlabel('log10([Total Bid])')
plt.ylabel('FRET Efficiency (%)')
print "----"
print "ic50: %f" % ic50()
print "nonspec: %f" % nonspec()
print "total: %f" % total()
print "f: %f" % f()
def plot_normalized_to_ref_curve():
data = bgsub_norm_averages
ref_index = 1
reference = data[data.keys()[ref_index]][VALUE]
# Normalize the reference to [0, 1]
reference = ((reference - np.min(reference)) /
(np.max(reference) - np.min(reference)))
plt.figure()
for i, conc_str in enumerate(data.keys()):
curve = data[conc_str][VALUE]
# Define the minval and maxval fit parameters
fit_min_val = fitting.Parameter(0.)
fit_max_val = fitting.Parameter(max(curve))
# Define the "model" func (just the normalization equation)
def normalization_eqn(y): return ((y - fit_min_val()) /
(fit_max_val() - fit_min_val()))
fitting.fit(normalization_eqn, [fit_max_val], np.array(reference),
np.array(curve))
curve = np.array(curve)
normalized_curve = normalization_eqn(np.array(curve))
plt.plot(normalized_curve)
def calc_pore_size_by_slope():
(fmax_arr, k1_arr, k2_arr, conc_list) = \
titration_fits.plot_two_exp_fits(bgsub_norm_wells, layout, plot=True)
titration_fits.plot_fmax_curve(fmax_arr, conc_list)
titration_fits.plot_k1_curve(k1_arr[:,1:], conc_list[1:])
titration_fits.plot_k2_curve(k2_arr[:,1:], conc_list[1:])
conc_lipos = 5.16
fmax_means = np.mean(fmax_arr, axis=0)
fmax_stds = np.std(fmax_arr, axis=0)
ratios = conc_list / conc_lipos
derivs = np.zeros(len(fmax_means)-1)
deriv_sds = np.zeros(len(fmax_means)-1)
for i in np.arange(1, len(fmax_means)):
if i == 1:
prev_mean = 0
prev_sd = 0
else:
prev_mean = fmax_means[i-1]
prev_sd = fmax_stds[i-1]
fmax_diff = fmax_means[i] - prev_mean
conc_diff = conc_list[i] - conc_list[i-1]
derivs[i-1] = fmax_diff / float(conc_diff)
deriv_sds[i-1] = np.sqrt(fmax_stds[i] ** 2 + prev_sd ** 2)
plt.figure()
#plt.errorbar(conc_list[1:], derivs, yerr=deriv_sds, marker='o', )
plt.errorbar(conc_list[1:], derivs, marker='o', )
plt.title('d[Fmax]/d[Bax]')
# Fit with exp dist?
exp_lambda = fitting.Parameter(1/20.)
def exp_dist(x):
return exp_lambda() * np.exp(-exp_lambda() * x)
(residuals, result) = fitting.fit(exp_dist, [exp_lambda], derivs,
conc_list[1:])
plt.plot(conc_list[1:], exp_dist(conc_list[1:]), color='g')
print "exp_lambda: %f" % exp_lambda()
# Fit with exp decay
exp_c0 = fitting.Parameter(0.005)
exp_k = fitting.Parameter(1/50.)
def exp_decay(x):
return exp_c0() * np.exp(-exp_k() * x)
(residuals, result) = fitting.fit(exp_decay, [exp_c0, exp_k],
derivs, conc_list[1:])
plt.plot(conc_list[1:], exp_decay(conc_list[1:]), color='r')
print "exp_c0: %f" % exp_c0()
print "exp_k: %f" % exp_k()
# We want to estimate Bax per pore, or pores per Bax, if Bax
# holes didn't exist. Basically we want to know the fraction release
# we would get if the amount of Bax approached 0. In the case of
# the exponential decay, this is the value of exp_c0.
# We convert exp_c0 from % release to number of pores:
# The derivative dfmax/dBax tells us how much more permeabilization we get
# per Bax. At low concentrations, we get a certain amount of release
# per Bax that is relatively unaffected by the Bax hole phenomenon.
print "exp_c0 * conc_lipos: %f" % (exp_c0() * conc_lipos)
def plot_fmax_curve(fmax_arr, conc_list):
fmax_means = np.mean(fmax_arr, axis=0)
fmax_sds = np.std(fmax_arr, axis=0)
# Plot of Fmax
plt.figure()
plt.errorbar(conc_list, fmax_means, yerr=fmax_sds / np.sqrt(3),
label='Data', linestyle='')
plt.ylabel(r'$F_{max}$ value (% Release)')
plt.xlabel('[Bax] (nM)')
plt.title(r'$F_{max}$')
# Try fitting the fmax values to a hill function
hill_vmax = fitting.Parameter(0.9)
kd = fitting.Parameter(100)
def hill(s):
return ((hill_vmax() * s) / (kd() + s))
fitting.fit(hill, [kd, hill_vmax], fmax_means, conc_list)
plt.plot(conc_list, hill(conc_list), 'r', linewidth=2, label='Hill fit')
plt.text(35, 0.93, '$K_D$ = %.2f' % kd(), fontsize=16)
plt.text(35, 0.99, '$V_{max}$ = %.2f' % hill_vmax(), fontsize=16)
# Try fitting the fmax values to an exponential function
exp_fmax = fitting.Parameter(1.0)
exp_k = fitting.Parameter(0.01)
def exp_func(s):
return exp_fmax() * (1 - np.exp(-exp_k()*s))
fitting.fit(exp_func, [exp_fmax, exp_k], fmax_means, conc_list)
plt.plot(conc_list, exp_func(conc_list), 'g', linewidth=2, label='Exp fit')
plt.legend(loc='lower right')
plt.show()
def plot_initial_slope_curves(k_arr, conc_list):
"""Plots the initial slopes as a function of concentration."""
k_means = np.mean(k_arr, axis=0)
k_sds = np.std(k_arr, axis=0)
plt.figure()
plt.errorbar(conc_list, k_means, yerr= k_sds / np.sqrt(3), color='r',
linestyle='', linewidth=2)
plt.title('Initial permeabilization rate')
plt.xlabel('[Bax] (nM)')
plt.ylabel('Rate $(\mathrm{sec}^{-1})$')
fmax = fitting.Parameter(1e-3)
k = fitting.Parameter(np.log(2)/300.)
def exp_func(s):
return fmax()*(1 - np.exp(-k()*s))
fitting.fit(exp_func, [fmax, k], k_means, conc_list)
plt.plot(conc_list, exp_func(conc_list), color='g', label='Exponential')
k_linear = fitting.Parameter(4e-7)
num_concs = 7
def linear(s):
return k_linear()*s
fitting.fit(linear, [k_linear], k_means[0:num_concs],
conc_list[0:num_concs])
plt.plot(conc_list, linear(conc_list), color='b',
label='Linear (first %d concs)' % num_concs)
plt.legend(loc='lower right')
plt.ylim([0, np.max(k_means) * 1.05])
return k_linear()
def plot_fret_exp_fits(donor_only):
# Plot each fret curve, F_DA/F_D
figure()
fret = collections.OrderedDict()
f_d = bgsub_timecourses['cBid 50 nM + 0 nM liposomes'][VALUE]
for i, conc_str in enumerate(bgsub_timecourses.keys()):
time = bgsub_timecourses[conc_str][TIME]
f_d_plus_a = bgsub_timecourses[conc_str][VALUE]
f = f_d_plus_a / donor_only
fret[conc_str] = []
fret[conc_str].append(time)
fret[conc_str].append(f)
plot(time, f, marker='o', linestyle='')
f0 = fitting.Parameter(1)
k = fitting.Parameter(0.001)
fmin = fitting.Parameter(0.6)
def exp_decay(t):
return f0()*np.exp(-k()*t) + fmin()
fitting.fit(exp_decay, [f0, k, fmin], f, time)
plot(time, exp_decay(time), marker='', color='gray')
ylabel('$F_{D+A} / F_A$')
xlabel('Time (sec)')
title("FRET timecourses")
return fret
def plot_liposome_titration_insertion_kinetics(model):
"""Plot the insertion kinetics of Bax over a liposome titration."""
lipo_concs = np.logspace(-2, 2, 40)
t = np.linspace(0, 12000, 100)
fmax_list = []
k_list = []
for lipo_conc in lipo_concs:
model.parameters['Vesicles_0'].value = lipo_conc
# Get the SS mBax value
#b_trans.model.parameters['Vesicles_0'].value = lipo_conc
#s = Solver(b_trans.model, t)
#s.run()
#max_mBax = s.yobs['mBax'][-1]
# Get the iBax curve
s = Solver(model, t)
s.run()
iBax = s.yobs['iBax'] / model.parameters['Bax_0'].value
plt.plot(t, iBax, 'r')
# Fit to single exponential
fmax = fitting.Parameter(0.9)
k = fitting.Parameter(0.01)
def single_exp(t):
return (fmax() * (1 - np.exp(-k()*t)))
fitting.fit(single_exp, [fmax, k], iBax, t)
plt.plot(t, single_exp(t), 'b')
fmax_list.append(fmax())
k_list.append(k())
plt.title('Inserted Bax with liposome titration')
plt.xlabel('Time')
plt.ylabel('Fraction of inserted Bax')
plt.show()
# Make plots of k and fmax as a function of lipo concentration
fmax_list = np.array(fmax_list)
k_list = np.array(k_list)
# k
plt.figure()
plt.plot(lipo_concs, k_list, 'ro')
plt.xlabel('Liposomes (nM)')
plt.ylabel('$k$')
plt.title("$k$ vs. Liposome conc")
plt.ylim([0, 0.0020])
plt.show()
# Fmax
plt.figure()
plt.plot(lipo_concs, fmax_list, 'ro')
plt.xlabel('Liposomes (nM)')
plt.ylabel('$F_{max}$')
plt.title("$F_{max}$ vs. Liposome conc")
plt.show()
def plot_normalized(bid_conc=0, bid_conc_for_normalization=0,
subtract_background=True, do_fit=False,
t0_val=None):
plt.figure()
k1_arr = []
fmax_arr = []
t0_arr = []
for conc_index in range(NUM_CONCS):
time = np.array(data[bid_conc][bax_concs[conc_index]][:].index.values,
dtype='float')
conc_data = data[bid_conc][bax_concs[conc_index]][:].values
f0 = data[bid_conc_for_normalization] \
[bax_concs[conc_index]][1].values[0]
if subtract_background: # from 0 Bax timecourse
bg = data[bid_conc][0][:].mean(axis=1).values
conc_data = np.subtract(conc_data.T, bg).T
f0 = f0 - bg[0]
conc_data = conc_data / float(f0)
# Plots the fits if desired
if do_fit:
fmax = fitting.Parameter(3.)
k1 = fitting.Parameter(0.0005)
def one_exp(t):
return 1. + fmax() * (1 - np.exp(-k1() * (t + t0())))
# Check if we're supposed to fit the t0 parameter or not
if t0_val is None:
t0 = fitting.Parameter(556)
fitting.fit(one_exp, [fmax, k1, t0],
np.mean(conc_data, axis=1), time)
else:
t0 = fitting.Parameter(t0_val)
fitting.fit(one_exp, [fmax, k1], np.mean(conc_data, axis=1),
time)
k1_arr.append(k1())
fmax_arr.append(fmax())
t0_arr.append(t0())
# Plot
plt.plot(time, one_exp(time), color='k')
plt.plot(time, np.mean(conc_data, axis=1),
label='Bax %s nM' % bax_concs[conc_index])
else:
plt.errorbar(time, np.mean(conc_data, axis=1),
yerr=np.std(conc_data, axis=1) / np.sqrt(3),
label='Bax %s nM' % bax_concs[conc_index])
box = plt.gca().get_position()
plt.gca().set_position([box.x0, box.y0, box.width * 0.8, box.height])
plt.legend(loc='upper left', prop=fontP, ncol=1,
bbox_to_anchor=(1, 1), fancybox=True, shadow=True)
plt.title(r'Normalized ($F/F_0$) by %s nM cBid $F_0$' % \
bid_conc_for_normalization)
plt.show()
if do_fit:
return (k1_arr, fmax_arr, t0_arr)
else:
return None
def plot_bax_titration_insertion_kinetics(model):
"""Plot the insertion kinetics of Bax over a Bax titration."""
bax_concs = np.logspace(-1, 3, 40)
t = np.linspace(0, 12000, 100)
fmax_list = []
k_list = []
for bax_conc in bax_concs:
model.parameters['Bax_0'].value = bax_conc
# Get the iBax curve
s = Solver(model, t)
s.run()
iBax = s.yobs['iBax'] / model.parameters['Bax_0'].value
plt.plot(t, iBax, 'r')
# Fit to single exponential
fmax = fitting.Parameter(0.9)
k = fitting.Parameter(0.01)
def single_exp(t):
return (fmax() * (1 - np.exp(-k()*t)))
fitting.fit(single_exp, [fmax, k], iBax, t)
plt.plot(t, single_exp(t), 'b')
fmax_list.append(fmax())
k_list.append(k())
plt.title('Inserted Bax with Bax titration')
plt.xlabel('Time')
plt.ylabel('Fraction of inserted Bax')
plt.show()
# Make plots of k and fmax as a function of lipo concentration
fmax_list = np.array(fmax_list)
k_list = np.array(k_list)
# k
plt.figure()
plt.plot(bax_concs, k_list, 'ro')
plt.xlabel('Bax (nM)')
plt.ylabel('$k$')
plt.title("$k$ vs. Bax conc")
plt.show()
# Fmax
plt.figure()
plt.plot(bax_concs, fmax_list, 'ro')
plt.xlabel('Bax (nM)')
plt.ylabel('$F_{max}$')
plt.title("$F_{max}$ vs. Bax conc")
plt.show()
def plot_fits(means, log_means, conc_list, log_concs):
# Ignore any NaNs
# Take mean of all dilution series
dilution_means = np.mean(means, axis=1)
nans = np.isnan(dilution_means)
dilution_means = dilution_means[nans==False]
dilution_stds = np.std(means[nans==False], axis=1)
# Take mean of all dilution series
infs = np.isinf(log_concs)
log_dilution_means = np.mean(log_means[(infs + nans) == False],
axis=1)
log_dilution_stds = np.std(log_means[(infs + nans) == False],
axis=1)
conc_list_no_nan = conc_list[nans == False]
conc_list_nonan_noinf = conc_list[(infs + nans) == False]
log_concs_nonan_noinf = log_concs[(infs + nans) == False]
# Fit with line
m = fitting.Parameter(900)
b = fitting.Parameter(0)
def linear(x):
return m()*x + b()
fitting.fit(linear, [m, b], dilution_means, conc_list_no_nan)
print "m: %f" % m()
print "b: %f" % b()
r_squared = fitting.r_squared(dilution_means, linear(conc_list_no_nan))
print "R^2: %f" % r_squared
figure()
errorbar(conc_list_no_nan, dilution_means, yerr=dilution_stds,
color='r', linestyle='')
plot(conc_list_no_nan, linear(conc_list_no_nan), color='r', label='Linear')
xlabel('[Fluorescein] (nM)')
ylabel('RFU')
title('Fits to Fluorescein titration')
legend(loc='lower right')
figure()
errorbar(log_concs_nonan_noinf, log_dilution_means, yerr=log_dilution_stds,
color='r', linestyle='')
plot(log_concs_nonan_noinf, np.log10(linear(conc_list_nonan_noinf)),
color='r', label='Linear')
legend(loc='lower right')
xlabel('log10([Fluorescein]) (nM)')
ylabel('log10(RFU)')
title('Fits to Fluorescein titration, log-log')
def plot_k_fmax_scaling(k_data, fmax_data, bid_ix, bax_concs,
plot_filename=None):
"""Plot Fmax and k vs. Bax concentration for a given Bid concentration."""
assert k_data.shape == fmax_data.shape, \
"k_data and f_max data must have same dimensions"
fig = plt.figure('exp_fits_k_fmax_var', figsize=(1.9, 1.5), dpi=300)
ax1 = fig.gca()
# Plot Fmax vs. concentration on the left-hand axis
ax1.plot(bax_concs, fmax_data[bid_ix, :], marker='o', markersize=3,
color='b', linestyle='')
ax1.set_xlabel('[Bax] (nM)')
ax1.set_ylabel('$F_{max}$', color='b')
ax1.set_xlim([0, 1100])
for tl in ax1.get_yticklabels():
tl.set_color('b')
ax1.set_xscale('log')
slope = fitting.Parameter(1.)
intercept = fitting.Parameter(1.)
def log_line(x):
log_concs = np.log10(x)
return slope() * log_concs + intercept()
fitting.fit(log_line, [slope, intercept], fmax_data[bid_ix, :],
bax_concs)
ax1.plot(bax_concs, log_line(bax_concs), color='b')
# Plot k vs. concentration on the right-hand axis
ax2 = ax1.twinx()
#ax2.set_xlim([0.05, 30])
ax2.plot(bax_concs, k_data[bid_ix,:], marker='o', markersize=3,
color='r')
ax2.set_ylabel(r'k (sec$^{-1} \times\ 10^{-5}$)', color='r')
for tl in ax2.get_yticklabels():
tl.set_color('r')
ax2.set_xlim([0, 1100])
ax2.set_ylim(5e-5, 3e-4)
ax2.set_yticks(np.linspace(5e-5, 3e-4, 6))
ax2.set_yticklabels([str(int(f)) for f in np.linspace(5, 30, 6)])
format_axis(ax1)
format_axis(ax2, yticks_position='right')
plt.subplots_adjust(left=0.18, bottom=0.19, right=0.75)
if plot_filename:
fig.savefig('%s.pdf' % plot_filename)
fig.savefig('%s.png' % plot_filename, dpi=300)
def plot_fit_donor_only():
# Fit the background (donor-only) to a line
m = fitting.Parameter(0)
b = fitting.Parameter(26000)
def linear(x):
return m()*x + b()
t = timecourse_averages['cBid 50 nM + 0 nM liposomes'][TIME]
x = timecourse_averages['cBid 50 nM + 0 nM liposomes'][VALUE]
fitting.fit(linear, [m, b], x, t)
figure()
plot(t, x)
plot(t, linear(t))
xlabel('Time (sec)')
ylabel('RFU')
title('Linear fit of Bid-only control')
return linear(t)
def plot_k_titration(fits):
bid_concs = []
k_means = []
k_ses = []
for cond_name in bid_fits.keys():
fits = bid_fits[cond_name]
bid_conc = float(cond_name.split(' ')[1])
bid_concs.append(bid_conc)
num_reps = fits.shape[0]
k_means.append(np.mean(fits[:,1]))
k_ses.append(np.std(fits[:,1]) / np.sqrt(num_reps))
bid_concs = np.array(bid_concs)
k_means = np.array(k_means)
plt.figure('k')
plt.errorbar(np.log10(bid_concs), k_means, yerr=k_ses, color='k')
kd1 = fitting.Parameter(1.)
k0 = fitting.Parameter(2e-5)
kmax1 = fitting.Parameter(2e-4)
slope = fitting.Parameter(1e-8)
def hill_func_line(concs):
return (k0() +
((kmax1() * concs) / (kd1() + concs)) +
(slope() * concs))
res = fitting.fit(hill_func_line, [kd1, k0, kmax1, slope],
np.array(k_means), np.array(bid_concs))
plt.figure('k')
plt.plot(np.log10(bid_concs), hill_func_line(bid_concs), color='b')
#plt.plot(np.log10(bid_concs), slope() * bid_concs, color='r')
#plt.plot(np.log10(bid_concs),
# k0() + ((kmax1()*bid_concs) / (kd1() + bid_concs)), color='g')
kd1 = fitting.Parameter(1.)
k0 = fitting.Parameter(7e-5)
kmax1 = fitting.Parameter(2e-4)
def hill_func_k(concs):
return (k0() + ((kmax1() * concs) / (kd1() + concs)))
res = fitting.fit(hill_func_k, [kd1, k0, kmax1],
np.array(k_means), np.array(bid_concs))
plt.figure('k')
plt.plot(np.log10(bid_concs), hill_func_k(bid_concs), color='r')
def normalize_fit(replicates):
"""
Find normalization parameters that minimize differences between replicates.
Chooses one (the first) trajectory as a reference, and then chooses
appropriate scaling parameters (min and max vals) to minimize the deviation
of the other trajectories from the reference trajectory.
Parameters
----------
replicates : list of numpy.array objects
Each entry in the list is a numpy.array containing the data for one
repeat of the experiment.
Returns
-------
list of numpy.array objects
The replicate data after normalization.
"""
normalized_replicates = []
# Choose the first replicate (arbitrarily) as the one to fit to,
# and normalize it to the range [0, 1]
reference = array(replicates[0])
reference = (reference - min(reference)) / (max(reference) - min(reference))
normalized_replicates.append(reference)
# For the other replicates, let min val and max val be fit so
# as to minimize deviation from the normalized first replicate
for replicate in replicates[1:len(replicates)]:
# Define the minval and maxval fit parameters
fit_min_val = Parameter(min(replicate))
fit_max_val = Parameter(max(replicate))
# Define the "model" func (just the normalization equation)
def normalization_eqn(repl): return ((repl - fit_min_val()) /
(fit_max_val() - fit_min_val()))
fit(normalization_eqn, [fit_min_val, fit_max_val], array(reference),
array(replicate))
replicate = array(replicate)
normalized_replicate = normalization_eqn(array(replicate))
normalized_replicates.append(normalized_replicate)
return normalized_replicates
def approach_1(wells, layout):
"""In this approach, we take the mean of each condition and then perform
a single fit to the mean. We then plot the value of each parameter
as a function of the concentration to look for patterns.
Note that the data is expected to be in raw format, not pre-averaged.
"""
(avgs, stderrs) = averages(wells, layout, stderr=True)
num_concs = len(layout.keys())
min_rfu_arr = np.zeros((num_concs, 2))
max_rfu_arr = np.zeros((num_concs, 2))
k_arr = np.zeros((num_concs, 2))
conc_list = np.zeros(num_concs)
plt.figure()
num_rows = 4
num_cols = 3
for i, conc_str in enumerate(layout.keys()):
print conc_str
conc = float(conc_str.split(' ')[1])
conc_list[i] = conc
time = avgs[conc_str][TIME][0:100]
curve = avgs[conc_str][VALUE][0:100]
plt.subplot(num_rows, num_cols, i+1)
plt.errorbar(time, curve, yerr=stderrs[conc_str][VALUE][0:100])
# Define the parameters
max_rfu = fitting.Parameter(np.max(curve))
min_rfu = fitting.Parameter(np.min(curve))
k = fitting.Parameter(1e-4)
def exp_func(t):
return (max_rfu() - min_rfu())*(1 - np.exp(-k()*t)) + min_rfu()
result = fitting.fit(exp_func, [max_rfu, min_rfu, k], curve, time)
cov_x = result[1]
plt.plot(time, exp_func(time), color='r')
plt.title(conc_str)
plt.xlabel('Time (sec)')
plt.ylabel('ANTS (RFU)')
max_rfu_arr[i] = [max_rfu(), np.sqrt(cov_x[0, 0])]
min_rfu_arr[i] = [min_rfu(), np.sqrt(cov_x[1, 1])]
k_arr[i] = [k(), np.sqrt(cov_x[2, 2])]
plt.tight_layout(pad=0, w_pad=0, h_pad=0)
# Plot values of fitted parameters
plt.figure()
plt.errorbar(conc_list, k_arr[:,0], yerr=k_arr[:,1], marker='o')
plt.figure()
plt.errorbar(conc_list, min_rfu_arr[:,0], yerr=min_rfu_arr[:,1], marker='o')
plt.figure()
plt.errorbar(conc_list, max_rfu_arr[:,0], yerr=max_rfu_arr[:,1], marker='o')
plt.figure()
plt.plot(conc_list, (max_rfu_arr[:,0] - min_rfu_arr[:,0])/min_rfu_arr[:,0],
marker='o')
plt.title('Percent increase')
import ipdb; ipdb.set_trace()
def calc_exp_fits(data_norm):
"""Fit each of the timecourses in the data matrix to an exponential func.
Parameters
----------
data_norm : np.array
Four-dimensional numpy array, with dimensions
[bid, bax, datatype (time or value), timepoints]
Returns
-------
tuple of two np.arrays
The first array contains the fitted k values, the second the fitted
fmax values. Both are of dimension [bid concs, bax_concs].
"""
fmax_data = np.zeros((len(bid_concs), len(bax_concs)))
k_data = np.zeros((len(bid_concs), len(bax_concs)))
fmax_sd_data = np.zeros((len(bid_concs), len(bax_concs)))
k_sd_data = np.zeros((len(bid_concs), len(bax_concs)))
for bid_ix in range(data_norm.shape[0]):
for bax_ix in range(data_norm.shape[1]):
t = data_norm[bid_ix, bax_ix, TIME, :]
v = data_norm[bid_ix, bax_ix, VALUE, :]
# Get an instance of the fitting function
k = fitting.Parameter(1e-4)
fmax = fitting.Parameter(3.)
def fit_func(t):
return 1 + fmax() * (1 - np.exp(-k()*t))
# Subtract 1 so that the curves start from 0
(residuals, result) = fitting.fit(fit_func, [k, fmax], v, t)
cov_matrix = result[1]
#(k, fmax) = fit.fit_timecourse(t, v)
k_data[bid_ix, bax_ix] = k()
fmax_data[bid_ix, bax_ix] = fmax()
#import ipdb; ipdb.set_trace()
# Get variance estimates
[k_sd, fmax_sd] = np.sqrt(np.diag(np.var(residuals, ddof=1) *
cov_matrix))
k_sd_data[bid_ix, bax_ix] = k_sd
fmax_sd_data[bid_ix, bax_ix] = fmax_sd
# Some diagnostic plots (commented out)
#plt.figure()
#plt.plot(t, v, 'k')
#plt.plot(t[:numpts], intercept + lin_fit[0] * t[:numpts], 'r')
#plt.figure()
#plt.plot(t, v / intercept - 1)
#plt.plot(t, fit.fit_func(t, [k, fmax]))
return (k_data, k_sd_data, fmax_data, fmax_sd_data)
def plot_k1_curve(k1_arr, conc_list):
k1_means = np.mean(k1_arr, axis=0)
k1_sds = np.std(k1_arr, axis=0)
plt.figure()
plt.errorbar(conc_list, k1_means, yerr= k1_sds / np.sqrt(3), color='r',
linestyle='', linewidth=2)
plt.title('$k_1$')
plt.xlabel('[Bax] (nM)')
plt.ylabel('$k_1\ (\mathrm{sec}^{-1})$')
# Fit with exponential-linear curve
vi = fitting.Parameter(0.05)
vf = fitting.Parameter(0.025)
tau = fitting.Parameter(0.02)
# Define fitting function
def biphasic(t):
return (vf()*t) + ( (vi() - vf()) *
((1 - np.exp(-tau()*t))/tau()) )
fitting.fit(biphasic, [vi, vf, tau], k1_means, conc_list)
plt.plot(conc_list, biphasic(conc_list), 'r', linewidth=2)
# Fit with "double-binding curve"
ksat = fitting.Parameter(0.0005)
knonsat = fitting.Parameter(20)
R0 = fitting.Parameter(20)
def double_binding(t):
return (R0() * t)/(ksat() + t) + (knonsat()*t)
fitting.fit(double_binding, [R0, ksat, knonsat], k1_means, conc_list)
plt.plot(conc_list, double_binding(conc_list), 'g', linewidth=2)
plt.text(400, 0.00025,
r'$\frac{R_0 Bax_0}{K_{sat} + Bax_0} + K_{nonsat} Bax_0$')
plt.text(400, 0.00020, r'$R_0$ = %.3g nM' % R0())
plt.text(400, 0.00015, r'$K_{sat}$ = %.3g nM' % ksat())
plt.text(400, 0.00010, r'$K_{nonsat}$ = %.3g nM' % knonsat())
plt.text(35, 0.00054,
r'$v_f + (v_i - v_f) \left(\frac{1 - e^{-\tau t}}{\tau}\right)$')
plt.text(35, 0.00049, r'$v_i$ = %.3g sec$^{-1}$ nM$^{-1}$' % vi())
plt.text(35, 0.00044, r'$v_f$ = %.3g sec$^{-1}$ nM$^{-1}$' % vf())
plt.text(35, 0.00039, r'$\frac{1}{\tau}$ = %.2f nM' % (1/tau()))
plt.title('Biphasic fit of $k_1$')
plt.show()
def fit_average_by_leastsq(self, reps):
"""Fit the average with an exponential model
Returns a dict with the parameter values for the fit.
"""
avg_trajectory = np.mean(reps, axis=0)
k_p = fitting.Parameter(np.log(2)/6000.)
initial_rfu_p = fitting.Parameter(avg_trajectory[0])
pct_increase_p = fitting.Parameter((avg_trajectory[-1]-avg_trajectory[0])/
avg_trajectory[0])
def exp_func(t):
return (pct_increase_p() * initial_rfu_p()) * (1 - np.exp(-k_p() * t)) + \
initial_rfu_p()
fitting.fit(exp_func, [k_p, initial_rfu_p, pct_increase_p], avg_trajectory, self.t)
return {'k': k_p(),
'initial_rfu': initial_rfu_p(),
'pct_increase': pct_increase_p()}
def fit_data(mcmc_set):
data = mcmc_set.chains[0].data
fmax_arr = np.zeros(len(data.columns))
k1_arr = np.zeros(len(data.columns))
k2_arr = np.zeros(len(data.columns))
bax_concs = np.array(data.columns, dtype='float')
for i, bax_conc in enumerate(bax_concs):
conc_data = data[bax_conc]
time = conc_data[:, 'TIME']
y = conc_data[:, 'MEAN']
fmax = fitting.Parameter(0.9)
k1 = fitting.Parameter(0.01)
k2 = fitting.Parameter(0.001)
def two_part_exp(t):
return (fmax() * (1 - np.exp(-k1() * (1 - np.exp(-k2()*t)) * t)))
fitting.fit(two_part_exp, [fmax, k1, k2], y, time)
fmax_arr[i] = fmax()
k1_arr[i] = k1()
k2_arr[i] = k2()
two_exp_fits_dict['data'] = (fmax_arr, k1_arr, k2_arr)
def fit_timecourse(self, time, y):
"""Fit a single timecourse with the desired function.
Parameters
----------
time : numpy.array
Vector of time values.
y : numpy.array
Vector of y-axis values (e.g., dye release).
Returns
-------
list of numbers
The best-fit parameter values produced by the fitting procedure.
"""
params = [fitting.Parameter(self.initial_guesses[i])
for i in range(len(self.initial_guesses))]
def fit_func_closure(t):
return self.fit_func(t, [p() for p in params])
fitting.fit(fit_func_closure, params, y, time,
log_transform=self.log_transform)
return [p() for p in params]
def fit_gouy_chapman_ols(bid_concs, fret_means, fret_ses):
lipid_conc = 129.8 # uM; 0.1 (mg/mL); ~1.5 nM liposomes
v = fitting.Parameter(6.) # charges per peptide chain
b = fitting.Parameter(1.) # dimensionless
pc = fitting.Parameter(1.) # partition coefficient
f = fitting.Parameter(0.35) # FRET efficiency
f0 = fitting.Parameter(0.095) # baseline quenching
r = np.logspace(-3, 0, 1000)
def cf(r):
return (r * alpha(r)) / pc()
def alpha(r):
log_alpha = 2 * v() * np.arcsinh(v() * b() * r)
return np.exp(log_alpha)
def fit_func(bid_concs):
cb = r * lipid_conc
cfr = cf(r)
ctot = cb + cfr
fracs_bound = cb / ctot
interp_fracs = np.interp(bid_concs, ctot, fracs_bound)
return f() * interp_fracs + f0()
fitting.fit(fit_func, [v, b, pc, f, f0], fret_means, bid_concs)
plt.figure()
plt.errorbar(np.log10(bid_concs), fret_means, yerr=fret_ses, linestyle='',
color='r')
bid_pred = np.logspace(-1, 3, 100)
plt.plot(np.log10(bid_pred), fit_func(bid_pred), color='k')
plt.xlabel('log10([Total Bid])')
plt.ylabel('FRET Efficiency (%)')
print f0()
print f()
print v()
print b()
print pc()
def estimate_background_residuals():
"""Perform a fit of each background trajectory separately and then
use these fits to estimate the variance of the residuals due to
measurement error."""
buffer_wells = extract(layout['Buffer'], timecourse_wells)
plot_all(buffer_wells)
(time_arr, buffer_reps) = get_replicates_for_condition(
timecourse_wells, layout, 'Buffer')
print "bg_init k_decay k_linear bg_bg"
num_timepoints = len(time_arr)
num_replicates = buffer_reps.shape[1]
residuals = np.zeros((num_timepoints, num_replicates))
param_matrix = np.zeros((num_replicates, 4))
for i in range(num_replicates):
plt.figure()
plt.plot(time_arr, buffer_reps[:,i])
bg_init = fitting.Parameter(1.)
k_decay = fitting.Parameter(1e-2)
bg_bg = fitting.Parameter(5.6)
k_linear = fitting.Parameter(1e-4)
def exp_decay(t):
return bg_init()*np.exp(-k_decay() * t) - k_linear()*t + bg_bg()
(residuals[:,i], result) = fitting.fit(exp_decay, [bg_init, k_decay, bg_bg, k_linear],
buffer_reps[:,i], time_arr)
plt.plot(time_arr, exp_decay(time_arr), linewidth=2, color='r')
print bg_init(), k_decay(), k_linear(), bg_bg()
param_matrix[i,:] = [bg_init(), k_decay(), k_linear(), bg_bg()]
plt.figure()
plt.hist(residuals.reshape(num_replicates * num_timepoints))
print "Mean"
print np.mean(residuals)
print "Variance"
print np.var(residuals)
print "SD"
print np.std(residuals)
print param_matrix
print "Mean"
print np.mean(param_matrix, axis=0)
print "SD"
print np.std(param_matrix, axis=0)
return np.var(residuals)
def fit_basic_binding_curve_ols(bid_concs, fret_means, fret_ses):
kd = fitting.Parameter(6.) # kd for binding lipo sites
f = fitting.Parameter(0.30) # FRET efficiency
f0 = fitting.Parameter(0.10) # baseline quenching
def fit_func(bid_conc):
frac_bound = bid_conc / (bid_conc + kd())
frac_dye = 10. / bid_conc
frac_dye_bound = frac_bound * frac_dye
return f() * frac_dye_bound + f0()
fitting.fit(fit_func, [kd, f], fret_means, bid_concs)
plt.figure('Bid FRET')
plt.errorbar(np.log10(bid_concs), fret_means, yerr=fret_ses, linestyle='',
color='r')
bid_pred = np.logspace(-1, 3, 100)
plt.plot(np.log10(bid_pred), fit_func(bid_pred), color='k')
plt.xlabel('log10([Total Bid])')
plt.ylabel('FRET Efficiency (%)')
print "Kd: %f" % kd()
print "f: %f" % f()
print "f0: %f" % f0()
def fit_fda_and_fd_curves(wells, fda, fd, fa, bg):
# Get the background averages
bg_avg = get_average_bg(wells, bg, plot=False)
fa_avg = get_average_fa(wells, fa, plot=False)
# Now, plot the curves, with fits
for conc_name in fda.keys():
plt.figure()
# FDA
for well_name in fda[conc_name]:
fda_value = np.array(wells[well_name][VALUE])
# Subtract the FA background:
fda_value = fda_value - fa_avg
# Do the fit
f0 = fitting.Parameter(3500.)
fmax = fitting.Parameter(2000.)
k = fitting.Parameter(1e-3)
def fit_func(t):
return fmax()*np.exp(-k()*t) + f0()
res = fitting.fit(fit_func, [f0, fmax, k], fda_value, time)
# Plot data and fit
plt.plot(time, fda_value)
plt.plot(time, fit_func(time), color='k', linewidth=2)
# FD
for well_name in fd[conc_name]:
fd_value = np.array(wells[well_name][VALUE])
# Subtract the BG background:
fd_value = fd_value - bg_avg
# Do the fit
f0 = fitting.Parameter(3500.)
fmax = fitting.Parameter(2000.)
k = fitting.Parameter(1e-3)
def fit_func(t):
return fmax()*np.exp(-k()*t) + f0()
res = fitting.fit(fit_func, [f0, fmax, k], fd_value, time)
# Plot data and fit
plt.plot(time, fd_value)
plt.plot(time, fit_func(time), color='k', linewidth=2)
def plot_requenching_result(req_results, plot_filename):
# Publication-quality figure of result
f_outs = req_results['f_outs']
q_ins = req_results['q_ins']
f_out_errs = req_results['f_out_errs']
q_in_errs = req_results['q_in_errs']
plt.figure(figsize=(1.5, 1.5), dpi=300)
plt.errorbar(f_outs, q_ins, xerr=f_out_errs, yerr=q_in_errs,
marker='o', markersize=2, color='k', linestyle='',
capsize=1)
ax = plt.gca()
#for i in range(len(f_outs)):
# ax.add_patch(patches.Rectangle((f_outs[i], q_ins[i]),
# f_out_errs[i], q_in_errs[i], alpha=0.5))
#plt.plot(f_outs[0:12], q_ins[0:12], marker='o', color='r', linestyle='')
#plt.plot(f_outs[12:24], q_ins[12:24], marker='o', color='g', linestyle='')
#plt.plot(f_outs[24:36], q_ins[24:36], marker='o', color='b', linestyle='')
plt.xlabel('$F_{out}$')
plt.ylabel('$Q_{in}$')
plt.xlim([0, 1])
plt.ylim([0, 0.5])
format_axis(ax)
const = fitting.Parameter(0.1)
def fit_func(x):
return const()
fitting.fit(fit_func, [const], q_ins, f_outs)
plt.hlines(const(), 0, 1, color='r')
#linfit = scipy.stats.linregress(f_outs, q_ins)
#f_out_pred = np.linspace(0, 1, 100)
#plt.plot(f_out_pred, (f_out_pred * linfit[0]) + linfit[1], color='r',
# alpha=0.5)
#plt.title('$Q_{in}$ vs. $F_{out}$')
plt.subplots_adjust(left=0.23, bottom=0.2)
plt.savefig('%s.pdf' % plot_filename)
plt.savefig('%s.png' % plot_filename, dpi=300)
def fit_timecourses(layout, timecourses, plot=True):
"""Fit all timecourses in the experiment."""
# We'll be doing a 3 parameter fit
num_params = 3
# This is where we'll store the results of the fitting
param_dict = collections.OrderedDict()
# Iterate over all of the conditions in the layout
for cond_name, rep_list in layout.iteritems():
# How many replicates for this condition?
num_reps = len(rep_list)
# Set up a plot to show the fits
if plot:
plt.figure(cond_name, figsize=(6, 2), dpi=150)
plots_per_row = 3
# Create an entry in the resulting param dict for this condition.
# Results will be stored in a matrix with shape (num_reps, num_params)
param_matrix = np.zeros((num_reps, num_params))
# Now iterate over every well in the replicate list
for rep_ix, well_name in enumerate(rep_list):
# Get the time and fluorescence vectors
time = timecourses[well_name][TIME]
value = timecourses[well_name][VALUE]
# Set up the fitting procedure: first define the parameters
fmax = fitting.Parameter(2.1)
f0 = fitting.Parameter(2.3)
k = fitting.Parameter(1.4e-4)
# Define the fitting function
def exp_func(t):
return (fmax() * f0()) * (1 - np.exp(-k() * t)) + f0()
# Do the fit
res = fitting.fit(exp_func, [fmax, k, f0], value, time)
# Save the values in the order: (fmax, k, f0)
param_matrix[rep_ix, 0] = fmax()
param_matrix[rep_ix, 1] = k()
param_matrix[rep_ix, 2] = f0()
# Plot into the appropriate subplot
if plot:
plt.subplot(1, num_plots_per_row, rep_ix+1)
plt.plot(time, value, color='r')
plt.plot(time, exp_func(time), color='k', linewidth=2)
plt.xlabel('Time (sec)')
plt.ylabel('RFU')
# Store the resulting parameter matrix in the dict entry for this cond
param_dict[cond_name] = param_matrix
return param_dict
def estimate_bg_release():
lipo_wells = extract(layout['Bax 0 nM'], timecourse_wells)
plot_all(lipo_wells)
(time_arr, lipo_reps) = get_replicates_for_condition(
timecourse_wells, layout, 'Bax 0 nM')
num_timepoints = len(time_arr)
num_replicates = lipo_reps.shape[1]
residuals = np.zeros((num_timepoints, num_replicates))
param_matrix = np.zeros((num_replicates, 2))
avg_bg = timecourse_averages['Buffer'][VALUE]
for i in range(num_replicates):
plt.figure()
plt.plot(time_arr, lipo_reps[:,i])
bg_sub = lipo_reps[:,i] - avg_bg
plt.plot(time_arr, bg_sub)
lipo_init = fitting.Parameter(20.)
k_app = fitting.Parameter(1e-2)
def linear(t):
return lipo_init() + k_app()*t
(residuals[:,i], result) = fitting.fit(linear, [lipo_init, k_app],
bg_sub, time_arr)
plt.plot(time_arr, linear(time_arr), linewidth=2, color='r')
print lipo_init(), k_app()
param_matrix[i,:] = [lipo_init(), k_app()]
plt.figure()
plt.hist(residuals.reshape(num_replicates * num_timepoints))
print "Mean"
print np.mean(residuals)
print "Variance"
print np.var(residuals)
print "SD"
print np.std(residuals)
print param_matrix
print "Mean"
print np.mean(param_matrix, axis=0)
print "SD"
print np.std(param_matrix, axis=0)
return
def plot_lipo_background(wells, layout):
"""Takes the lipo background well timecourses and plots the
fluorescence values of the initial points as a function of lipo
concentration. Shows that the liposome background fluorescence
is strictly linear.
"""
init_vals = np.zeros(len(layout.keys()))
conc_list = np.zeros(len(layout.keys()))
for i, cond_name in enumerate(layout):
conc = float(cond_name.split(" ")[1])
well_name = layout[cond_name][0]
init_val = wells[well_name][VALUE][0]
init_vals[i] = init_val
conc_list[i] = conc
# Fit the liposome background to a line
print "Fitting liposome background"
m = fitting.Parameter(1.0)
b = fitting.Parameter(1.0)
def linear(s):
return m() * s + b()
result = fitting.fit(linear, [m, b], init_vals, conc_list)
plt.figure()
plt.plot(conc_list, init_vals, marker="o", linestyle="", color="b")
plt.plot(conc_list, linear(conc_list), color="r")
plt.xlabel("Liposome concentration (mg/ml)")
plt.ylabel("RFU")
plt.title("Liposome background fluorescence at t=0")
ax = plt.gca()
ax.set_xscale("log")
print "m: %f" % m()
print "b: %f" % b()
return [m(), b()]
def approach_3(timecourse_wells, layout):
"""Global fit of all curves by least squares."""
# Get the average trajectories, with standard errors
(avgs, stderrs) = averages(timecourse_wells, layout, stderr=True)
#(avgs, stderrs) = averages(timecourse_wells, layout, stderr=False)
# Build full matrix of all timecourse averages and SDs
num_concs = len(layout.keys())
num_timepoints = 100
time_arr = np.zeros((num_timepoints, num_concs))
avg_arr = np.zeros((num_timepoints, num_concs))
std_arr = np.zeros((num_timepoints, num_concs))
conc_list = np.zeros(num_concs)
for i, conc_str in enumerate(layout.keys()):
conc = float(conc_str.split(' ')[1])
conc_list[i] = conc
time_arr[:, i] = avgs[conc_str][TIME][:num_timepoints]
avg_arr[:, i] = avgs[conc_str][VALUE][:num_timepoints]
std_arr[:, i] = stderrs[conc_str][VALUE][:num_timepoints]
# Starting value for highest concentration
min_initial = np.min(avg_arr[0,:])
max_initial = avg_arr[0,0]
max_conc = conc_list[0]
min_final = np.min(avg_arr[-1,:])
max_final = avg_arr[-1,0]
min_int = fitting.Parameter(min_initial)
min_slope = fitting.Parameter((max_initial - min_initial) / max_conc)
max_int = fitting.Parameter(min_final)
max_slope = fitting.Parameter((max_final - min_final) / max_conc)
k = fitting.Parameter(1.2e-4)
avg_arr_lin = avg_arr.T.reshape(num_timepoints * num_concs)
def exp_func(t, conc, min_int, min_slope, max_int, max_slope, k):
max_val = max_slope * conc + max_int
min_val = min_slope * conc + min_int
return (max_val - min_val) * (1 - np.exp(-k * t)) + min_val
def exp_matrix(t):
result = np.zeros((num_timepoints*num_concs))
for i, conc in enumerate(conc_list):
result[i*num_timepoints:(i+1)*num_timepoints] = \
exp_func(t, conc, min_int(), min_slope(),
max_int(), max_slope(), k())
return result
fitting.fit(exp_matrix, [min_int, min_slope, max_int, max_slope, k],
avg_arr_lin, time_arr[:,0])
plt.figure()
plt.plot(conc_list, avg_arr[0,:], marker='o')
plt.plot(conc_list, [min_slope()*conc + min_int() for conc in conc_list], marker='o')
plt.figure()
num_rows = 4
num_cols = 3
def well_exp(initial_val, t):
return 0.05*initial_val*(1 - np.exp(-1.5e-4 * t)) + initial_val
for i, conc_str in enumerate(layout.keys()):
print conc_str
conc = float(conc_str.split(' ')[1])
conc_list[i] = conc
time = avgs[conc_str][TIME][0:100]
curve = avgs[conc_str][VALUE][0:100]
plt.subplot(num_rows, num_cols, i+1)
plt.errorbar(time, curve, yerr=stderrs[conc_str][VALUE][0:100], color='b')
plt.plot(time, exp_func(time, conc, min_int(), min_slope(),
max_int(), max_slope(), k()), color='r')
plt.plot(time, well_exp(curve[0], time), color='g')
plt.title(conc_str)
import ipdb; ipdb.set_trace()
plt.figure()
k_list = []
# Perform titration
mgtp_concs = np.arange(1, 15) * 1000 # nM (1 - 15 uM)
for mgtp_conc in mgtp_concs:
# Titration of labeled GTP:
model.parameters['mGTP_0'].value = mgtp_conc
sol.run()
# Fit to an exponential function to extract the pseudo-first-order rates
k = fitting.Parameter(1.)
def expfunc(t):
return 500 * (1 - np.exp(-k() * t))
res = fitting.fit(expfunc, [k], sol.yexpr['HRAS_mGXP_'], t)
# Plot data and fits
plt.plot(t, sol.yexpr['HRAS_mGXP_'], color='b')
plt.plot(t, expfunc(t), color='r')
# Keep the fitted rate
k_list.append(k())
# Plot the scaling of rates with mGTP concentration
plt.figure()
plt.plot(mgtp_concs, k_list, marker='o')
plt.ylim(bottom=0)
# Figure 3:
# A constant amount of labeled GDP