def mean_lags(tracks):
"""Calculate mean lag in velocity and turning angle of track(s)"""
means = []
for _, track in tracks.groupby(track_identifiers(tracks)):
lags = np.mean(track[['Velocity', 'Turning Angle']].diff()**2)
if not np.isnan(lags).any():
means.append(lags)
return np.mean(means, axis=0)
def _split_at_skip(tracks, jump_threshold, verbose):
"""Split track if timestep is missing or too long"""
if 'Time' not in tracks.columns:
return
if not tracks.index.is_unique:
tracks.reset_index(drop=True, inplace=True)
if 'Track_ID' in tracks.columns:
max_track_id = tracks['Track_ID'].max()
else:
max_track_id = 0
for criterium, track in tracks.groupby(track_identifiers(tracks)):
timesteps = track['Time'].diff()
skips = ((timesteps - timesteps.min())/timesteps.min()).round()
if skips.max() > 0:
index = track.index
if 'Track_ID' in track.columns:
tracks.loc[index, 'Orig. Track_ID'] = track['Track_ID']
skip_sum = skips.fillna(0).cumsum()
tracks.loc[index, 'Track_ID'] = max_track_id + 1 + skip_sum
max_track_id += max(skip_sum) + 1
if verbose:
print(' Warning: Split track {} with non-uniform timesteps.'
.format(criterium))
if jump_threshold is None:
return
for criterium, track in tracks.groupby(track_identifiers(tracks)):
positions = track[['X', 'Y', 'Z']]
dr = positions.diff()
dr_norms = np.linalg.norm(dr, axis=1)
skips = dr_norms > jump_threshold
if skips.max() > 0:
index = track.index
if 'Track_ID' in track.columns:
tracks.loc[index, 'Orig. Track_ID'] = track['Track_ID']
skip_sum = skips.cumsum()
tracks.loc[index, 'Track_ID'] = max_track_id + 1 + skip_sum
max_track_id += max(skip_sum) + 1
if verbose:
print(' Warning: Split track {} with jump > {}um.'
.format(criterium, jump_threshold))
def plot_arrest(tracks, condition='Condition', arrest_velocity=3, save=False,
context='notebook'):
"""Plot velocity aligned to minimum and distribution of arrested steps"""
if 'Displacement' not in tracks.columns:
tracks = analyze(tracks)
if condition not in tracks.columns:
tracks[condition] = 'Default'
sns.set(style='ticks', context=context)
fig, axes = plt.subplots(1, 2, figsize=(8, 5.5))
axes[0].set_xlabel('Time to minimum')
axes[0].set_ylabel('Velocity')
axes[1].set_xlabel(r'Consecutive steps below {} $\mu$m/min'.format(arrest_velocity))
axes[1].set_ylabel('Proportion')
for i, (cond, cond_tracks) in enumerate(tracks.groupby(condition)):
velocities = pd.Series()
arrested_segment_lengths = []
for _, track in cond_tracks.groupby(track_identifiers(cond_tracks)):
min_index = track['Velocity'].argmin()
track_velocities = pd.Series(
track['Velocity'].values, track['Time'] - track.loc[min_index, 'Time'])
velocities = velocities.append(track_velocities.dropna())
arrested = track['Velocity'] < arrest_velocity
arrested_segments = np.split(arrested, np.where(np.diff(arrested))[0] + 1)
arrested_segment_lengths.extend([sum(segment)
for segment in arrested_segments
if sum(segment) > 0])
velocities.index = np.round(velocities.index, 5) # Handle non-integer 'Times'
arrestats = velocities.groupby(velocities.index).describe().unstack()
color = sns.color_palette(n_colors=i+1)[-1]
axes[0].plot(arrestats.index, arrestats['50%'], color=color)
axes[0].fill_between(arrestats.index, arrestats['25%'], arrestats['75%'],
color=color, alpha=0.2)
axes[0].fill_between(arrestats.index, arrestats['min'], arrestats['max'],
color=color, alpha=0.2)
sns.distplot(arrested_segment_lengths, bins=np.arange(1,
max(arrested_segment_lengths) + 1) - 0.5,
norm_hist=True, kde=False, color=color, ax=axes[1])
axes[0].set_xlim([-3, 3])
axes[1].get_xaxis().set_major_locator(plt.MaxNLocator(integer=True))
sns.despine()
plt.tight_layout()
if save:
conditions = [cond.replace('= ', '')
for cond in tracks[condition].unique()]
plt.savefig('Arrest_' + '-'.join(conditions) + '.png', dpi=300)
else:
plt.show()
def plot_dr(raw_tracks, save=False, condition='Condition', context='notebook'):
"""Plot the differences in X, Y (and Z) to show biases"""
tracks = raw_tracks.copy()
_uniquize_tracks(tracks)
_split_at_skip(tracks)
dimensions = [dim for dim in ['X', 'Y', 'Z'] if dim in tracks.columns]
differences = pd.DataFrame()
for _, track in tracks.groupby(track_identifiers(tracks)):
differences = differences.append(track[dimensions].diff().dropna())
if 'Track_ID' in differences.columns:
differences = differences.fillna(track['Track_ID'].iloc[0])
else:
differences['Track_ID'] = track['Track_ID'].iloc[0]
sns.set(style="ticks", palette='deep', context=context)
fig, axes = plt.subplots(ncols=3, figsize=(15.5,5.5))
plt.setp(axes, yticks=[])
plt.setp(axes, xticks=[])
axes[0].set_title(r'$\Delta \vec r$')
axes[0].set_xticks([0])
axes[0].set_xticklabels([r'$0$'])
for dimension in dimensions:
sns.kdeplot(differences[dimension], shade=True, ax=axes[0])
axes[1].set_title('Joint Distribution')
axes[1].set_xlabel(r'$\Delta x$')
axes[1].set_ylabel(r'$\Delta y$')
axes[1].axis('equal')
axes[1].set_xlim([differences['X'].quantile(0.1), differences['X'].quantile(0.9)])
axes[1].set_ylim([differences['Y'].quantile(0.1), differences['Y'].quantile(0.9)])
sns.kdeplot(differences[['X', 'Y']], shade=False, cmap='Greys', ax=axes[1])
axes[2].set_title(r'$\Delta \vec r$ Lag Plot')
axes[2].axis('equal')
axes[2].set_xlabel(r'$\Delta r_i(t)$')
axes[2].set_ylabel(r'$\Delta r_i(t+1)$')
for i, dim in enumerate(dimensions):
color = sns.color_palette()[i]
for _, track in differences.groupby('Track_ID'):
axes[2].scatter(track[dim], track[dim].shift(), facecolors=color)
sns.despine()
plt.tight_layout()
if save:
conditions = [cond.replace('= ', '')
for cond in tracks[condition].unique()]
plt.savefig('dr_' + '-'.join(conditions) + '.png', dpi=300)
else:
plt.show()
def _uniquize_tracks(tracks, verbose):
"""Cluster tracks, if not unique"""
if 'Time' not in tracks.columns:
return
tracks['Orig. Index'] = tracks.index
if not tracks.index.is_unique:
tracks.reset_index(drop=True, inplace=True)
if 'Track_ID' in tracks.columns:
max_track_id = tracks['Track_ID'].max()
else:
max_track_id = 0
for identifiers, track in tracks.groupby(track_identifiers(tracks)):
if sum(track['Time'].duplicated()) != 0:
n_clusters = track['Time'].value_counts().max()
track = track.copy()
index = track.index
if 'Track_ID' in track.columns:
tracks.loc[index, 'Orig. Track_ID'] = track['Track_ID']
clusters = AgglomerativeClustering(n_clusters).fit(
track[['X', 'Y', 'Z']])
track.loc[:, 'Cluster'] = clusters.labels_
if sum(track[['Cluster', 'Time']].duplicated()) != 0:
clusters = AgglomerativeClustering(n_clusters).fit(
track[['Orig. Index']])
track.loc[:, 'Cluster'] = clusters.labels_
if sum(track[['Cluster', 'Time']].duplicated()) == 0:
tracks.loc[index, 'Track_ID'] = max_track_id+1+clusters.labels_
max_track_id += n_clusters
pd.set_option('display.max_rows', 1000)
if verbose:
print(' Warning: Split non-unique track {} by clustering.'
.format(identifiers))
else:
tracks.drop(index, inplace=True)
if verbose:
print(' Warning: Delete non-unique track {}.'
.format(identifiers))
def plot_situation(tracks, n_tracks=6*3, n_dcs=50, tcz_volume=0.524e9/400,
min_distance=0, min_distance_std=200/10, zoom=1, t_detail=None, save=False,
context='notebook'):
"""Plot some T cell tracks, DC positions and T cell zone volume"""
sns.set(style='ticks', context=context)
_ = plt.figure(figsize=(8, 5.5))
gs = gridspec.GridSpec(2,3)
space_ax = plt.subplot(gs[:,:-1], projection='3d')
time_ax = plt.subplot(gs[0,-1])
reach_ax = plt.subplot(gs[1,-1])
plt.locator_params(nbins=6)
space_ax.set_title('{} T Cell Tracks & {} DCs'.format(n_tracks, n_dcs))
n_conditions = len(tracks['Condition'].unique())
palette = itertools.cycle(sns.color_palette())
if min_distance_std != 0:
moved_tracks = tracks.copy()
for id in tracks['Track_ID'].unique():
moved_tracks.loc[moved_tracks['Track_ID'] == id, ['X', 'Y', 'Z']] += \
np.random.randn(3)*min_distance_std
else:
moved_tracks = tracks
for i, (cond, cond_tracks) in enumerate(moved_tracks.groupby('Condition')):
choice = np.random.choice(cond_tracks['Track_ID'].unique(),
n_tracks/n_conditions)
chosen_tracks = cond_tracks[cond_tracks['Track_ID'].isin(choice)]
for _, track in chosen_tracks.groupby(track_identifiers(chosen_tracks)):
if t_detail:
track = track[track['Time'] <= t_detail*60]
if n_conditions > 1:
color = sns.color_palette(n_colors=i+1)[-1]
else:
color = next(palette)
space_ax.plot(track['X'].values, track['Y'].values, track['Z'].values,
color=color)
tcz_radius = (3*tcz_volume/(4*np.pi))**(1/3)
ratio = (min_distance/tcz_radius)**3
r = tcz_radius*(ratio + (1 - ratio)*np.random.rand(n_dcs))**(1/3)
theta = np.random.rand(n_dcs)*2*np.pi
phi = np.arccos(2*np.random.rand(n_dcs) - 1)
dcs = pd.DataFrame({
'X': r*np.sin(theta)*np.sin(phi),
'Y': r*np.cos(theta)*np.sin(phi),
'Z': r*np.cos(phi)})
space_ax.scatter(dcs['X'], dcs['Y'], dcs['Z'], c='y')
r = (3*tcz_volume/(4*np.pi))**(1/3)
for i in ['x', 'y', 'z']:
circle = Circle((0, 0), r, fill=False, linewidth=2)
space_ax.add_patch(circle)
art3d.pathpatch_2d_to_3d(circle, z=0, zdir=i)
time_ax.set_xlabel('Time within Lymph Node [h]')
time_ax.set_ylabel('Probab. Density')
reach_ax.set_xlabel(r'Maximal Reach [$\mu$m]')
reach_ax.set_ylabel('Probab. Density')
def residence_time(track): return track['Time'].diff().mean()/60*len(
track[np.linalg.norm(track[['X', 'Y', 'Z']], axis=1) < r])
for i, (cond, cond_tracks) in enumerate(moved_tracks.groupby('Condition')):
color = sns.color_palette(n_colors=i+1)[-1]
residence_times = [residence_time(track)
for _, track in cond_tracks.groupby('Track_ID')]
if not all(time == residence_times[0] for time in residence_times):
sns.distplot(residence_times, kde=False, norm_hist=True, ax=time_ax,
label=cond, color=color)
max_reaches = [max(np.linalg.norm(track[['X', 'Y', 'Z']], axis=1))
for _, track in cond_tracks.groupby('Track_ID')]
sns.distplot(max_reaches, kde=False, norm_hist=True, ax=reach_ax,
label=cond, color=color)
time_ax.set_yticks([])
time_ax.axvline(np.median(residence_times), c='0', ls=':')
sns.despine(ax=time_ax)
reach_ax.set_yticks([])
reach_ax.legend()
reach_ax.axvline(tcz_radius, c='0', ls=':')
sns.despine(ax=reach_ax)
equalize_axis3d(space_ax, zoom)
plt.tight_layout()
if save == True:
save = 'situation.png'
if save:
plt.savefig(save, dpi=300)
else:
plt.show()
def summarize(tracks, arrest_velocity=3, skip_steps=4):
"""Summarize track statistics, e.g. mean velocity per track"""
if 'Displacement' not in tracks.columns:
tracks = analyze(tracks)
print('\nSummarizing track statistics')
summary = pd.DataFrame()
for i, (_, track) in enumerate(tracks.groupby(track_identifiers(tracks))):
if 'Track_ID' in track.columns:
summary.loc[i, 'Track_ID'] = track.iloc[0]['Track_ID']
if 'Condition' in track.columns:
summary.loc[i, 'Condition'] = track.iloc[0]['Condition']
else:
summary.loc[i, 'Condition'] = 'Default'
if 'Sample' in track.columns:
summary.loc[i, 'Sample'] = track.iloc[0]['Sample']
summary.loc[i, 'Mean Velocity'] = track['Velocity'].mean()
summary.loc[i, 'Mean Turning Angle'] = track['Turning Angle'].mean()
if 'Plane Angle' in track.columns:
summary.loc[i, 'Mean Plane Angle'] = track['Plane Angle'].mean()
summary.loc[i, 'Track Duration'] = \
track['Time'].iloc[-1] - track['Time'].iloc[0]
summary.loc[i, 'Arrest Coefficient'] = \
len(track[track['Velocity'] < arrest_velocity])/ \
len(track['Velocity'].dropna())
if 'Z' in track.columns:
positions = track[['X', 'Y', 'Z']]
ndim = 3
else:
positions = track[['X', 'Y']]
ndim = 2
summary.loc[i, 'Motility Coefficient'] = np.pi* \
track['Displacement'].iloc[-1]/(2*ndim)/track['Track Time'].max()
dr = positions.diff()
dr_norms = np.linalg.norm(dr, axis=1)
summary.loc[i, 'Confinement Ratio'] = track['Displacement'].iloc[-1] \
/dr_norms[1:].sum()
summary.loc[i, 'Corr. Confinement Ratio'] = track['Displacement'].iloc[-1] \
/dr_norms[1:].sum()*np.sqrt(track['Track Time'].max())
summary.loc[i, 'Mean Sq. Velocity Lag'] = np.mean(
track['Velocity'].diff()**2)
summary.loc[i, 'Mean Sq. Turn. Angle Lag'] = np.mean(
track['Turning Angle'].diff()**2)
if len(track) > skip_steps + 1:
dot_products = np.sum(dr.shift(-skip_steps)*dr, axis=1)
norm_products = dr_norms[skip_steps:]*dr_norms[:-skip_steps]
turns = np.arccos(dot_products.iloc[1:-skip_steps]/norm_products[1:])
summary.loc[i, 'Max. Turn Over {} Steps'.format(skip_steps + 1)] = \
max(turns)
summary.loc[i, 'Turn Time'] = track.loc[turns.idxmax(), 'Time']
cross_product = np.cross(dr.shift(-skip_steps).loc[turns.idxmax()],
dr.loc[turns.idxmax()])
normal_vec = cross_product/np.linalg.norm(cross_product)
summary.loc[i, 'Skew Lines Distance'] = abs(np.sum(
(positions.shift(-skip_steps).loc[turns.idxmax()] - \
positions.loc[turns.idxmax()])*normal_vec))
hull = ConvexHull(track[['X', 'Y', 'Z']])
summary.loc[i, 'Scan. Area/Step'] = hull.area/len(track)
summary.loc[i, 'Scan. Vol./Step'] = hull.volume/len(track)
if 'Surface Area (µm2)' in track.columns:
summary.loc[i, 'Mean Surface Area (µm2)'] = track['Surface Area (µm2)'].mean()
if 'Volume (µm3)' in track.columns:
summary.loc[i, 'Mean Volume (µm3)'] = track['Volume (µm3)'].mean()
if 'Surface Area (µm2)' in track.columns and 'Volume (µm3)' in track.columns:
summary.loc[i, 'Mean Sphericity'] = (np.pi**(1/3) \
*(6*track['Volume (µm3)'])**(2/3)/track['Surface Area (µm2)']).mean()
for cond, cond_summary in summary.groupby('Condition'):
print(' {} tracks in {} with {} timesteps in total.'.format(
cond_summary.__len__(), cond,
tracks[tracks['Condition'] == cond].__len__()))
return summary
def plot_tracks(raw_tracks, summary=None, draw_turns=True, n_tracks=25,
condition='Condition', context='notebook', save=False):
"""Plot tracks"""
tracks = raw_tracks.copy()
_uniquize_tracks(tracks)
if type(summary) == pd.core.frame.DataFrame:
skip_steps = int(next(word
for column in summary.columns
for word in column.split() if word.isdigit()))
if summary is not None and draw_turns:
alpha = 0.33
else:
alpha = 1
if condition not in tracks.columns:
tracks[condition] = 'Default'
n_conditions = len(tracks[condition].unique())
sns.set(style='ticks', context=context)
fig = plt.figure(figsize=(12,12))
if 'Z' in tracks.columns:
ax = fig.add_subplot(111, projection='3d')
else:
ax = fig.add_subplot(111, aspect='equal')
labels = []
for i, (cond, cond_tracks) in enumerate(tracks.groupby(condition)):
if summary is not None and draw_turns:
cond_summary = summary[summary[condition] == cond]
max_turn_column = next(column for column in summary.columns
if column.startswith('Max. Turn'))
if len(cond_tracks['Track_ID'].unique()) > n_tracks/n_conditions:
choice = cond_summary.sort_values(max_turn_column, ascending=False)\
['Track_ID'][:int(n_tracks/n_conditions)]
cond_tracks = cond_tracks[cond_tracks['Track_ID'].isin(choice)]
elif len(cond_tracks['Track_ID'].unique()) > n_tracks/n_conditions:
choice = np.random.choice(cond_tracks['Track_ID'].unique(),
n_tracks/n_conditions, replace=False)
cond_tracks = cond_tracks[cond_tracks['Track_ID'].isin(choice)]
color = sns.color_palette(n_colors=i+1)[-1]
for j, (_, track) in enumerate(cond_tracks.groupby(track_identifiers(cond_tracks))):
labels.append(cond)
track_id = track['Track_ID'].iloc[0]
if 'Z' in tracks.columns:
ax.plot(track['X'].values, track['Y'].values, track['Z'].values,
color=color, alpha=alpha, label=track_id, picker=5)
else:
ax.plot(track['X'].values, track['Y'].values,
color=color, alpha=alpha, label=track_id, picker=5)
if summary is not None and draw_turns:
turn_time = cond_summary[cond_summary['Track_ID'] == track_id]['Turn Time']
turn_loc = track.index.get_loc(
track[np.isclose(track['Time'], turn_time.values[0])].index.values[0])
turn_times = track['Time'][turn_loc - 1:turn_loc + skip_steps]
turn = track[track['Time'].isin(turn_times)]
if 'Z' in tracks.columns:
ax.plot(turn['X'].values, turn['Y'].values, turn['Z'].values,
color=color)
else:
ax.plot(turn['X'].values, turn['Y'].values, color=color)
def on_pick(event):
track_id = event.artist.get_label()
if summary is not None:
print(summary[summary['Track_ID'] == float(track_id)]
[['Track_ID', 'Condition', 'Mean Velocity', 'Track Duration']])
else:
print('Track_ID: ' + track_id)
fig.canvas.mpl_connect('pick_event', on_pick)
if 'Z' in tracks.columns:
equalize_axis3d(ax)
else:
sns.despine()
handles, _ = ax.get_legend_handles_labels()
unique_entries = OrderedDict(zip(labels, handles))
ax.legend(unique_entries.values(), unique_entries.keys())
plt.tight_layout()
if save:
conditions = [cond.replace('= ', '')
for cond in tracks[condition].unique()]
plt.savefig('Tracks' + '-'.join(conditions) + '.png', dpi=300)
else:
plt.show()
def analyze(raw_tracks, uniform_timesteps=True, min_length=6, jump_threshold=None,
verbose=True):
"""Return dataframe with velocity, turning angle & plane angle"""
print('\nAnalyzing tracks')
tracks = raw_tracks.copy()
if 'Time' not in tracks.columns:
print(' Warning: no time given, using index!')
tracks['Time'] = tracks.index
if not tracks.index.is_unique: # For inplace analysis!
tracks.reset_index(drop=True, inplace=True)
else:
_uniquize_tracks(tracks, verbose)
if uniform_timesteps:
_split_at_skip(tracks, jump_threshold, verbose)
if not verbose and 'Orig. Track_ID' in tracks.columns:
print(' Warning: Some tracks were split, use verbose=True for more info.')
n_i = tracks.Track_ID.unique().size
for criterium, track in tracks.groupby(track_identifiers(tracks)):
if len(track) < min_length:
tracks.drop(track.index, inplace=True)
if verbose:
print(' Warning: Delete track {} with {} timesteps.'
.format(criterium, len(track)))
else:
tracks.loc[track.index, 'Track Time'] = \
(track['Time'] - track['Time'].iloc[0]).round(4)
if 'Z' in track.columns:
positions = track[['X', 'Y', 'Z']]
else:
positions = track[['X', 'Y']].copy()
positions['Z'] = 0
tracks.loc[track.index, 'Displacement'] = \
np.linalg.norm(positions - positions.iloc[0], axis=1)
dr = positions.diff()
dr_norms = np.linalg.norm(dr, axis=1)
tracks.loc[track.index, 'Velocity'] = dr_norms/track['Time'].diff()
dot_products = np.sum(dr.shift(-1)*dr, axis=1)
norm_products = dr_norms[1:]*dr_norms[:-1]
tracks.loc[track.index, 'Turning Angle'] = \
np.arccos(dot_products[:-1]/norm_products)
tracks.loc[track.index, 'Plane Angle'] = np.nan
n_vectors = np.cross(dr, dr.shift())
n_norms = np.linalg.norm(n_vectors, axis=1)
dot_products = np.sum(n_vectors[1:]*n_vectors[:-1], axis=1)
norm_products = n_norms[1:]*n_norms[:-1]
angles = np.arccos(dot_products/norm_products)
cross_products = np.cross(n_vectors[1:], n_vectors[:-1])
cross_dot_dr = np.sum(cross_products[2:]*dr.as_matrix()[2:-1],
axis=1)
cross_norms = np.linalg.norm(cross_products[2:], axis=1)
signs = cross_dot_dr/cross_norms/dr_norms[2:-1]
if 'Z' in track.columns:
tracks.loc[track.index[2:-1], 'Plane Angle'] = signs*angles[2:]
else:
tracks.loc[track.index[2:-1], 'Plane Angle'] = angles[2:]
n_f = tracks.Track_ID.unique().size
if not verbose and n_f != n_i:
print(' Warning: Some tracks were deleted, use verbose=True for more info.')
return tracks
