tutorial_10-Modeling_parameters_optimization


Experimental data assessment and model parameters optimisation

Data preparation

The first step to generate three-dimensional (3D) models of a specific genomic regions is to filter columns with low counts and with no diagonal count in order to remove outliers or problematic columns from the interaction matrix. The particles associated with the filtered columns will be modelled, but will have no experimental data applied.

Here we load the data previous data already normalised.


In [1]:
from pytadbit import load_chromosome
from pytadbit.parsers.hic_parser import load_hic_data_from_bam

In [2]:
crm = load_chromosome('results/fragment/chr3.tdb')

In [3]:
B, PSC = crm.experiments

In [4]:
B, PSC


Out[4]:
(Experiment mouse_B (resolution: 100 kb, TADs: 96, Hi-C rows: 1601, normalized: None),
 Experiment mouse_PSC (resolution: 100 kb, TADs: 118, Hi-C rows: 1601, normalized: None))

Load raw data matrices, and normalized matrices


In [5]:
base_path = 'results/fragment/{0}_both/03_filtering/valid_reads12_{0}.bam'
bias_path = 'results/fragment/{0}_both/04_normalizing/biases_{0}_both_{1}kb.biases'
reso = 100000
chrname = 'chr3'
cel1 = 'mouse_B'
cel2 = 'mouse_PSC'

In [6]:
hic_data1 = load_hic_data_from_bam(base_path.format(cel1),
                                   resolution=reso,
                                   region='chr3',
                                   biases=bias_path.format(cel1, reso / 1000),
                                   ncpus=8)
hic_data2 = load_hic_data_from_bam(base_path.format(cel2),
                                   resolution=reso,
                                   region='chr3',
                                   biases=bias_path.format(cel2, reso / 1000),
                                   ncpus=8)


  (Matrix size 1601x1601)                                                      [2018-10-16 16:31:07]

  - Parsing BAM (101 chunks)                                                   [2018-10-16 16:31:07]
     .......... .......... .......... .......... ..........     50/101
     .......... .......... .......... .......... ..........    100/101
     .                                                         101/101

  - Getting matrices                                                           [2018-10-16 16:31:25]
     .......... .......... .......... .......... ..........     50/101
     .......... .......... .......... .......... ..........    100/101
     .                                                         101/101


  (Matrix size 1601x1601)                                                      [2018-10-16 16:31:38]

  - Parsing BAM (101 chunks)                                                   [2018-10-16 16:31:38]
     .......... .......... .......... .......... ..........     50/101
     .......... .......... .......... .......... ..........    100/101
     .                                                         101/101

  - Getting matrices                                                           [2018-10-16 16:31:56]
     .......... .......... .......... .......... ..........     50/101
     .......... .......... .......... .......... ..........    100/101
     .                                                         101/101


In [7]:
B.load_hic_data([hic_data1.get_matrix(focus='chr3')])
B.load_norm_data([hic_data1.get_matrix(focus='chr3', normalized=True)])

PSC.load_hic_data([hic_data2.get_matrix(focus='chr3')])
PSC.load_norm_data([hic_data2.get_matrix(focus='chr3', normalized=True)])

It is a good practice to check that the data is there:


In [8]:
crm.visualize(['mouse_B', 'mouse_PSC'], normalized=True, paint_tads=True)


Focus on the genomic region to model.


In [9]:
crm.visualize(['mouse_B', 'mouse_PSC'], normalized=True, paint_tads=True, focus=(300, 360))


Data modellability assessment via MMP score

We can use the Matrix Modeling Potential (MMP) score (Trussart M. et al. Nature Communication, 2017) to identify a priori whether the interaction matrices have the potential of being use for modeling. The MMP score ranges from 0 to 1 and combines three different measures: the contribution of the significant eigenvectors, the skewness and the kurtosis of the distribution of Z-scores.


In [7]:
from pytadbit.utils.three_dim_stats import mmp_score

In [9]:
mmp_score(hic_data1.get_matrix(focus='chr3:30000000-36000000'), savefig='images/mmp_score.png')


Out[9]:
(0.8049955427707568, 0.788681592624057, 0.6737285903820441, 0.9006937517389604)

Data Transformation and scoring function

This step is automatically done in TADbit. A a weight is generated for each pair of interactions proportional to their interaction count as in formula:

$$weight(I, J) = \frac{\sum^N_{i=0}{\sum^N_{j=0}{(matrix(i, j))}}}{\sum^N_{i=0}{(matrix(i, J))} \times \sum^N_{j=0}{(matrix(I, j))}}$$

The raw data are then multiplied by this weight. In the case that multiple experiments are used, the weighted interaction values are normalised using a factor (default set as 1) in order to compare between experiments. Then, a Z-score of the off-diagonal normalised/weighted interaction is calculated as in formula:

$$zscore(I, J) = \frac{log_{10}(weight(I, J) \times matrix(I, J)) - mean(log_{10}(weight \times matrix))}{stddev(log_{10}(weight \times matrix))}$$

The Z-scores are then transformed to distance restraints. To define the type of restraints between each pair of particles. we need to identified empirically three optimal parameters (i) a maximal distance between two non-interacting particles (maxdist), (ii) a lower-bound cutoff to define particles that do not interact frequently (lowfreq) and (iii) an upper-bound cutoff to define particles that do interact frequently (upfreq). In TADbit this is done via a grid search approach.

The following picture shows the different component of the scoring funtion that is optimised during the Monte Carlo simulated annealing sampling protocol. Two consecutive particles are spatially restrained by a harmonic oscillator with an equilibrium distance that corresponds to the sum of their radii. Non-consecutive particles with contact frequencies above the upper-bound cutoff are restrained by a harmonic oscillator at an equilibrium distance, while those below the lower-bound cutoff are maintained further than an equilibrium distance by a lower bound harmonic oscillator.

Optimization of parameters

We need to identified empirically (via a grid-search optimisation) the optimal parameters for the mdoelling procedure:

  • maxdist: maximal distance assosiated two interacting particles.
  • upfreq: to define particles that do interact frequently (defines attraction)
  • lowfreq: to define particles that do not interact frequently ( defines repulsion)
  • dcutoff: the definition of "contact" in units of bead diameter. Value of 2 means that a contact will occur when 2 beads are closer than 2 times their diameter. This will be used to compare 3D models with Hi-C interaction maps.

Pairs of beads interacting less than lowfreq (left dashed line) are penalized if they are closer than their assigned minimum distance (Harmonic lower bound). Pairs of beads interacting more than ufreq (right dashed line) are penalized if they are further apart than their assigned maximum distance (Harmonic upper bound). Pairs of beads which interaction fall in between lowfreq and upfreq are not penalized except if they are neighbours (Harmonic)

In the parameter optimization step we are going to give a set of ranges for the different search parameters. For each possible combination TADbit will produce a set of models.

In each individual model we consider that two beads are in contact if their distance in 3D space is lower than the specified distance cutoff. TADbit builds a cumulative contact map for each set of models as shown in the schema below. The contact map is then compared with the Hi-C interaction experiment by means of a Spearman correlation coefficient. The sets having higher correlation coefficients are those that best represents the original data.


In [30]:
opt_B = B.optimal_imp_parameters(start=300, end=360, n_models=40, n_keep=20, n_cpus=8, 
                                   upfreq_range=(0, 0.6, 0.3),
                                   lowfreq_range=(-0.9, 0, 0.3),
                                   maxdist_range=(1000, 2000, 500), 
                                   dcutoff_range=[2, 3, 4])


Optimizing 61 particles
  num scale	kbending	maxdist	lowfreq	upfreq	dcutoff	correlation
  1   0.01 	0       	1000   	-0.9   	0     	4      0.1699
  1   0.01 	0       	1000   	-0.9   	0     	3      0.5399
  1   0.01 	0       	1000   	-0.9   	0     	2      0.8703
  2   0.01 	0       	1000   	-0.9   	0.3   	4      0.3357
  2   0.01 	0       	1000   	-0.9   	0.3   	3      0.6376
  2   0.01 	0       	1000   	-0.9   	0.3   	2      0.8724
  3   0.01 	0       	1000   	-0.9   	0.6   	4      0.5638
  3   0.01 	0       	1000   	-0.9   	0.6   	3      0.735
  3   0.01 	0       	1000   	-0.9   	0.6   	2      0.8341
  4   0.01 	0       	1000   	-0.6   	0     	4      0.1719
  4   0.01 	0       	1000   	-0.6   	0     	3      0.5417
  4   0.01 	0       	1000   	-0.6   	0     	2      0.87
  5   0.01 	0       	1000   	-0.6   	0.3   	4      0.3357
  5   0.01 	0       	1000   	-0.6   	0.3   	3      0.6375
  5   0.01 	0       	1000   	-0.6   	0.3   	2      0.8716
  6   0.01 	0       	1000   	-0.6   	0.6   	4      0.5647
  6   0.01 	0       	1000   	-0.6   	0.6   	3      0.7367
  6   0.01 	0       	1000   	-0.6   	0.6   	2      0.8347
  7   0.01 	0       	1000   	-0.3   	0     	4      0.1816
  7   0.01 	0       	1000   	-0.3   	0     	3      0.546
  7   0.01 	0       	1000   	-0.3   	0     	2      0.8663
  8   0.01 	0       	1000   	-0.3   	0.3   	4      0.3644
  8   0.01 	0       	1000   	-0.3   	0.3   	3      0.6365
  8   0.01 	0       	1000   	-0.3   	0.3   	2      0.8657
  9   0.01 	0       	1000   	-0.3   	0.6   	4      0.56
  9   0.01 	0       	1000   	-0.3   	0.6   	3      0.7333
  9   0.01 	0       	1000   	-0.3   	0.6   	2      0.8327
  10  0.01 	0       	1000   	0      	0     	4      0.1641
  10  0.01 	0       	1000   	0      	0     	3      0.538
  10  0.01 	0       	1000   	0      	0     	2      0.8642
  11  0.01 	0       	1000   	0      	0.3   	4      0.3614
  11  0.01 	0       	1000   	0      	0.3   	3      0.6382
  11  0.01 	0       	1000   	0      	0.3   	2      0.8704
  12  0.01 	0       	1000   	0      	0.6   	4      0.5564
  12  0.01 	0       	1000   	0      	0.6   	3      0.7318
  12  0.01 	0       	1000   	0      	0.6   	2      0.8283
  13  0.01 	0       	1500   	-0.9   	0     	4      0.2911
  13  0.01 	0       	1500   	-0.9   	0     	3      0.6079
  13  0.01 	0       	1500   	-0.9   	0     	2      0.8816
  14  0.01 	0       	1500   	-0.9   	0.3   	4      0.4308
  14  0.01 	0       	1500   	-0.9   	0.3   	3      0.6946
  14  0.01 	0       	1500   	-0.9   	0.3   	2      0.8797
  15  0.01 	0       	1500   	-0.9   	0.6   	4      0.6034
  15  0.01 	0       	1500   	-0.9   	0.6   	3      0.7661
  15  0.01 	0       	1500   	-0.9   	0.6   	2      0.8226
  16  0.01 	0       	1500   	-0.6   	0     	4      0.3003
  16  0.01 	0       	1500   	-0.6   	0     	3      0.6033
  16  0.01 	0       	1500   	-0.6   	0     	2      0.8842
  17  0.01 	0       	1500   	-0.6   	0.3   	4      0.4306
  17  0.01 	0       	1500   	-0.6   	0.3   	3      0.7004
  17  0.01 	0       	1500   	-0.6   	0.3   	2      0.8773
  18  0.01 	0       	1500   	-0.6   	0.6   	4      0.5964
  18  0.01 	0       	1500   	-0.6   	0.6   	3      0.7633
  18  0.01 	0       	1500   	-0.6   	0.6   	2      0.8305
  19  0.01 	0       	1500   	-0.3   	0     	4      0.3134
  19  0.01 	0       	1500   	-0.3   	0     	3      0.6251
  19  0.01 	0       	1500   	-0.3   	0     	2      0.8817
  20  0.01 	0       	1500   	-0.3   	0.3   	4      0.4292
  20  0.01 	0       	1500   	-0.3   	0.3   	3      0.6976
  20  0.01 	0       	1500   	-0.3   	0.3   	2      0.882
  21  0.01 	0       	1500   	-0.3   	0.6   	4      0.6027
  21  0.01 	0       	1500   	-0.3   	0.6   	3      0.7641
  21  0.01 	0       	1500   	-0.3   	0.6   	2      0.831
  22  0.01 	0       	1500   	0      	0     	4      0.3061
  22  0.01 	0       	1500   	0      	0     	3      0.6216
  22  0.01 	0       	1500   	0      	0     	2      0.886
  23  0.01 	0       	1500   	0      	0.3   	4      0.4257
  23  0.01 	0       	1500   	0      	0.3   	3      0.6979
  23  0.01 	0       	1500   	0      	0.3   	2      0.881
  24  0.01 	0       	1500   	0      	0.6   	4      0.6047
  24  0.01 	0       	1500   	0      	0.6   	3      0.7652
  24  0.01 	0       	1500   	0      	0.6   	2      0.826
  25  0.01 	0       	2000   	-0.9   	0     	4      0.4056
  25  0.01 	0       	2000   	-0.9   	0     	3      0.701
  25  0.01 	0       	2000   	-0.9   	0     	2      0.8931
  26  0.01 	0       	2000   	-0.9   	0.3   	4      0.5013
  26  0.01 	0       	2000   	-0.9   	0.3   	3      0.7637
  26  0.01 	0       	2000   	-0.9   	0.3   	2      0.8874
  27  0.01 	0       	2000   	-0.9   	0.6   	4      0.6337
  27  0.01 	0       	2000   	-0.9   	0.6   	3      0.7854
  27  0.01 	0       	2000   	-0.9   	0.6   	2      0.8334
  28  0.01 	0       	2000   	-0.6   	0     	4      0.4141
  28  0.01 	0       	2000   	-0.6   	0     	3      0.6998
  28  0.01 	0       	2000   	-0.6   	0     	2      0.8956
  29  0.01 	0       	2000   	-0.6   	0.3   	4      0.5109
  29  0.01 	0       	2000   	-0.6   	0.3   	3      0.7621
  29  0.01 	0       	2000   	-0.6   	0.3   	2      0.8843
  30  0.01 	0       	2000   	-0.6   	0.6   	4      0.6316
  30  0.01 	0       	2000   	-0.6   	0.6   	3      0.7835
  30  0.01 	0       	2000   	-0.6   	0.6   	2      0.8361
  31  0.01 	0       	2000   	-0.3   	0     	4      0.4018
  31  0.01 	0       	2000   	-0.3   	0     	3      0.6995
  31  0.01 	0       	2000   	-0.3   	0     	2      0.8917
  32  0.01 	0       	2000   	-0.3   	0.3   	4      0.5099
  32  0.01 	0       	2000   	-0.3   	0.3   	3      0.7648
  32  0.01 	0       	2000   	-0.3   	0.3   	2      0.8847
  33  0.01 	0       	2000   	-0.3   	0.6   	4      0.6404
  33  0.01 	0       	2000   	-0.3   	0.6   	3      0.7922
  33  0.01 	0       	2000   	-0.3   	0.6   	2      0.8407
  34  0.01 	0       	2000   	0      	0     	4      0.3912
  34  0.01 	0       	2000   	0      	0     	3      0.6963
  34  0.01 	0       	2000   	0      	0     	2      0.8929
  35  0.01 	0       	2000   	0      	0.3   	4      0.5051
  35  0.01 	0       	2000   	0      	0.3   	3      0.7564
  35  0.01 	0       	2000   	0      	0.3   	2      0.8858
  36  0.01 	0       	2000   	0      	0.6   	4      0.6377
  36  0.01 	0       	2000   	0      	0.6   	3      0.7839
  36  0.01 	0       	2000   	0      	0.6   	2      0.8392

In [31]:
opt_B.plot_2d(show_best=10)


Refine optimization in a small region:


In [32]:
opt_B.run_grid_search(upfreq_range=(0, 0.3, 0.3), lowfreq_range=(-0.9, -0.3, 0.3),
                       maxdist_range=[1750], 
                       dcutoff_range=[2, 3],
                       n_cpus=8)


Optimizing 61 particles
  num scale	kbending	maxdist	lowfreq	upfreq	dcutoff	correlation
  1   0.01 	0       	1750   	-0.9   	0     	3      0.6576
  1   0.01 	0       	1750   	-0.9   	0     	2      0.8908
  2   0.01 	0       	1750   	-0.9   	0.3   	3      0.7223
  2   0.01 	0       	1750   	-0.9   	0.3   	2      0.8791
  3   0.01 	0       	1750   	-0.6   	0     	3      0.6603
  3   0.01 	0       	1750   	-0.6   	0     	2      0.889
  4   0.01 	0       	1750   	-0.6   	0.3   	3      0.7223
  4   0.01 	0       	1750   	-0.6   	0.3   	2      0.8827
  5   0.01 	0       	1750   	-0.3   	0     	3      0.6543
  5   0.01 	0       	1750   	-0.3   	0     	2      0.8909
  6   0.01 	0       	1750   	-0.3   	0.3   	3      0.731
  6   0.01 	0       	1750   	-0.3   	0.3   	2      0.8823

In [33]:
opt_B.plot_2d(show_best=5)



In [34]:
opt_B.run_grid_search(upfreq_range=(0, 0.3, 0.3), lowfreq_range=(-0.3, 0, 0.1),
                       maxdist_range=[2000, 2250], 
                       dcutoff_range=[2],
                       n_cpus=8)


  xx   	0.01 	0       	2000   	-0.3   	0     	4      	0.4018
  xx   	0.01 	0       	2000   	-0.3   	0.3   	2      	0.8847
Optimizing 61 particles
  num scale	kbending	maxdist	lowfreq	upfreq	dcutoff	correlation
  1   0.01 	0       	2000   	-0.2   	0     	2      0.8933
  2   0.01 	0       	2000   	-0.2   	0.3   	2      0.8842
  3   0.01 	0       	2000   	-0.1   	0     	2      0.8945
  4   0.01 	0       	2000   	-0.1   	0.3   	2      0.8866
  xx   	0.01 	0       	2000   	0      	0     	4      	0.3912
  xx   	0.01 	0       	2000   	0      	0.3   	2      	0.8858
  5   0.01 	0       	2250   	-0.3   	0     	2      0.8944
  6   0.01 	0       	2250   	-0.3   	0.3   	2      0.886
  7   0.01 	0       	2250   	-0.2   	0     	2      0.8952
  8   0.01 	0       	2250   	-0.2   	0.3   	2      0.8876
  9   0.01 	0       	2250   	-0.1   	0     	2      0.8954
  10  0.01 	0       	2250   	-0.1   	0.3   	2      0.8881
  11  0.01 	0       	2250   	0      	0     	2      0.895
  12  0.01 	0       	2250   	0      	0.3   	2      0.8855

In [35]:
opt_B.plot_2d(show_best=5)



In [36]:
opt_B.run_grid_search(upfreq_range=(0, 0.3, 0.1), lowfreq_range=(-0.3, 0, 0.1),
                       n_cpus=8,
                       maxdist_range=[2000, 2250], 
                       dcutoff_range=[2])


  xx   	0.01 	0       	2000   	-0.3   	0     	4      	0.4018
Optimizing 61 particles
  num scale	kbending	maxdist	lowfreq	upfreq	dcutoff	correlation
  1   0.01 	0       	2000   	-0.3   	0.1   	2      0.8905
  2   0.01 	0       	2000   	-0.3   	0.2   	2      0.8877
  xx   	0.01 	0       	2000   	-0.3   	0.3   	2      	0.8847
  xx   	0.01 	0       	2000   	-0.2   	0     	2      	0.8933
  3   0.01 	0       	2000   	-0.2   	0.1   	2      0.8857
  4   0.01 	0       	2000   	-0.2   	0.2   	2      0.8864
  xx   	0.01 	0       	2000   	-0.2   	0.3   	2      	0.8842
  xx   	0.01 	0       	2000   	-0.1   	0     	2      	0.8945
  5   0.01 	0       	2000   	-0.1   	0.1   	2      0.8892
  6   0.01 	0       	2000   	-0.1   	0.2   	2      0.8854
  xx   	0.01 	0       	2000   	-0.1   	0.3   	2      	0.8866
  xx   	0.01 	0       	2000   	0      	0     	4      	0.3912
  7   0.01 	0       	2000   	0      	0.1   	2      0.8895
  8   0.01 	0       	2000   	0      	0.2   	2      0.8846
  xx   	0.01 	0       	2000   	0      	0.3   	2      	0.8858
  xx   	0.01 	0       	2250   	-0.3   	0     	2      	0.8944
  9   0.01 	0       	2250   	-0.3   	0.1   	2      0.8941
  10  0.01 	0       	2250   	-0.3   	0.2   	2      0.8876
  xx   	0.01 	0       	2250   	-0.3   	0.3   	2      	0.886
  xx   	0.01 	0       	2250   	-0.2   	0     	2      	0.8952
  11  0.01 	0       	2250   	-0.2   	0.1   	2      0.8917
  12  0.01 	0       	2250   	-0.2   	0.2   	2      0.8893
  xx   	0.01 	0       	2250   	-0.2   	0.3   	2      	0.8876
  xx   	0.01 	0       	2250   	-0.1   	0     	2      	0.8954
  13  0.01 	0       	2250   	-0.1   	0.1   	2      0.8974
  14  0.01 	0       	2250   	-0.1   	0.2   	2      0.8855
  xx   	0.01 	0       	2250   	-0.1   	0.3   	2      	0.8881
  xx   	0.01 	0       	2250   	0      	0     	2      	0.895
  15  0.01 	0       	2250   	0      	0.1   	2      0.8948
  16  0.01 	0       	2250   	0      	0.2   	2      0.8876
  xx   	0.01 	0       	2250   	0      	0.3   	2      	0.8855

In [37]:
opt_B.plot_2d(show_best=5)



In [38]:
opt_B.get_best_parameters_dict()


Out[38]:
{'dcutoff': 2.0,
 'kbending': 0.0,
 'kforce': 5,
 'lowfreq': -0.1,
 'maxdist': 2250.0,
 'reference': '',
 'scale': 0.01,
 'upfreq': 0.1}

For the other replicate, we can reduce the space of search:


In [39]:
opt_PSC = PSC.optimal_imp_parameters(start=300, end=360, n_models=40, n_keep=20, n_cpus=8, 
                                     upfreq_range=(0, 0.3, 0.1),
                                     lowfreq_range=(-0.3, -0.1, 0.1),
                                     maxdist_range=(2000, 2250, 250), 
                                     dcutoff_range=[2])


Optimizing 61 particles
  num scale	kbending	maxdist	lowfreq	upfreq	dcutoff	correlation
  1   0.01 	0       	2000   	-0.3   	0     	2      0.925
  2   0.01 	0       	2000   	-0.3   	0.1   	2      0.9242
  3   0.01 	0       	2000   	-0.3   	0.2   	2      0.9199
  4   0.01 	0       	2000   	-0.3   	0.3   	2      0.9119
  5   0.01 	0       	2000   	-0.2   	0     	2      0.9243
  6   0.01 	0       	2000   	-0.2   	0.1   	2      0.924
  7   0.01 	0       	2000   	-0.2   	0.2   	2      0.9191
  8   0.01 	0       	2000   	-0.2   	0.3   	2      0.9122
  9   0.01 	0       	2000   	-0.1   	0     	2      0.9255
  10  0.01 	0       	2000   	-0.1   	0.1   	2      0.9231
  11  0.01 	0       	2000   	-0.1   	0.2   	2      0.9194
  12  0.01 	0       	2000   	-0.1   	0.3   	2      0.911
  13  0.01 	0       	2250   	-0.3   	0     	2      0.9274
  14  0.01 	0       	2250   	-0.3   	0.1   	2      0.9239
  15  0.01 	0       	2250   	-0.3   	0.2   	2      0.918
  16  0.01 	0       	2250   	-0.3   	0.3   	2      0.9126
  17  0.01 	0       	2250   	-0.2   	0     	2      0.9262
  18  0.01 	0       	2250   	-0.2   	0.1   	2      0.9239
  19  0.01 	0       	2250   	-0.2   	0.2   	2      0.9172
  20  0.01 	0       	2250   	-0.2   	0.3   	2      0.9122
  21  0.01 	0       	2250   	-0.1   	0     	2      0.9287
  22  0.01 	0       	2250   	-0.1   	0.1   	2      0.9243
  23  0.01 	0       	2250   	-0.1   	0.2   	2      0.9172
  24  0.01 	0       	2250   	-0.1   	0.3   	2      0.9144

In [40]:
opt_PSC.plot_2d(show_best=5)



In [41]:
opt_PSC.get_best_parameters_dict()


Out[41]:
{'dcutoff': 2.0,
 'kbending': 0.0,
 'kforce': 5,
 'lowfreq': -0.1,
 'maxdist': 2250.0,
 'reference': '',
 'scale': 0.01,
 'upfreq': 0.0}