Manual Reconstruction from integral data

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Mathematical reconstruction of the material-density distribution using integral albedo data

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  • Regulatory network reconstruction using an integral additive model with flexible kernel functions.
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Contact the seller - opens in a new window or tab and request post to your location. Postage cost can't be calculated. Please enter a valid postcode. There are 2 items available. Please enter a number less than or equal to 2. Select a valid country. Please enter up to 7 characters for the postcode. Domestic dispatch time. Will usually dispatch within 5 working days of receiving cleared payment - opens in a new window or tab. Nevertheless, we found that this algorithm performs reasonably well in many cases, particularly for relatively sparse networks. We compared performances of the differential and integral inference models using various artificial systems producing simulated data and three experimental datasets from [ 16 ].

As available experimental datasets are typically limited in size, we explored models where the number of free fit parameters was small. In each case we had one free parameter per link. This also equalizes the degrees of freedom in the compared differential and integral inference models. The delta-function model described in the Methods section was not applied because all tested systems demonstrate behavior continuous in time. To appreciate how our predictions are far from random, we also applied the integral model with the zero-degree polynomial kernel to infer network structures from the permuted data, i.

In the first set of experiments the model used for network inference was that used for data generation. Artificial regulatory networks were generated with random and scale-free topologies. For random topology, any two nodes are connected with the probability p independent from the other connections.

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We used the growing network with redirection algorithm [ 19 ] to generate networks with scale-free topology. The number of nodes in the generated networks was 20; the probability p for the random networks was equal to 0. We demonstrate examples of networks undergoing random topology Fig. A set of first-order ordinary differential equations 1 was used to simulate time series.

The parameters w ij were randomly generated from the uniform distribution in the interval [-1;1]. The background levels b i were set to zero and the initial states y i t 0 were set to 1 for all nodes. Examples of node artificial networks. We used the fourth-order Runge-Kutta formula [ 20 ] to numerically solve differential equations 1.

The solution was built on time points uniformly spaced over the interval [0;10]. The resulting time series were sampled to produce 20 time points to approach the quality of experimental data. We split the original point time series into 20 intervals of 50 points. At each interval the output time point was randomly selected. This led to a time series with non-homogeneous random time intervals between subsequent measurements.

Each of 20 intensity values was statistically distorted. The distorted value was generated as a Gaussian random variable with the mean equal to the true value and standard deviation proportional to the true value. The coefficient of proportionality — noise-to-signal level — was set to 0.

Regulatory network reconstruction using an integral additive model with flexible kernel functions

As time series were simulated using a set of first-order ordinary differential equations, the corresponding inference model is either the differential model Eq. Although the single exponential kernel may also be used in this case, it is clearly non-adequate and therefore it was not tested. We reconstructed the networks from the generated time series using the forward selection procedure.

Other possible performance measures, such as negative predictive value or specificity, are not relevant for sparse networks when the forward selection procedure is used for reconstruction.

Reconstruction from Integral Data

During first iterations of the fitting procedure the number of TN largely exceeds the number of TP leveling the difference between reconstruction models. We averaged the dependence of PPV and Se on the total number of links over runs of the simulation procedure. A different network structure, different link parameters, different time sampling and different noise realizations were generated at each run. We used two mathematical models for real biological systems yeast glycolysis [ 11 ] and the MAPK cascade [ 12 ] to test the performance of the developed inference models for more realistic systems.

The network structures and SBML files used for simulations are also available from our web page [ 22 ]. We stress that we used these modules as they were originally developed, i. Mathematical representations and kinetic parameters of the models can be viewed in JDesigner. We used JDesigner to integrate the models on time points spaced uniformly over the interval [0;1] for yeast glycolysis and [0;] for the MAPK cascade.

Two data distorting steps were performed as before: we left 20 time points at random time intervals, and added Gaussian noise with noise-to-signal level equal to 0.

Examples of time series used for the inference are available on our web page [ 22 ]. The resulting curves were averaged over runs of the simulation procedure. The simulation procedure generated different time sampling and different realizations of noise at each run, whereas the network structure, kinetic laws and kinetic parameters remained the same. To demonstrate applicability of the developed approach to real experimental data, we used the yeast Saccharomyces cerevisiae cell cycle microarray time series dataset [ 16 ]. As others did [ 23 - 25 ], we selected a part of the yeast cell cycle pathway available from KEGG [ 26 ] Fig.

Taking into account that fitting models are very approximate, it may not be always reasonable to require perfect fitting quality. Therefore we investigated the performance PPV and Se of the inference models as a function of the number of generated links.

Integral of e^x/x vs. integral of 1/ln(x), non-elementary but special!

As for the artificial systems, we compared here performances of the differential and integral inference models. We also applied DBN approach to infer network structures from the experimental datasets.

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We used the Banjo software [ 27 ] to perform Bayesian inference. For analysis, we selected the alpha and elu datasets as only these two datasets were measured at equidistant time points. The latter is prerequisite for Banjo. To run Banjo we used the same input settings as given in [ 21 ].

Finally, we performed an additional comparison of the differential and integral inference models based on an independent set of artificial data described in [ 21 ]. Briefly, 20 random gene networks with an average in-degree per gene of 2 were generated. For each network, time-series data time points were simulated using linear ordinary differential equations. Each data point was statistically distorted with noise-to-signal ratio equal to 0. In our analysis we first sampled the point time series to produce point time series, which were then used for network reconstruction.

As the network structures are known, we built the dependencies of PPV and Se on the number of generated links for each network. The obtained dependencies were further averaged over 20 networks. The developed algorithms for the network inference were implemented in the software package NETI, freely available from our web page [ 22 ].

We found that the integral model was superior to the differential model for both scale-free and random topologies, demonstrating higher predictive power and sensitivity. The networks with scale-free topology were reconstructed with greater accuracy i. In this case, the inference procedure needed more links to reproduce the simulated time series. Many of those links were false positives, decreasing the PPV values. The better performance for the scale-free networks can be due to the fact that they had fewer nodes that simultaneously regulated another node.

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Therefore, the fitting procedure has fewer chances to incorrectly select a node as a regulator. Despite the correspondence between data producing models and network inference models, reconstruction was not perfect. There are various reasons for that. Network topology: a random and b scale free. Inference models: differential blue and integral with the zero-degree polynomial kernel red.