controllability and observability

Controllability [Control Bootcamp]

http://databookuw.com/page-2/page-4/

 

http://databookuw.com/databook.pdf

Controllability gramian. 

Adjacency matrix in Yuan for directed graphs

Adjacency matrix in Yuan exact controllability paper use column_node -> row_node, which is a mirror of the conventional row_node -> column_node.  These mirrored adjacency matrix for direct graphs do not change controllability analysis using the matrix based method, it seems to me.

controllability and mirror effect of adjacency matrix

using the matrix eigen value genometric multiplicity, the adjacentcy matric matrix can be mirrored, and its control nodes should not be changed, based on my intuitive understanding.

multiplicity of eigen value

The number of linearly independent eigenvectors associated with a given eigen value $lambda$ is called the geometric multiplicity of eigenvalue $lambda$.

tissue specific gene regulation

How does these affect aging, controllability? Some tissue aging fast and some slow?

http://www.cell.com/cell-reports/abstract/S2211-1247%2817%2931418-3#.We-APbV2phM.facebook

Highlights

• •Regulatory network connections are more tissue specific than nodes (genes and transcription factors)
• •Tissue-specific function is not solely regulated by transcription factor expression
• •Tissue-specific genes assume bottleneck positions in their corresponding networks
• •Tissue specificity is driven by context-dependent, non-canonical regulatory paths

data for network changes during aging

Protein biogenesis machinery is a driver of replicative aging in yeast https://elifesciences.org/articles/08527 . This paper gives a 'directed graph' that can be used for controllability analysis. 

CYCLOP: rapamycin data

controversies on driver node identificaiton

In a PONE paper on driver node in cancer networks, a reader commented on different ways to identify driver nodes.

http://journals.plos.org/plosone/article/comment?id=10.1371/annotation/fa7b59e2-c5b0-4e34-b2bc-da5d4eef0ee5


Nacher J, Akutsu T (2012) Dominating scale-free networks with variable scaling exponent: heterogeneous networks are not difficult to control - Abstract - New Journal of Physics - IOPscience. New Journal of Physics 14.

There more recent papers on controllability outside of Liu and Barabasi group. 

Controllability analysis, yeast PPI

Liu11 code input format

TextWrangler replace ',' in pairs.csv to white space, save as 'pair.txt'.

applejack:run.ppi hqin$ Parse pairs.txt 

pairs.txt

# N= 4478, E= 13667

In total it takes 0.20 s

applejack:run.ppi hqin$ head pairs.txt.nodemap 

0 YMR047C

1 YGR218W

2 YOR128C

3 YCR067C

4 YLL028W

5 YHR188C

6 YMR292W

7 YDR414C

8 YNL078W

9 YHR187W

applejack:run.ppi hqin$ ControllabilityAnalysis pairs.txt > _typscript.txt 

_typescript.txt

Read Standard Elist:

N= 4478

E= 13667

<k> = 6.10406

Number of driver nodes: Nd= 2295

Control profile (Ns, Ne, Ni)=(1321,0,974)

Nd= 2295

Type-I node classification: (Ncn, Nrn, Non)=(460,1961,2057)

Type-II node classification: (Ncn, Nrn, Non)=(1321,1526,1631)

Link classification: (Ncl, Nrl, Nol)=(796,6079,6792)

Clustering Coefficient: <LCC>= 0.0806148; GCC=0.0604306; <LCC>_d=0.0403074; GCC_d=0.0302153

0 72348.1

1 53025.5

2 5971.48

3 10

4 3242.97

5 705.874

6 0

7 7391.21

8 2264.2

... ... 

... ... 

<Betweeness Centrality> = 1085.74<Closeness Centrality> = 1085.74

<Hub Centrality> = 0.00255414

<Authority Centrality> = 0.00232173

In total it takes 4.87 s.

applejack:run.ppi hqin$ ll

total 1328

-rw-r--r--  1 hqin  staff    42K Feb 15 06:51 _typscript.txt

lrwxr-xr-x  1 hqin  staff    17B Feb 15 06:48 pairs.txt -> ../data/pairs.txt

-rw-r--r--  1 hqin  staff   107K Feb 15 06:49 pairs.txt.elist.b

-rw-r--r--  1 hqin  staff   117K Feb 15 06:49 pairs.txt.elist.t

-rw-r--r--  1 hqin  staff   240K Feb 15 06:51 pairs.txt.linktype

-rw-r--r--  1 hqin  staff    56K Feb 15 06:49 pairs.txt.nodemap

-rw-r--r--  1 hqin  staff    80K Feb 15 06:51 pairs.txt.nodetype

-rw-r--r--  1 hqin  staff   213B Feb 15 06:51 pairs.txt.output

applejack:run.ppi hqin$ wc -l pairs.txt.linktype 

   13668 pairs.txt.linktype

applejack:run.ppi hqin$ head pairs.txt.linktype 

#source  target  LinkType 

YMR047C YGR218W 1

YOR128C YCR067C 2

YLL028W YHR188C 1

Q: Liu11 treat PPI as directed graph. How did PNAS16 paper deal with this problem? 

Reference
Liu 2011 Nature.

*** Control systems engineering, control theory, Laplace transform, observability,

A control system has an input, a process, and an output. It can be open loop or closed loop. Open loop systems do not monitor or correct the output. Closed loop systems can monitor output and make adjustments.

linear time-invariant differential equation

Transfer function is another way of mathematically modeling a system.  Transfer function can be derived from the linear, time-invariant differential equation using Laplace transform. Transfer function can only be used for linear systems. (Lapalace transformation was developed as a technique to solve differential equations).

State-space representation is another model for systems and is suitable for non-linear systems.
Essentially, state-space model change nth-order differential equation into n simultaneous first-order equations. It seems to me that the state-space model is the mostly used ODE modeling methods in systems biology.

Test signals with different waveforms can be used to study systems.

The basic analysis of a system is to evaluate the time response of a system.

A sensitivity analysis can yield the percentage of change in a specification as a function of a change in a system parameter.

In biology, many ODEs has nonlinear terms with product of variables. So, transfer function cannot be applied, but state-space method can be used.

Controllability and Observability are well understood in continuous time-invariant linear state-space model, see https://en.wikipedia.org/wiki/State-space_representation#State_variables 

Stability: a system is stable if every bounded input yields a bounded output. So, does aging changes a stable gene network into an unstable network?

Observability: If the initial state vector x(t0) can be found from input u(t) and output y(t) over a finite interval of time from t0, the system is observable; otherwise it is unobservable. 
Observability is the ability to deduce state variables from knowledge of input u(t) and output y(t).