are cells dynamically critical

Stuart A. Kauffman Nature, Max Planck Dec. 2006

Are cells dynamically critical

Stuart A. Kauffman, iCore [email protected]

Institute for Biocomplexity and Informatics, University of Calgary, CanadaDepartment of Physics & Astronomy of the University of Calgary, Canada

University of Calgary 2500 University Drive NW, T2N 1N4

CANADAhttp://www.ibi.ucalgary.ca/


1. Cancer stem cells have been discovered in breast, prostate, skin, blood, brain and other tumors.

2. Cancer stem cells are capable of persistent self renewal and differentiating into other cells of limited proliferation potential in the tumor.

3. Recent evidence suggests that specific subsets of genes are abnormally expressed in cancer stem cells.

4. Aim of cancer stem cell therapy is to kill, stop the proliferation of, or causedifferentiation of cancer stem cells.

5. IBI lab is focusing on high throughput and specific siRNA, small molecules, and expression vector library screening.

Cancer Stem Cell Therapy


Genetic Regulatory Networks

Transcriptome Yeast regulates 6500 genes

Transcriptome in Humans regulates about 25000 genes

Transcriptome plus Protein signaling network is a parallel processing non linear stochastic dynamical system.

Systems Biology seeks the integrated behavior of this system within and between cells.

Systems Biology also seeks possible general laws.

A possible general law is that cells are dynamically critical.


Boolean networks as models of GRN’s


Boolean Networks basins of attraction

All basins of attraction belong to the same network realization with K=2 and N=15.


Ordered, Critical and Chaotic Behaviour

Order ChaosCritical


Derrida Curve and criticality

(000) --- > (001)

dT=1/3 dT+1=2/3

(100) --- > (100)

Derrida curves of RBN's with

k=1,2,3,4,5. Analytical results.

Normalized Hamming distance


Reasons why cells “should” be critical

.Cells must bind reliable past discriminations to future reliable actions.

. In the order regime convergence in state space forgets past distinctions.

.In the chaotic regime small noise yields divergence in state space trajectories precluding reliable action.

.Critical regime with near parallel flow optimizes capacity to bind past and future.


Reasons why cells “should” be critical Critical Cells maximize correlated behaviour of genes over time. Hence can carry out most complex coordinated behaviour


Reasons why cells “should” be critical

. Critical Boolean Networks maximize robustness to mutations.

. Numerical experiments: add random gene, connected at random, to the existing network, with random logic.

.Hypothesis: Cell types correspond to attractors

.Results: critical networks maximize the probability that such mutations leave all existing attractors intact and occasionally add new attractors.

. Thus, critical networks optimize robustness of cell types to mutations and the capacity to evolve new cell types.


Capacity to evolve to criticality

Selection on Boolean Networks to play mismatch and random games converged from chaotic and ordered regimes to critical behaviour

Selection for capacity to coordinate classification of time varying signal converged to critical regime.


Evidence that cells are critical

. Reka Albert Boolean model of early Drosophila development is critical by derrida criteria.

. Elena Alvarez-Buylla model of floral development in Arabidopsis is critical by derrida criteria.

.



Known e Coli. Network is critical, with random Boolean functions (1500 genes).



Known Yeast. Network is critical, with random Boolean functions (3500 genes).



Hela 48 hourly time point gene array data.

(Lempel-Ziv analysis)


Evidence that cells are critical: Avalanche size distribution

250 Yeast deletion mutants and the distribution of number of genes that alter activity is a power law with slope -1.5 (characteristic of critical networks).

Dotted line: power law with slope -1.5.Solid line: data from Hughes et. al.

Logarithmic binning. A gene is considered differentially expressed if its expression has changed more than 4 fold.

n - avalanche size

P(Z=n) - number of times avalanche size was observed divided by the number of observations.


Boolean D iscretization

A B

Ternary D iscretization

Boolean Netw orks Ternary Netw orks

C D

Evidence that cells are critical: normalized compression distance (NCD) analysis

A : binary macrophage data.

B : ternary macrophage data

C : RBNs with K = 1,2,3,4

D : ternary nets with K = 1, 1.5, 2, 3, 4

(for ternary nets, K=1.5 is critical)

Analysis of Toll-like receptors by gene array


Serra stuff

Chi-square distance between experimental data and theoretical prediction, concerning 6 more frequent avalanches

Comparison between experimental data and analytical calculus of 6 more frequent avalanches (theta = 7, lambda = 6/7)


More Serra stuff


1. Cancer stem cells have been discovered in breast, prostate, skin, blood, brain and other tumors.

2. Cancer stem cells are capable of persistent self renewal and differentiating into other cells of limited proliferation potential in the tumor.

3. Recent evidence suggests that specific subsets of genes are abnormally expressed in cancer stem cells.

4. Aim of cancer stem cell therapy is to kill, stop the proliferation of, or causedifferentiation of cancer stem cells.

5. IBI lab is focusing on high throughput and specific siRNA, small molecules, and expression vector library screening.

Cancer Stem Cell Therapy


• Approaches to discovering structure and logic of genetic regulatory nets.

1. ChIP-Chip.2. Inference of transcription factor binding

sites.3. Inference of structure and logic from time

series gene expression data.4. Promoter bashing. 5. Data base integration.


Basins of attraction

A. Wuensche. Basins of attraction in network dynamics:A conceptual framework for biomolecular networks. In G. Schlosser and G. Wagner, editors, Modularity in Development and Evolution. (in press) University of Chicago Press, Chicago, 2002.


1) Generate network and Boolean functions2) Generate a random initial state3) Generate a path of states (affected by noise)4) Infer the network with pairwise MI and DPI5) Apply post-inference engine6) Results Analysis

Memory requirements:

N (number genes), R (number runs), k (connectivity) Path of States ~ O(N.R)K functions ~ (N.2k)Memory usage before inference engine ~ O(N2+N.R+N.2k)Adjacency Matrix ~ O(N2) Inference engine: O(2.S.N2+N2)

Limits: 50.000 genes, 20 inputs/gene.

IADGRN


Mutual Information Threshold

.With N genes we will compute about N^2 pair wise mutual information's.

.Given M samples, we require an MI threshold (Bickel, Samuelsson and Andrecut) such that on the order of 1 false positive will be found.

Pairwise Mutual Information distribution of randomly generated binary time series

Increasing number of nodes, 100 experiments per data point.


Mean Mutual Information and Inference ability

Expected Pairwise Mutual Information of Random Boolean Functions Given Different Input Distributions

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5 6 7 8 9 10

Regular

Poisson

Exponential

MI distribution of all Boolean Functions as a function of k

1E-154

1E-143

1E-132

1E-121

1E-110

1E-99

1E-88

1E-77

1E-66

1E-55

1E-44

1E-33

1E-22

1E-11

1

k1

k2

k3

k4

k5

k6

k7

k8

k9


Inferring from random Boolean networks time series with random Boolean functions and monotonic functions,

for increasing connectivity

30.000 nodes Boolean networks, 1000 independent state transitions, 100 experiments per data point.


Inferring from random Boolean networks time series with monotonic functions, for increasing experimental

noise in measuring genes expression level

30.000 nodes Boolean networks, 1000 independent state transitions, 100 experiments per data point.


Inferring from random Boolean networks continuous time series with Monotonic and random functions, for increasing k

30.000 nodes Boolean networks, 1000 continuous state transitions, 100 experiments per data point.

Boolean functions were inferred at 90%, independent of k.


Predicting inferability

. Yeast Network, 3459 genes, exponential input distribution: Medusa network.

. Using 600 independent state transitions and mutual information threshold we predicted that we could infer 33% of the regulatory connections and in fact predicted 34% with no false positives.

. Future: Inferring Stochastic Genetic Networks with array noise.

. Long term aim is to use gene expression time series from real cells and be able to estimate inferability of network’s structure and logic.

. Inferring personalized structure and logic of cancer stem cell aberrant circuitry for therapy.

are cells dynamically critical

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