Life of a cell Cells live in complex environments and can sense many different signals: Physical parameters Physical parameters Biological signaling molecules Biological signaling molecules Nutrients or harmful chemicals Nutrients or harmful chemicals Internal state of the cell Internal state of the cell Cell response is producing appropriate proteins that act on the internal or external environment
Transcription factors Cells use transcription factors to represent environmental states. Designed to switch rapidly between active & inactive. Regulate the rate of transcription of genes: Change the probability per unit time that RNAp binds to the promoter and creates an mRNA molecule. Change the probability per unit time that RNAp binds to the promoter and creates an mRNA molecule. Can be activators or repressors.
Transcription network Transcription factors are encoded by genes, which are regulated by transcription factors, which are regulated by transcription factors … Transcription networks describe all the regulatory transcription interactions in a cell
Nodes: genes Directed edges: transcriptional regulation Sign on edged: activation or repression Network input: environmental signals
Input function - activator Input function – strength of the effect of a t.f on the transcription rate of target gene. Hill function: Logical function:
Input function - repressor Hill function: Logical function:
Multi dimensional input functions All activators present: At least one activator present: Non Boolean:
Dynamics and response time Single edge in a network: Production of Y is balanced by protein degradation and dilution: Change in concentration of Y: Steady state:
Detecting network motifs Looking for meaningful network patterns with statistical significance. Network Motif – Patterns that occur in the real network significantly more often than in randomized network. Idea: these patterns have been preserved over evolutionary timescale against mutations that randomly change edges.
Erdos-Renyi random networks Same number of nodes and edges. Directed edges assigned at random. N nodes N 2 possible edges. Probability edge position is occupied:
Autoregulation – A network motif Autoregulation – regulation of a gene by its own product. Graph: a self edge. Example E.coli graph has 40 self edges, 34 of them are repressors (negative autoregulation). Is that significant?
Autoregulation – the statistics What is the probability of having k self edges in an ER network? One self edge:P self =1/N k self edges:
Statistics – cont. In our E. coli network: N=424, E=519 Difference in STD units:
Why negative autoregulation? Dynamics of X: At early times: Steady state:
Negative Autoregulation Response time: Evolutionary selection on β and K
Negative auto vs. simple Mathematically controlled comparison Best of both worlds: rapid production and desired steady state
Robustness to production fluctuations Production rate β fluctuates over time. Twin cells differ in production rate of all proteins in O(1) up to O(10). Repression threshold K is more fixed. Simple regulation is affected strongly by β: Negative autoregulation is not:
Sub graphs in ER networks Probability edge position is occupied: P=E/N 2 Occurrences of sub graph G(n,g) in an ER network: Mean connectivity:λ=E/N
Three-node patterns There are 13 possible sub-graphs with 3 nodes Feed forward loop XY Z Feedback loop XY Z
Feed-Forward is a network motif The feed-forward loop (FFL) is a strong motif. The only motif of the 13 possible 3-node patterns 3 node feedback Feed forward loop 042 E. Coli 0.6± ±1.3 (Z=31) ER networks 0.2±0.6 7±5 (Z=7) Degree preserving random nets
C1-FFL with AND logic
C1-FFL equations For transcription factor Y: For gene Z:
C1-FFL as a delay element Consider the response to 2 steps of signal S x : ON step – S x is absent and then appears. ON step – S x is absent and then appears. OFF step – S x is present and then disappears. OFF step – S x is present and then disappears. Assumption: S Y is always present.
Delay following ON step ON step Production of Y* accumulation of Y* Y*threshold Production of Z
C1-FFL + AND graphs
C1-FFL + OR logic - Example Sign-sensitive delay in the OFF step: X* can activate gene Z by itself, but both X* and Y* have to fall below their K Z levels for the activation to stop. Allows maintaining expression even if signal momentarily lost.
I1-FFL Two parallel but opposing paths: the direct path activates Z and the other represses Z. Z shows high expression when X* is bound and low expression when Y* is bound. Use: pulse generator & fast response time.
I1-FFL equations Accumulation of Y: For gene Z: X*, Y*
I1-FFL equations – cont. Y* represses ZZ production at β’ z
I1-FFL response time Half of steady state is reached during the fast stage: F – repression coefficient. The larger the coefficient (the stronger the repression) the shorter the response time.
I1-FFL - example Galactose system in E. coli Low expression of Gal genes when Glu present. When both are absent Gal genes have low but significant expression (“getting ready”). When Gal appears – full expression of Gal genes
Other FFL types The other 6 types of FFL are rare in transcription networks. Some of the lack responsiveness to one of the signals. Example: I4-FFL
I4-FFL vs. I1-FFL Z st – I4 Z st – I1 SYSYSYSY SxSxSxSx low β’ z /a z high β z /a z 01 low β’ z /a z 11