Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Boolean Networks Edda Klipp Humboldt-Universitt Berlin SS 2010 Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin One Network, Different Models gene agene bgene cgene d C A D B A B + + repression activation transcription translation gene protein ab cd Directed graphs V = {a,b,c,d} E = {(a,c,+),(b,c,+), (c,b,-),(c,d,-),(d,b,+)} ab cd Boolean network a(t+1) = a(t) b(t+1) = (not c(t)) and d(t) c(t+1) = a(t) and b(t) d(t+1) = not c(t) ab cd Bayesian network p(xa)p(xa) p(xb)p(xb) p(x c |x a,x b ), p(x d |x c ), Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Simplification of Gene Expression Regulation Gene mRNA Protein Gene mRNA Protein Transcription Factor ABCDEFG Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Boolean Network Boolean network is - a directed graph G(V,E) characterized by - the number of nodes (genes): N - the number of inputs per node (regulatory interactions): k AB C E D F G N=7, k A =0, k B =1, k C =2, in-degrees Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Boolean Logic (George Boole, ) Each gene can assume one of two states: expressed (1) or not expressed (0) Background: Not enough information for more detailed description Increasing complexity and computational effort for more specific models Replacement of continuous functions (e.g. Hill function) by step function Boolean models are discrete (in state and time) and deterministic. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Dynamics of Boolean NetworkS The dynamics are described by rules: if input value/s at time t is/are...., then output value at t+1 is.... AB A(t)B(t+1) Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Boolean Models: Truth functions in output p p not p rule AB B(t+1) = not (A(t)) rule 2 A(t)B(t+1) Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Dynamics of Boolean Networks with k=1 Linear chain ABCD A fixed (no input). Rules 0 and 3 not considered (since independence of input). A(t) B(t+1) B(t+1) C(t+2) C(t+2) D(t+3) The system reaches a steady state after N-1 time steps. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Dynamics of Boolean Networks with k=1 Ring AB CD A B Again: Rules 0 and 3 not considered (since independence of input). A(t+1)=B(t) B(t+1)=A(t) Both rule 1 A BA BA BA B A(t+1)=not B(t) B(t+1)=A(t) Both rule 1 A BA BA BA B Fixpoint or cycle of length 2 depending on initial conditions Cycle of length 4 independent of initial conditions. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Boolean Models: Truth functions k=2 input output p q A C C(t+1) = not (A(t)) and B(t) rule 4 B p=A(t), q=B(t) Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Boolean Network Boolean networks have always a finite number of possible states: 2 N and, therefore, a finite number of state transitions: AB C E D F G N=7, 2 7 states, theoretically possible state transition Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Attractor The trajectory connects the successive states for increasing time. An attractor is a region of a dynamical system's state space that the system can enter but not leave, and which contains no smaller such region (a special trajectory). Fixpoint cycle of length 1 Cycles of length L Basin of attraction: is the surrounding region in state space such that all trajectories starting in that region end up in the attractor. Bifurcation: appearance of a boarder separating two basins of attraction. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Example Network as Boolean Model gene agene bgene cgene d C A D B A B + + repression activation transcription translation gene protein ab cd Boolean network a(t+1) = a(t) b(t+1) = (not c(t)) and d(t) c(t+1) = a(t) and b(t) d(t+1) = not c(t) Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Example Network as Boolean Model ab cd Boolean network a(t+1) = a(t) b(t+1) = (not c(t)) and d(t) c(t+1) = a(t) and b(t) d(t+1) = not c(t) 0000 0000 Steady state: 1010 Cycle: 1000 1001 1101 1111 1010 1000 Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Nave Reconstruction of Boolean Models If it is known -the number of vertices, N, and -the number of inputs per vertex, k, -As well as a sufficient set of successive states, one can reconstruct the network List - List for each vertex all possible input combinations - List all respective outputs Experiments: - Delete after every experiment all wrong entries of the list Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Nave Reconstruction of Boolean Models AB N=2, k=1 Input Output A(A),B(A) A B rule Inout rule Input Output A(B),B(B) A B rule Input Output A(A),B(B) A B rule Input Output A(B),B(A) A B rule AB 1 2 Experimente. InOutA B Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Random Boolean Networks If the rules for updating states are unknown select rules randomly N nodes pN (N-1) edges Rule 2 Rule 0 Rule 1 Rule 2 Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Kauffmans NK Boolean Networks An NK automaton is an autonomous random network of N Boolean logic elements. Each element has K inputs and one output. The signals at inputs and outputs take binary (0 or 1) values. The Boolean elements of the network and the connections between elements are chosen in a random manner. There are no external inputs to the network. The number of elements N is assumed to be large. S.A. Kauffman, 1969, J Theor Biol. Metabolic Stability and Epigenesis in Randomly Constructed Genetic Nets S. A. Kauffman. The Origins of Order: Self-Organization and Selection in Evolution, Oxford University Press, New York, S.A. Kauffman, 2003, PNAS, Random Boolean Network Models and the Yeast Transcriptional Network Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Kauffmans NK Boolean Networks An automaton operates in discrete time. The set of the output signals of the Boolean elements at a given moment of time characterizes a current state of an automaton. During an automaton operation, the sequence of states converges to a cyclic attractor. The states of an attractor can be considered as a "program" of an automaton operation. The number of attractors M and the typical attractor length L are important characteristics of NK automata. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Kauffmans Boolean Network Fundamental question: require metabolic stability and epigenesis the genetic regulatory circuits to be precisely constructed?? Has fortunate evolutionary history selected only nets of highly ordered circuits which alone insure metabolic stability; Or are stability and epigenesis, even in nets of randomly interconnected regulatory circuits, to be expected as the probable consequence of as yet unknown mathematical laws? Are living things more akin to precisely programmed automata selected by evolution, or to randomly assembled automata? Note: cellular differentiation despite identical sets of genes Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Kauffmans Boolean Network Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Further Properties K connections: 2 2 K Boolean input functions Nets are free of external inputs. Once, connections and rules are selected, they remain constant and the time evolution is deterministic. Earlier work by Walker and Ashby (1965): same Boolean functions for all genes: Choice of Boolean function affects length of cycles: and yields short cycles, exclusive or yields cycles of immense length Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Further Properties: Cycles State of the net: Row listing the present value of all N elements (0 or 1) Finite number of states (2 N ) as system passes along a sequence of states from an arbitrary initial state, it must eventually re-enter a state previously passed a cycle Cycle length: number of states on a re-enterant cycle of behavior Cycle of length 1 equilibrial state Transient (or run-in) length: number of state between initial states and entering the cycle Confluent: set of states leading to or being part of a cycle Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Further Properties: Number of Cycles Such a net must contain at least one cycle, it may have more. Their number can be counted just be releasing the net from different initial states No state can diverge on to two different states, no state can be on two different cycles Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Further Properties: Number of Cycles (a) A net of three binary elements, each of which receives inputs from the other two. The Boolean function assigned to each element is shown beside the element. (b) All possible states of the 3-element net are shown in the left 3 x 8 matrix below T. The subsequent state of the net at time T+ 1, shown in the matrix on the right, is derived from the inputs and functions shown in (a). (c) A kimatograph showing the sequence of state transitions leading into a state cycle of length 3. All states lie on one confluent. There are three run-ins to the single state cycle. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Example: Net with N=10 Periodic attractor (yellow) and basin of attraction (cyan) Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Example: Net with N=10 The entire state space of an RBN with 10 nodes. Note: Self connections do not appear so a period-1 attractor appears to have no outputs although each network state must have exactly one output. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Further Properties: Distance Distance compares two states of the net Can be defined as the number of genes with different values in two states. For example N=5: state (00000) and state (00111) differ in the value of three elements This is used as measure of dissimilarity between - subsequent states on a transient - subsequent states on a cycle - cycles Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Totally Connected Nets, K=N Is like random mapping of a finite set of numbers into itself. Expected length of cycle is E.g. net with N=200 states expected cycle length ~ Compare to Hubbels age of the universe: If every transition would take only a second. Such networks are biologically impossible Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin One Connected Nets, K=1 Either one cycle of length N Or a number of disconnected cycles for the full systems state cycles lengths are lowest common multiples of the individual loop lenghts the state cycle length becomes easily very large Again biologically not feasible Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Two Connected Nets, K=2 Kauffman studied networks of N = 15, 50, 64,, 400, 1024,.., 8191 Nets of 1000 elements possess ~ states 16 Boolean functions Study of cycle length (surprisingly short) Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Two Connected Nets, K=2: Cycle Length (a) A histogram of the lengths of state cycles in nets of 400 binary elements which used all 16 Boolean functions of two variables equiprobably. The distribution is skewed toward short cycles. (b) A histogram of the lengths of state cycles in nets of 400 binary elements which used neither tautology nor contradiction, but used the remaining 14 Boolean functions of 2 variables equiprobably. The distribution is skewed toward short cycles. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Two Connected Nets, K=2: Cycle Length Log median cycle length as a function of log N, in nets using all 16 Boolean functions of two inputs (all Boolean functions used), and in nets disallowing these two functions (tautology and contradiction not used). The asymptotic slopes are about 0.3 and 0.6. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin K=2: Transient Lengths A scattergram of run-in length and cycle length in nets of 400 binary elements using neither tautology nor contradiction. Run-in length appears uncorelated with cycle length. A log/log plot was used merely to accommodate the data. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin K=2: Number of Cycles A histogram of the number of cycles per net in nets of 400 elements using neither tautology nor contradiction, but the remaining Boolean functions of two inputs equiprobably. The median is 10 cycles per net. The distribution is skewed toward few cycles. Expected number of cycles: Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Noise One unit of noise may be introduced by arbitrarily changing the value of a single gene for one time moment. The system may return to the cycle perturbed or run into a different cycle. In a net of size N there are just N states which differ from any state in the value of just one gene Consider a net with several cycles: By perturbing all states on each cycle (distance 1) one obtains a matrix listing all cycles and how often they are reached from another one. By dividing all cells by the rows totals transition probabilities The matrix is a Markov chain. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Noise: for the Example ab cd Boolean network a(t+1) = a(t) b(t+1) = (not c(t)) and d(t) c(t+1) = a(t) and b(t) d(t+1) = not c(t) Cycle 0000 Steady state: 0101 Cycle 1010 Cycle: 1000 1001 1101 1111 1010 1000 C1 C2 C1 C2 Transition Matrix Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Noise (a) A matrix listing the 30 cycles of one net and the total number of times one unit of perturbation shifted the net from each cycle to each cycle. The system generally returns to the cycle perturbed. Division of the value in each cell of the matrix by the total of its row yields the matrix of transition probabilities between modes of behavior which constitute a Markov chain. The transition probabilities between cycles may be asymmetric. (b) Transitions between cycles in the net shown in (a). The solid arrows are the most probable transition to a cycle other than the cycle perturbed, the dotted arrows are the second most probable. The remaining transitions are not shown. Cycles 2, 7, 5 and 15 form an ergodic set into which the remaining cycles flow. If all the transitions between cycles are included, the ergodic set of cycles becomes: 1, 2, 3, 5, 6, 12, 13, 15, 16. The remainder are transient cycles leading into this single ergodic set-. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Noise The total number of cycles reached from each cycle after it was perturbed in all possible ways by one unit of noise correlated with the number of cycles in the net being perturbed. The data is from nets using neither tautology nor contradiction, with N = 191, and 400. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Application to Cell Cycle Logarithm of cell replication time in minutes against logarithm of estimated number of genes for various single cell organisms and cell types. Solid lines: connects medium replication times of bacteria, protozoa, chicken, mouse, dog, and man. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Application to Cellular Differentiation The logarithm of the number of cell types is plotted against the logarithm of the estimated number of genes per cell, and the logarithm of the median number of state cycles is plotted against logarithm N. The observed and theoretical slopes are about 0.5. Scale: 2 x lo6 genes per cell = 6 x 10-12g DNA per cell. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Yeast: Cell Cycle Signaling Interaction Overview over the cell cycle network including the connections to the downstream MAPKs of the pheromone and HOG pathways. Solid lines represent direct and indirect biochemical implications, dotted lines denote other empirical observations such as co-expression and implications from cell division. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Rules Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Analysis Results A: Core cell cycle attractor states with basins of attraction. All initial conditions with disabled osmotic stress and pheromone converge on this cyclic attractor. B: Average values of core cell cycle components over time out of 100 simulations, when the system is updated assynchronously. C: Partially linear differential equation simulation of selected species demonstrating the timing of core cell cycle components. D: Comparison with 1000 random Boolean models. Our model is more likely to exhibit the original attractor when perturbed than random networks. However, our significance threshold of p < 0.05 is not reached. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin No Stress Stress Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Differentiation and Reprogramming Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Developmental Potency Each state is characterized by the expression of specific markers. Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Cellular Reprogramming State 1State 2State 3 . Promises: Personalized treatment Development of individual tissues and organs without ESC Personal disease models Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Pluripotency Network Literature Search Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Refined Putative Regulatory Network Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Control of the Pluripotency Network by miRNAs in a Boolean Approach miR145* = not OCT4 miR21* = not SOX2 KLF4* = KLF4 and not miR145 NANOG* = (OCT4 and SOX2 or NANOG or FGF2) and not miR21 OCT4* = (OCT4 and SOX2 or NANOG) and not (miR21 or miR145) SOX2* = (OCT4 and SOX2 or NANOG) and not (BMP4 or miR21 or miR145) FGF2* = FGF2 BMP4* = BMP4 Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin The Boolean Approach miR145* = not OCT4 miR21* = not SOX2 KLF4* = KLF4 and not miR145 NANOG* = (OCT4 and SOX2 or NANOG or FGF2) and not miR21 OCT4* = (OCT4 and SOX2 or NANOG) and not (miR21 or miR145) SOX2* = (OCT4 and SOX2 or NANOG) and not (BMP4 or miR21 or miR145) FGF2* = FGF2 BMP4* = BMP4 Oct4 1 0 Sox2 1 0 Nanog BMP4 0 FGF2 0 1 Klf4 0 miR miR Time t t+1 t+2 t t+1 t+2 t+3 OFF ON Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin The Entire Statespace of the Network Pluripotency related states Green arrows indicate the steady states related to the differentiated states of the cell N=8 nodes 2 8 =256 states =65536 possible state transitions Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Transitions for Differentiation and Reprogramming Possible transitions between states that differ in the expression of 1 gene only Pluripotency related statesDifferentiated states Humboldt- Universitt Zu Berlin Edda Klipp, Humboldt-Universitt zu Berlin Properties and Simulation Results Model jumps to differentiated state if BMP4 is switched on Activin addition leads to differentiated state, but induces a different lineage It is possible to reverse differentiation just by activation of the TGF pathway Stochastic Simulation of Boolean Network