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Chrisantha T. Fernando^{1,5}, Anthony M.L. Liekens^{2}, Lewis Bingle^{1}_{,}^{ }Christian Beck^{4}, Thorsten Lenser^{4}, Dov Stekel^{1}, Jon Rowe^{3} ^{1 }Systems Biology Centre, University of Birmingham, Birmingham, B15 2TT, UK ^{2}TU/e Techniche Universiteit Eindhoven, the Netherlands ^{3 }School of Computer Science, University of Birmingham, Birmingham, B15 2TT, UK ^{4 }Bio Systems Analysis Group, Friedrich Schiller University Jena, Germany ^{5}MRC National Institute for Medical Research, Mill Hill, London, London NW7 1AA, UK Supplementary Material 1. ProteinProtein Interactions and Modifications to Hebbian Learning Figure S1. Additional synthetic components improve the robustness of the Hebbian learning circuits. See text. As described in Fritz et al (2007), proteinprotein interactions can be designed to modify synthetic gene circuits. Hebbian learning is known to be inherently unstable, and various techniques are known in neuroscience to deal with runaway positive feedback. In the main paper we have used simple transcription factor decay to prevent runaway positive feedback. However, figure S1 shows a range of optional extras for a synthetic Hebbian circuit. (A) Autocatalytic p and w_{j} transcription factors may permit the maintenance of memory over more generations. The gating of this positive autoregulation by a growthphase dependent sigma factor ensures that weights and outputs do not increase during stationary phase. (B) Subtractive and (C) multiplicative normalization of weights^{8} can be implemented by heterodimer (p + u_{j}) and homodimer (p + p) mediated proteolysis of the transcription factor w_{j}. The later allows competition for limited ‘weight resources’. (C) The BCM rule (Dayan & Abbott, 2001)^{ }can be implemented by using an extra molecule m to represent the sliding average of p or u_{j}, which then combines with u_{j}_{ }and p to degrade weights. 1.1 Presynaptic rule (not shown in Figure S1) …S1 can be implemented simply by adding a constant decay to the weight equal to , that is zeroth order w.r.t. weight. This can be done if u_{j} is a saturated enzyme that degrades w at a constant rate, or if u linearly activates an enzyme that does this job. It should be noted that the presynaptic rule is unstable. For then weights grow without bound, whereas for weights decay to zero. The postsynaptic rule is a minor modification, with decay of weights in proportion to rather than . 1.2. Saturating weight dependence rule (Figure S1B). …S2 works by reducing the increase in weights as weights go to w^{max}, and actually decreasing weights if the weight goes above w^{max}. Here the w_{ij} transcription factor is destroyed at rate in the first order. This can be implemented if a heterodimer acts to degrade w_{ij} directly. 1.3. Oja’s rule (Figure S1C) …S3 has a very similar form, except that here the homodimer v^{2}_{ }is responsible for the first order decay of w_{ij}. See that S3 differs from S2 in that the decay is transmitted to all weights via v even if that particular weight did not cause the production of v, thus implementing competition 1.4. Weight consolidation rule (not shown in figure S1) …S4 ( and ) can be implemented by introducing a autocatalytic promotor for w_{ij} with Hillcoefficient for w_{ij} binding of n = 2. The strength of the autocatalytic promotor of w_{ij} is equal to . At the same time there must be decay of w_{ij} at a first order constant rate and a decay of w_{ij} at a third order rate ! This is a complex process to implement, but it is conceivable that such processes could evolve if selection existed for consolidation of transcription factor cytoplasmic memory. 1.5. BCM rule (Figure S1D) The rule is described by: …S5 is a zero crossing function with positive slope for example v_{i}(v_{i} ). is a shortterm average of output activity and is given for example by …S6 Let us first implement the sliding threshold 10.1. This is simply a TF that is activated by the output v and decays at constant rate n_{v}. The production of w_{ij} is at rate v_{i}^{2}u_{j} and w_{ij} decays at rate vu_{j}. Here, three species must come together to destroy w_{ij}, v, , and u_{j}. This rate must be 0^{th} order, i.e. independent of the concentration of w_{ij}. can be implemented as another molecule m (Figure S1D), that is activated by v with Hill Coefficient = 2, and decays at 1^{st} order with the same time constant as its production. 2. Evolution of Circuits In Silico For the In silico optimization we expanded the SBMLevolver of Thorsten Lenser, which implements a new 2layer evolutionary algorithm. The model in the main paper is a subsequent version optimized by hand, but based on the evolved parameters below. The upper level evolves the structure of the network using genetic programming while the lower level uses a evolution strategy to optimize only the parameter of the network. For the upper level we had a (10+10)elitism strategy over 500 generations, that means in each generation we selected, based on the fitness rank, the 10 best individuals as parents for the next generation and created 10 new offspring. As we have used the network in figure S3 as the starting network and don’t want to mutate this, the offspring are just a copy of the parents. In the lower level we had a (2+2)elitism strategy over 10 generations. After all we received a network with the output you can see in figure 4 in the original paper. The values of the signals  remember they are repressor  is 200 if they are inactive and 105 if they are active. For the parameter of the network remember the equations: …S7 and …S8
Table S1. The parameter of the evolved network. The concentrations at the beginning are 3.942(w1), 0.021 (w2) and 0.0002 (p) To find this learning network we had to define a new fitness function. To make the explanation of the function easier we have to make a definition. We split one experiment into 9 sections:
So we define as the value of protein O at the last timestep in section j in the test case, analogous as value of O in section j in the control case. Then our fitness function looks like this: As the SBMLevolver tries to find the smallest fitness possible, it will reduce all added parts and terms over the slash and increase the values under the slash. The first 4 punishments have no influence in the learning of the network, they should lead to a reasonable network only. So we used the following:
We also added two punishments, which have a influence in the learning of the fitness:
This helps the evolutionary algorithm to move in the space, but will have no influence in the final result, because the premises will not be true. The c_{const} in the main function is for regulation of the strength of the fraction. The higher c the weightier is the main function in relation to the punishments. In our case we used also 100000 for c_{const}. 3. Evolution of circuits in vivo. Figure S2 shows a proposed outline protocol for evolution of Hebbian learning circuits in vivo. There are two phases of testing, the control and the test phase. In the control phase we do not pair the two stimuli and require no learning. In the test phase we pair two stimuli and require the learning of the association. A facs machine is used to select for no GFP expression, and high GFP expression respectively. Simulated evolution in silico was used to design an effective fitness function. Figure S2. The overall protocol for artificial selection of plasmid gene circuits is shown. After circuits have been cloned in E. coli we will undertake high throughput selection using a classical conditioning task, to evolve bacterial colonies capable of robust conditioning. A FACS machine will be used to undertake selection using a 2phase fitness function. 4. Simple model of MAPK implementation The reactions describing the MAPK circuit are shown below: with the corresponding differential equations: …32* Figure S3. U1Copy and U2Copy represent input chemicals. Initially W11 starts at 0.1, and V11 starts at 0. So that U1 initially activates the output, by U2 does not. After pairing of U1 and U2, output P11 is produced just with U2 input alone. This was not the case before pairing. A classical conditioning experiment is shown in figure S3. The Cellerator^{TM} file (Bruce Shapiro) for Mathematica is available on online, and was used to simulate this system. These equations represent mass action kinetics, although more complex models are possible. The basic organization of Hebbian learning can be appreciated. There are two positive feedback loops, coupled by a common output signal. Both output and input must be present for weight increase. Weights and outputs are both double phosphorylated protein kinases. The single phosphorylated and unphosphorylated forms are intermediates in the circuit, without any enzymatic activity of their own. The system implements weight decay because w_{11} and v_{11} is dephosphorylated slowly back to w_{01} and v_{01} respectively. 5. References Dayan P, Abbott LF (2001) Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. (The MIT Press) 