Computational Models of Bar1 Negative Feedback
Part 1: Negative Feedback and Noise Reduction
Background: Noise can be defined as extracellular or intracellular perturbation which affects signal transduction as well as gene expression[1]. A signaling pathway can include mechanisms that reduce noise. Such reduction in biological noise may be necessary for pathway functionality. For example, the variety of T cell receptors (TCR) in a population of T cells generates a wide range of responses to a particular antigen, yet somehow, a very specific, low noise, response is achieved through its biological circuitry [2]. The regulatory mechanisms downstream of TCR include negative feedback which allows stimulation signals to be controlled, guaranteeing appropriate responses to external perturbations. Also, noise is regarded as a quantitative index of variability within a population, which can be shown by the distribution of population response when stimulated. We wonder if adding a negative feedback to the production of extracellular communication factor will reduce population noise.
Circuit Diagram: We have added the Bar1 gene, which is activated by alpha factor in the primary circuit [3]. Bar1 is a protease that cleaves alpha factor in the extracellular zone. So, alpha factor inhibits the accumulation of itself by producing Bar1, which forms a negative feedback for extracellular alpha factor. We can tune the strength of negative feedback with different alpha factor responsive promoters.
Modeling Result: Here is the data plot using a stochastic model based on Chemical Langevin Equation [4]. The data along the whole time period is integrated into histograms on the right. From the results, we find that strains with negative feedback loop have lower noise (Upper panel), while strains without it have higher noise (Lower panel).
Part 2: Bimodal Response
Background: We explored another behavior pattern, bimodal response, which means when stimulated, the whole cell group differentiates into two subgroups - one of them activated while the other one remains in OFF state.
Circuit Design: This is the circuit we designed to achieve bimodal response. Based on the primary secrete-sense circuit, we add a positive feedback, and a degradation factor Bar1. The strength of positive feedback can be tuned by doxycycline concentration, and the strength of degradation factor can be tuned by changing pTEF promoter in front of Bar1, which we characterized experimentally.
Modeling Formula: In order to describe the dynamic of extracellular alpha factor concentration, we build a deterministic model based on this formula.
The first two terms represents generation of alpha factor: the first is the basal synthesis, and the second describes the positive feedback for generating alpha factor. We use sigmoidal form based on our pre-experiment results. The last two terms represents degradation of alpha factor. In this circuit, Bar1 is constitutively expressed, and the degradation of alpha factor by Bar1 is of first-order. The last term is natural degradation.
If we define this formula as the driving force, then the negative value of its integration can be defined as potential.
Modeling Results: Assuming the basal synthesis of alpha factor is zero, we plot the secretion term and degradation term against concentration of alpha factor respectively.
Tuning the parameters to change the strength of positive feedback and degradation term, when extracellular alpha factor concentration stops changing at last, there are three fixed points – among which two are stable and one is unstable judged from Jacobian matrix. By calculating the potential (defined above), we correspond two of fixpoints to the two potential barriers where cells may gather. Thus a bimodal response appears.
Parameter Tuning: In this model, the strength of positive feedback changes by tuning V, while the strength of Bar1 degradation changes by tuning deltaBar1. So, we set a range for these two parameters to see under which condition bimodal response appears. This figure shows tuning deltaBar1, the slope of the degradation line. For different values, the pattern is distinctive. Along with its decrease, the potential difference between the barrier and that of the well decreases, which means cells may have a bigger probability to jump out of the well by stochastic events, causing the bimodal pattern to collapse.
This figure shows the results of tuning V, the highest value of secretion curve. Also, along with its decrease, the trend is similar with changing deltaBar1, resulting in difficulty maintaining bimodal response within a cell population.
Kilobots
What are Kilobots?
Kilobots are small robots designed by the Self-Organizing Research Group at Harvard University to operate in large groups[5]. When programmed, the kilobots can cooperate through local interactions to perform collective behaviors such as swarming and self-assembly. They can flash LED lights, transmit and receive electronic signals to and from each other, and move using two vibration modules. For our purposes, we only need to use the LED light to report robot response.
Why model with Kilobots?
With their simple design, we hope to produce complex collective behavioral models that mimic our yeast circuits. By programming Kilobots to behave as our yeast cells, we hope to produce predicted community responses such as converging or diverging behaviors.
In our models, we utilized their ability to sense and secrete signals to produce community responses. A Kilobot can be in one of five states: NONE*, LOW, MED, HIGH, or MAX, which describe how frequently the cell transmits signals. A Kilobot’s state may vary depending on its program and the states of nearby Kilobots.
A Kilobot’s internal state is reported using its LEDs, which correspond to the following key:
- None - Blue
- Low - Cyan
- Medium - White
- High - Orange
- Maximum - Red
*Kilobots in the NONE state send signals at a very low frequency; this is so that if isolated, the Kilobot will still be able to self-signal.
Circuit Components
i_sense_rand_bar
This code allows Kilobots (yeast cells) to sense and transmit electronic signals (mating factor alpha) that will influence other Kilobots’ states and may influence its own state (self-sensing). The Kilobots are randomized into different initial states, representing different initial constitutive expressions. There are five possible states, which describe the Kilobot’s output: NONE* [blue], LOW [cyan], MED [white], HIGH [orange], and MAX [red]. At each state, the cell lights up in the assigned color and transmits signals [value = 0] at the assigned frequency, NONE being the lowest level of frequency and MAX being highest. If a Kilobot senses enough signals transmitted by other Kilobots, it will go up in stage to the next highest level (so from NONE to LOW to MED to HIGH to MAX), until it reaches the MAX stage, at which point it will stay red. If the Kilobots sense enough signals transmitted by a i_bar1 Kilobot [value =1], it will go down a stage.
i_bar1
This code is meant to mimic yeast cells that produce Bar1, which degrades mating alpha factor and can cause negative feedback to reduce activity in nearby cells. The Kilobot will flash magenta every time it transmits a signal [value = 1], which may inhibit nearby bar1-sensitive Kilobots to reduce signal if exposed for long enough.
i_sense_average_bar_rand
This code allows Kilobots (yeast cells) to sense and transmit electronic signals (mating factor alpha) that will influence other Kilobots’ states and may influence its own state (self-sensing) in order to converge to a medium level of expression. The Kilobots are randomized into different initial states, representing different initial constitutive expressions. There are five possible states, which describe the Kilobot’s output: NONE [blue], LOW [cyan], MED [white], HIGH [orange], and MAX [red]. At each state, the cell lights up in the assigned color and transmits signals [value = 0] at the assigned frequency, NONE being the lowest level of frequency and MAX being highest. If a Kilobot senses enough signals transmitted by other Kilobots, it will go up in stage to the next highest level (so from NONE to LOW to MED). If the Kilobot initially is at a HIGH or MAX state, it will gradually reduce its expression until it matches that of other cells at the MED.
Results
ConvergeHigh
When all the Kilobots were programmed with code 1, they each started at different initial states, but they all reached maximum level of expression very quickly. Note that robots in the center seem to turn red faster while the slowest robot to turn red is in the bottom corner, where it gets less signal due to having fewer adjacent neighbors.
This progression from random to maximum is similar to how a noisy but highly expressing yeast strain would respond to mating factor alpha.
Diverging Bar1
In this clip, seven Kilobots were given program 1 and three were given program 2 (flashing magenta) to mimic Bar1 cells. This produced a diverging response, in which robots further away from the Bar1 Kilobots still reached maximum expression level, but those closer to the Bar1 Kilobots stayed at low levels of expression, eventually settling at a medium level of expression.
This diverging response may be similar to what happens when different strains of yeast with and without bar1 are co-cultured.
RandomtoBimodal
In this clip, half of the Kilobots were programmed with code 1 and the other half were programmed with code 2. The initial state is similar to what happens in the first scenario (ConvergeHigh), in that initial response is random and noisy; however, the robots have clearly differentiated into medium and high levels of expression (white and red, respectively) by the end.
This diverging yet coordinated response may be similar to what happens when two noisy strains of yeast are intermixed.
LowtoBimodal
In this clip, half of the Kilobots were programmed with code 1 and the other half were programmed with code 2, this time with a few modifications such that they all started at the lowest level of expression (blue for NONE). Again, the robots ultimately produced a bimodal response, with half of the robots emitting white light for medium expression and the other half emitting red light for maximum expression.
This diverging response may model what happens when two strains of yeast with the same initial response but different secondary responses are mixed together and induced with mating factor alpha.
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