Stem Cell Reprogramming Colin McDonnell 1 Introduction Since the demonstration of adult cell reprogramming by Yamanaka in 2006, there has been immense research interest in modelling and understanding the complicated un differentiation process. The addition of four transcription factors, or nucleic acid encoding the same, can fully un differentiate a mature adult somatic cell in a matter of days. These un differentiated cells can then be reprogrammed to a new state, providing a source of individualized pluripotent stem cells for use in autologous cell replacement therapies, which have the potential to treat or cure Alzheimer's disease, Parkinson's disease, cardiovascular disease, and diabetes, among others. Here, we study a degenerate case involving just two proteins X 1 and X 2, representing an arbitrary pair of the four Yamanaka factors. Each positively autoregulates itself and negatively regulates the other, as shown in Figure 1. Figure 1: Each protein activates itself and represses the other. Without positive autoregulation, this circuit reduces to a standard bi stable toggle switch. We develop a model that demonstrates both tri stability within certain parameter ranges and the existence of a weakly stable un differentiated state.
2 Tri stable ODE Model The behavior of the system is modelled using standard ODEs as follows. 2.1 Parameters We see that X 1 has some basal expression a 01, a positive autoregulation term with coefficient a 1, a X 2 dependent repression term with coefficient b 1, and a degradation term with constant δ 1. Similar terms exist for the ODE governing X 2 expression. The autoregulation and repression terms follow a Hill model with dissociation constants of K 1 and K 2 and cooperativity constants of n and m. These are all the parameters of the system. 2.2 Conditions for tri stability We now examine the qualitative parameter ranges over which this system demonstrates tristable behavior. For now, we assume zero basal expression ( a 01 = a 02 = 0 ) and a fully symmetric system, that is a 1 = a 2, b 1 = b 2, etc. We therefore eliminate the subscripts for simplicity. a The positive autoregulation constant a is related to the x 3 behavior of the system required for 5 equilibria to emerge. As a approaches zero, the system degenerates to the standard bi stable genetic toggle switch, which is incapable of tristable behavior. Thus a should be the same order of magnitude as b to avoid bi stability. b K n = m Increasing b actually increases the stability of the un differentiated state. As it gets much larger than a however, there is a tipping point where instantaneously goes unstable and bi stable behavior emerges. The dissociation constants have the opposite effect of a. Increasing K acts as a damper on the x 3 behavior, thus it should be small relative to a. Though any increase in K can be counteracted by an appropriate increase in a, the absolute values of the equilibria in terms of X 1 and concentrations shift outwards. Decreasing the cooperativity exponent damps x 3 behavior and, if sufficiently large, results in a non linear change from tri stability to mono stability. Increasing cooperativity also results in X 2
nonlinear behavior, even resulting in tetra stability for sufficiently large values. There is a happy medium around 2 where tristable behavior can exist. δ The degradation coefficient is inherently a damping term. At large values the system is mono stable. As the value decreases, the system becomes briefly bi stable before the onset of tri stability. After this onset, increasing δ increases the stability of. 2.3 Parameter selections After some trial and error, we settled on a set of parameter values that resulted in tri stable behavior while maximizing ease of implementation and analysis. The results of nullcline stability analysis are shown in Figure 2. a = 2 b =. 5 K = 1 n = m = 2 δ = 1 Figure 2: Nullclines for given parameter values. Note the 5 equilibria, three of which are stable. 3 Noise mediated Differentiation un differentiated cells frequently differentiate due to stochastic noise in the system during the creation of induced PSCs. Thus, to make this model realistic, the un differentiated state (the equilibrium point that falls on the line y = x above) should be less stable than the differentiated states S 1 and S 2. The parameter set above does not fulfill this criterion, as a transient perturbation in X 1 of over 1 nm is required to differentiate from to S 1, though a perturbation of approximately. 4nM is sufficient to revert back to. 3.1 Decreasing Stability To rectify this, we define a new set of parameters with a shallower potential well for. We changed only a.
a = 1.6 b =. 5 K = 1 n = m = 2 δ = 1 Figure 3: Nullclines for new parameter values. Note that the perturbation 5 equilibria, three of which are stable. Note that the necessary perturbation to differentiate is now much smaller, and the stability of differentiated states is far greater, requiring a perturbation of up to. 8nM. The state occurs at concentrations X 1 = X 2 = 1.1, S 1 occurs at [ X 1 = 1.65, X 2 = 0.2], and S 2 occurs at [ X 1 = 0.2, X 2 = 1.65]. By simulating the system with these parameters and introducing Gaussian white noise, we were able to see spontaneous differentiation occur. Here are plots resulting from the simulation with a intensity I 0 = 1 and standard deviation σ = 2. Figure 4: Time series data of and according to the model with the described X 1 X 2 parameters and the introduction of randomly generated Gaussian noise.
We see that, from an initial condition of, the system spontaneously differentiates into S 1 or S 2 within tens of hours, after which is reliably stays at the differentiated equilibrium for over 100 hours. 4 Reprogramming Strategies Now that the parameterized model is sufficiently reflective of reality, we begin looking into strategies for reverting differentiated cells back to reliably and easily. We propose two strategies. 4.1 Tuning cooperativity It is possible to consistents un differentiate a cell in state or by transiently decreasing the cooperativity of the positive autoregulation. These exponents are proportional to the number of ligand binding sites on the X i promoter available for transcription factor. Thus, this value can be changed by designing a transcription factor N that binds to the domain but activates protein expression to a lesser degree than a bound X 1 protein. In our lumped model, this would decrease the net cooperativity, as there are fewer available sites for X 1 to bind to and activate its own promoter. By delivering a transient pulse of N that degrades over time, we can temporarily change the cooperativity then have it return to it s default value of 2. We show the results of a nullcline stability analysis of the system for different values of cooperativity. S 1 S 2 Figure 5: Nullclines for cooperativity of 1.67 (red) and 1.87 (green). Notice how the red nullcline is more linear than the green, eliminating the possibility of tri stability. At 1.87, we see the green nullcline is tangent to itself: the onset of tri stability in the system.
As the system moves from n = 1.67 through n = 1.86 and back to the natural value n = 2, the operating point of the system returns back to before noise induced fluctuations again cause differentiation to occur. The behavior of the system as it experienced this transient change in cooperativity is shown below. Figure 6: The time series data showing the concentrations of X 1 and X 2. The system was initialized to S 1 and experienced a transient pulse of neutral transcription factor N that decreased the effective cooperativity n eff to 1.5 at t = 100 hrs. As N was degraded and diluted, the deviation n died off exponentially with half life of As the system underwent a transient decrease in n eff 7 hours. n i eff, note how the system was briefly restored to for approximately 24 hours before redifferentiating. In the first two trials, the system returned to its original equilibrium, and in the third trial the system switched states. 4.2 Transient overexpression The above methodology required protein engineering, which is notoriously difficult. To make the process of reprogramming more easily implemented, we describe an alternative method involving transient overexpression of X 1 and X 2. This can be implemented by introducing mrna that encodes these proteins. Over time the mrna will be degraded and diluted, but not before being many rounds of translation. This results in an exponentially decaying transient pulse of protein to the system. In our model, this is represented as a transient increase in our basal production rates a 0 = a 01 = a 02.
We have conducted nullcline analysis on the system at different levels of basal expression. a 0 Figure 7: Nullclines of the system at different basal production rates. At a 0 =. 1, the system is monostable, thus, for sufficiently large basal expression, the operating point of the system will migrate back towards an un differentiated state. At a 0 =. 05, we see the onset of tri stability, as before in the cooperativity case. At a basal rate of zero, we return to the original nullclines of our system. We consider a transient pulse of mrna delivered at time t = 100 that translates into a spike in a 0 of. 1 that dies off with half life of 34 hours. Below are the results of a simulation without noise. Figure 8: un differentiation through transient overexpression of X 1 and X 2. The system becomes temporarily monostable. We also simulated the system with Gaussian noise, again with intensity standard deviation σ = 2. I 0 = 1 and
Figure 9: un differentiation through transient overexpression of X 1 and X 2. The system becomes temporarily monostable, resulting in un differentiation in spite of the strong stochastic noise. This demonstrates the feasibility of reverting a cell back to through transient overexpression that can be implemented via any standard transfection mechanism. 5 Stability with Feedback One problem remains to be solved. Though we have successfully reverted the cells to an un differentiated state, they will still spontaneously re differentiate from stochastic fluctuations in concentration. An additional mechanism must be put in place to enhance the stability properties of relative to and. There are many mechanisms that can accomplish this without dramatically changing the absolute values of. We consider two: genome editing with Crispr and high gain feedback. 5.1 Genome editing There are myriad ways to use genome editing with Crispr to change the parameters such that enjoys stronger stability properties. As we mentioned in Section 2.2, increasing or increases the stability of the un differentiated state. We can use a b Crispr to identify the and promoters, after which we can excise and insert X 1 X 2 sections of DNA. We can modify the and parameters by inserting stronger a b ligand binding sites to increase the strength and duration of the positive autoregulation and repression effects. S 1 S 2
Alternatively, we can insert a strand of DNA into the genome that constitutively produces X 1 and X 2. We conducted nullcline analysis for a basal expression rate of 0.4nM/hr. Recall from the previous section that. 5nM/hr is the onset of tri stability, and as the basal expression rate decreases, gets progressively less stable. Figure 10: Increasing the stability of through Crispr mediated constitutive basal expression of X 1 and X 2. Note that large perturbations are required to knock the system from to a differentiated state. All these genomic changes can be reversed or mitigated through another application of Crispr designed to excise the inserted DNA and return the genome to its unmodified state. However, this is difficult to implement with high fidelity because Crispr has poor success rates and may have unanticipated effects on the cells viability or ability to re differentiate. 5.2 High gain Feedback We can implement a high gain feedback loop to amplify the stability of. Using an understood mechanism such as phosphorylation as a source of gain in the system, we can implement a feedback loop as shown in Figure 11. Figure 11: High gain feedback loop. G represents a gain in the system, implemented transcriptional activation or phosphorylation processes.
The output y of this system is the concentration of X i. By using a sufficiently high 1 gain, we can have the output concentration approach K and experience arbitrarily small impact from noise and other effects. By tuning K, we can implement the system such that is the feedback loop s desired value. Any perturbation from this value will result in a large corrective effect, conferring strong stability properties to the un differentiated cells. 6 Conclusion We have described here a realistic two dimensional model describing the process of cell differentiation and re programming. Based on a Hill model of repression and positive autoregulation, we derived a system of ODEs that could be tuned to demonstrate spontaneous differentiation, as seen in true mammalian cells. We then describe two strategies for reverting these cells to an un differentiated state: by decreasing cooperativity and by transiently overexpressing the two proteins X 1 and X 2. We proceed to describe strategies of stabilizing this un differentiated state through genome editing and through the introduction of a high gain negative feedback loop. We hope this model can be generalized to the four dimensions indicated by the Yamanaka factors and shed light on the complicated processes at play during the creation of induced pluripotent stem cells.