Introduction
Reconciling Kinetic and Equilibrium Models of Bacterial Tran-
Abstract
The study of transcription remains one of the centerpieces of modern biology, with implications in settings from development to metabolism to evolution to disease. In particular, our understanding of the simplest gene circuit is sufficiently sophisticated, both experimentally and theoretically, that it has become possible to carefully distinguish between different conceptual pictures of the operation of this regulatory system.
Introduction
In this paper, we show how seven different models of this so-called simple repression motif, based on equilibrium and kinetic reasoning, can be used to derive predicted gene expression levels and shed light on the often surprising past success of equilibrium models. For example, do we need to worry about all or even any of the steps that.
Mean Gene Expression
In this simplest model, depicted as (1) in Figure 2.1(B), the promoter is idealized as existing in one of two states, either repressor bound or repressor unbound. We point out that the same generalization can be quite easily incorporated into any of our models in Figure 2.1 by simply rescaling the suppressor copy number 𝑅 in the equilibrium models, or equivalently 𝑘+.
Beyond Means in Gene Expression
One might further guess that this accumulation would lead to super-Poissonian noise in the mRNA distribution over a population of cells. Once in the inactive state, the system remains there for an average time 1/𝑘+ before returning to the active state and repeating the process.
Finding the “right” model: Bayesian parameter inference
To do this for the case of the bursty promoter, we generated negative bionomial-distributed mRNA counts for each of the parameter samples shown in Figure 2.3(A). From our samples of the posterior distribution, plotted in Figure 2.4(A), we generate many replicated data using a random number generator.
Discussion and future work
The surfaces in (A) and the curves in (B) are the nullclines of the state variables, and their intersections, marked in red in (B), are the steady-state solutions of the system. An example of the three scenarios (only cooperative XapR, only cooperative promoter, no cooperativity) can be seen in Figure 3.8. Betancourt, "The prior can often only be understood in the context of the probability", Entropy.
The plots in the main part do not show the magnitude of the vector fields.
Theoretical Investigation of a Genetic Switch for Metabolic Adap-
Abstract
The expression of membrane transporters in metabolic pathways is often upregulated by the transporter substrate. Dynamic systems analysis and stochastic simulations show that membrane transport makes the model system bistable in certain parameter regimes. We find that the negative feedback from the degrading enzyme does not significantly disturb the positive feedback from the membrane transporter.
Fundamentally, this work investigates how a stable genetic switch for a set of enzymes is obtained from transcriptional autoactivation of a membrane transporter by its substrate.
Introduction
A classic example of a genetic switch is a system in which two repressor proteins each regulate the transcription of the other [1, 2] (illustrated schematically in Fig. 3.1). Here, one steady state is high expression of the first protein and low expression of the second, and the second steady state is the opposite. This switch allows the system to have a memory: if something induces expression of one of the proteins, the system will remain in this state until a significant perturbation occurs.
The key feature of the type of system we are investigating is the indirect activation of the.
Experimental motivation
Experimentally, we found that the level of xapABa expression between cells is bimodal and that the system appears to be bistable (see next section). Note that the fluorescence scale of the middle panel is not comparable to the other two and that the xanthosine concentrations chosen are different. Working in the ΔxapAB background, we measured the expression level of our reporter when driven by two constructs: the wild-type xapAB promoter (Figure 3.2B, left panel) and the xapAB promoter with the upper binding site removed (Figure 3.2B, right). plate).
It is clear that removing one
Model
P] denotes the polymerase concentration, and Δ𝐸coop stands for the interaction energy of the two XapR dimers. The parameters are respectively the interaction energy of the two XapR dimers Δ𝐸coop and the dissociation constants𝐾XapR and𝐾P from XapR and polymerase to the promoter. With [m] being the mRNA concentration, 𝑟m the transcription rate, 𝛾m the mRNA decay rate, and 𝑝active the probability that the promoter is in the active state, we get.
Here, 𝑤𝑖 stands for the thermodynamic weight of the ith state in the order in which they are listed in Fig. 3.4.
Results and discussion
In the following, we will discuss some interesting properties of the system that can be observed through phase portraits. The output of the stochastic simulations agrees well with the concentrations at fixed points in the deterministic phase portraits. Therefore, we additionally show the approximate position of the two peaks in the bimodal distribution.
The positions of the two peaks in the bimodal distributions are indicated by smaller points, connected by dashed lines.
Conclusion
This can be converted to the matrix form of the master equation shown in Eq. For our particular case, all the data sets from [8] used in this paper have O (103) data points. In Fig. B.8, the results mentioned in the hysteresis section of the main text are shown.
For both, the mean of the distribution is indicated by a line in the appropriate color.
An efficient representation for statistical mechanics of multi-
Introduction
An exact solution specifies the probability distribution over available states of the system as a function of time, given some initial state, but finding such a solution is rarely possible. In some cases, although an exact calculation of the full distribution may be impossible, an exact analytical calculation of the distribution's moments is possible (eg [5, 6]). For systems with a variable number of (classical) identical particles, one new alternative formulation of a master equation uses Fock spaces, the vector spaces normally associated with quantum field theories, which provide a natural representation of the variable particle number.
Although our method is motivated by these Fock space methods for nonequilibrium problems, and although a treatment of the complete nonequilibrium problem would be the ultimate goal, we have found that even a simpler equilibrium formalism remains full of subtleties and surprises.
Formalism
With a (discrete) internal index ranging from 1 to 𝑁, is an explicit representation of the vacuum state. With the basics of the formalism in hand, we now turn to the construction of the. Then the total free energy of a complex is the sum of the free energies of all its features.
Recall that in constructing the sumi state we counted separately states that differed only in their internal indices.
Derivations for non-bursty promoter models
With these definitions, we can collapse Eq. Simple, albeit tedious, algebra confirms that Eqs. A.11 for the special case of non-equilibrium model 2 in Fig. 2.1, the governing chemical equations for all non-equilibrium models in Fig. 2.1, except for model 5, can be formulated in this form. To begin with, we will find the promoter state probabilities h ®𝑚0i from Eq. A.12 by summing all mRNA copy numbers𝑚, resulting in For non-equilibrium model 1 in Figure 2.1, we have already shown the full master equation in Eq. A.6, but for completeness we reprint it as A.49).
Then we need to calculate ®𝑚iffra Eq. A.71), where we labeled the components of h ®𝑚i as 𝑚𝐸 and 𝑚𝑃 since they are the average mRNA counts conditional on the system residing in the empty or polymerase-bound state, respectively.
Bursty promoter models - generating function solutions and numerics 129
In section 2.4 of the main text we derive this functional form for the burst size distribution. What this implies is that for a geometrically distributed burst size, we have an average burst size of the shape. Let us now look at the general form of the derivative for our generating function in Eq.
Throughout this appendix we will refer to the appropriate notation for probability distributions of the form
Model choices, simplifications and assumptions
Thereby we can obtain the same form of 𝑝active that we work with in the main part by including the interaction between polymerase and XapR by𝜖coop. For much higher intra- than extracellular xanthosin concentrations, the difference in the chemical potential of xanthosin across the membrane can dominate that of the protons and there is a net efflux. The latter therefore does not need to be explicitly included in the kinetic model (it is implicitly part of the turnover rate).
Instead, it can also bind a new proton and substrate and transport them in the opposite direction (steps.
Parameter estimation
To find 𝐾𝜒A, we further assume that the steep part of the induction curve lies in the biologically relevant xanthosine regime, which is estimated experimentally, see the section on [c]aaa above. It becomes clear from the figure that the position of the steep region is strongly affected by Δ𝜖x, and a change of 1 in the latter leads to approximately a change of 101 in the former. The main plot shows the probability that XapR is in the active state as a function of the logarithm of the xanthosine concentration for Δ𝜖x as shown,𝐾𝜒A =102, and𝐾IA=102.
Changing𝐾𝜒A by one order of magnitude, on the other hand, shifts all the curves by one order of magnitude in the corresponding direction to the left or right.
Additional plots and explanations of the results
2D phase portraits as a vector representation for the standard parameter set used in the main text. We observe the same qualitative behavior, but with stronger fluctuations around the mean and larger variations in the time to reach the mean in the dynamic equilibrium state. As an example, Figure B.6A shows the time evolution of one run of the crack simulation for the same conditions (ie, parameters and initial values) as the time evolution presented in the main text.
The parameters used are the same as in the table in the main text, the only exception being the extracellular xanthosine concentration, which was chosen to be [c]a =25 (recall [c]a .= 𝑐.
Chemical master equation
All other parameters are as in the table in the main text, and the simulation was run 1000 times for 106 seconds each (simulated time) and started with an intracellular mRNA, protein, and xanthosine count of 0. Except for [c]a, the parameters are same as in the table in the main text. For the distributions, the simulations were run 1000 times for 106 seconds each and started at the highest fixed-point intracellular mRNA, protein, and xanthosine counts in the corresponding phase portraits.
When compared with the distributions in the main text, where the simulation had starting concentrations of 0, a clear hysteresis effect can be observed.
Experimental materials and methods
As can be seen from the equation, none of the propensities depend explicitly on time, and therefore we can use the hybrid algorithm between classical Gillespie and τ-jump that we relied on. Reverse primer for pKD4 with homology toxapABRlocus xapABR-seq-fwd cgggtcgttagctcagttggta upstream of xapR, to. For the data presented in this work, expression was maximized using 10 ng/ml of the TetR inducer anhydrotetracycline.
Dandanell, “Specificity and Topology of the Escherichia coli Xanthosine Permease, a Representative of the NHS Subfamily of the Major Facilitator Superfamily,” Journal of Bacteriology.
