IARA

MODEL: METAL COLLECTION

INTRODUCTION

The protagonist of the Iara system for metal accumulation consists on a modified Metal Binding Peptide (MBP), capable of irreversibly binding to Hg²⁺ ions. Our MBP is similar to the the Peking system (BBa_K346004) for metal accumulation, in which a modified MerR protein from Mer family is used for mercury retention. However, our MBP has two additional domains attached to its ends: at it's C-terminal end, a HlyA secretion signal, which is recognized by a Type I Secretion System, allows the secretion of our MBP; also, at it's N-terminal end, a Cellulose Binding Domain (CBD) allows the attachment of our MBP to biofilm cellulose.

In this model, we analyse the effects of secretion and constitutive expression in both mercury accumulation and survivability of bacteria.

ASSUMPTIONS

To model the activity of the Iara System (BBa_K3280002) and to compare it to the Peking System, we did numerical simulations starting from a kinetic description of the systems. To do that, we assembled a system of ordinary differential equations to describe the evolution of the concentrations of each involved chemical species. The model bases itself in the following assumptions:

The initial system conditions involve a fixed number of cells in a mercury-free medium to which a limited amount of mercury is added. Therefore, variations due to population growth or spontaneous cell death are neglected. The total set of cells is approximated to a single wrapper with constant volume and area (like a huge cell). In order to avoid concentration corrections, it's also assumed that the external volume is equal to the cell volume.
The biological membrane is permeable to mercury, and the flow of mercury due to diffusion follows Flick's first law: $$J = -D\frac{\delta \phi}{\delta x},$$ where $J$ is the diffusion flow ([quantity] / [area] [time]), $D$ is the diffusion coefficient and $ \frac {\delta \phi} {\delta x} $ is the spatial concentration gradient. The contribution of mercury diffusion through the biological membrane can be approximated in terms of the difference in concentrations between the external and internal environment, so that $$ \left (\frac{d[Hg_{(in)}]}{dt} \right)_{D}=a([Hg_{(ex)}]-[Hg_{(in)}]) = -\left (\frac{d[Hg_{(ex)}]}{dt}\right)_{D},$$ where $ a $ is a constant and $ [Hg _ {(in)}] $ and $ [Hg _ {(ex)}] $ correspond to the internal and external mercury concentrations, respectively.
Free metal binding peptide (MBP) production within the wrapper occurs at a constant rate $ b $, so that $$\varnothing \overset{b}{\rightarrow} M$$
Each MBP is capable of capturing two mercury molecules. The mercury binding process is irreversible and occurs in two steps. The reaction rate of both steps are equal to each other and take the same value for both MBPs (Iara and Peking). $$ M + Hg \overset{a}{\rightarrow} MHg^{(1)}$$ $$ MHg^{(1)} + Hg \overset{a}{\rightarrow} MHg^{(2)}$$
In the case of the Iara system, the free MBP produced within the wrapper is exported at a constant rate $g $. Once out of the wrap, free MBPs can bind to external medium mercury in the same way as internal mercury. $$ M_{in} \overset{g}{\rightarrow} M_{ex} $$
If the internal mercury concentration reaches a threshold, the system is disrupted, thus simulating a cell death caused by mercury toxicity.

MODELLING

PEKING SYSTEM

For the Peking system, the chemical equations are:

$$\textrm{Hg}_{(\textrm{ex})} \overset{a}{\rightarrow} \textrm{Hg}_{(\textrm{in})}$$
$$\varnothing \overset{b}{\rightarrow} \textrm{M}_{(\textrm{in})}$$
$$\textrm{M}_{(\textrm{in})}+\textrm{Hg}_{(\textrm{in})}\overset{\lambda}{\rightarrow} \textrm{MHg}_{(\textrm{in})}^{(1)}$$
$$\textrm{MHg}_{(\textrm{in})}^{(1)}+\textrm{Hg}_{(\textrm{in})}\overset{\lambda}{\rightarrow} \textrm{MHg}_{(\textrm{in})}^{(2)}$$

Note that equation (I) is not a chemical equation itself, since the contribution of diffusion to the variation of mercury concentrations is given by Fick's first law.

Taking Flick's law and the chemical equations of the system, we can gauge the differential equations that describe the temporal variation of the concentration of each chemical species in the system:

$$\frac{d}{dt}[\textrm{Hg}_{(in)}]= a([\textrm{Hg}_{(ex)}]-[\textrm{Hg}_{(in)}])$$
$$\frac{d}{dt}[\textrm{Hg}_{(ext)}]=a([\textrm{Hg}_{(ex)}]-[\textrm{Hg}_{(in)}])-\lambda[\textrm{M}_{(in)}][\textrm{Hg}_{(in)}]-\lambda[\textrm{MHg}_{(in)}^{(1)}][\textrm{Hg}_{(in)}]$$
$$\frac{d}{dt}[\textrm{M}_{(in)}]=b-\lambda[\textrm{M}_{(in)}][\textrm{Hg}_{(in)}]$$
$$\frac{d}{dt}[\textrm{MHg}_{(in)}^{(1)}]=\lambda[\textrm{M}_{(in)}][\textrm{Hg}_{(in)}]-\lambda[\textrm{MHg}_{(in)}^{(1)}][\textrm{Hg}_{(in)}]$$
$$\frac{d}{dt}[\textrm{MHg}_{(in)}^{(2)}]=\lambda[\textrm{MHg}_{(in)}^{(1)}][\textrm{Hg}_{(in)}]$$

IARA SYSTEM

The chemical equations that describe the Iara system are similar to those of the Peking system, with the addition of interactions resulting from MBP export to the extracellular environment:

$$\textrm{Hg}_{(\textrm{ex})} \overset{a}{\rightarrow} \textrm{Hg}_{(\textrm{in})}$$
$$\varnothing \overset{b}{\rightarrow} \textrm{M}_{(\textrm{in})}$$
$$\textrm{MHg}_{(\textrm{in})}^{(1)}+\textrm{Hg}_{(\textrm{in})}\overset{\lambda}{\rightarrow} \textrm{MHg}_{(\textrm{in})}^{(2)}$$
$$\textrm{M}_{(in)} \overset{g}{\rightarrow}\textrm{M}_{(ex)}$$
$$\textrm{M}_{(ex)}+\textrm{Hg}_{(ex)} \overset{\lambda}{\rightarrow}\textrm{MHg}_{(ex)}^{(1)}$$
$$\textrm{MHg}_{(ex)}^{(1)} + \textrm{Hg}\overset{\lambda}{\rightarrow} \textrm{MHg}_{(ex)}^{(2)}$$

Taking Flick's law and the chemical equations of the system, one can gauge the differential equations that describe the temporal variation of the concentration of each chemical species in the system:

$$\frac{d}{dt}[\textrm{Hg}_{(in)}]= a([\textrm{Hg}_{(ex)}]-[\textrm{Hg}_{(in)}]) - \lambda[\textrm{M}_{(ex)}][\textrm{Hg}_{(ex)}]-\lambda[\textrm{MHg}_{(ex)}^{(1)}][\textrm{Hg}_{(ex)}]$$
$$\frac{d}{dt}[\textrm{Hg}_{(ext)}]=a([\textrm{Hg}_{(ex)}]-[\textrm{Hg}_{(in)}])-\lambda[\textrm{M}_{(in)}][\textrm{Hg}_{(in)}]-\lambda[\textrm{MHg}_{(in)}^{(1)}][\textrm{Hg}_{(in)}]$$
$$\frac{d}{dt}[\textrm{M}_{(in)}]=b-\lambda[\textrm{M}_{(in)}][\textrm{Hg}_{(in)}]-g[\textrm{M}_{(in)}]$$
$$\frac{d}{dt}[\textrm{MHg}_{(in)}^{(1)}]=\lambda[\textrm{M}_{(in)}][\textrm{Hg}_{(in)}]-\lambda[\textrm{MHg}_{(in)}^{(1)}][\textrm{Hg}_{(in)}]$$
$$\frac{d}{dt}[\textrm{MHg}_{(in)}^{(2)}]=\lambda[\textrm{MHg}_{(in)}^{(1)}][\textrm{Hg}_{(in)}]$$
$$\frac{d}{dt}[\textrm{M}_{(ex)}]=g[\textrm{M}_{(in)}]-\lambda[\textrm{M}_{(ex)}][\textrm{Hg}_{(ex)}$$
$$\frac{d}{dt}[\textrm{MHg}_{(ex)}^{(1)}=\lambda[\textrm{M}_{(ex)}][\textrm{Hg}_{(ext)}]-\lambda[\textrm{MHg}_{(ex)}^{(1)}][\textrm{Hg}_{(ex)}]$$
$$\frac{d}{dt}[\textrm{MHg}_{(in)}^{(2)}]=\lambda[\textrm{MHg}_{(ex)}^{(1)}][\textrm{Hg}_{(ex)}]$$

SIMULATIONS

In order to perform a qualitative analysis of the models, the fourth-order Runge-Kutta method was used to numerically simulate both systems.

COLLECTED METAL

TEST 1: WITHOUT INITIAL MBP CONCENTRATION

The $ \lambda $ and $ g $ parameters were taken as $ \lambda = g = L = 1.0 $, varying the values of the rate $a$ associated with the diffusion of $ \textrm {Hg} ^ {\textrm {2 +}} $ therough the membrane and MBP's production rate $b $. In this test, the initial conditions taken were such that the bacterium starts MBP production only when in the presence of Hg for both systems.

Initial Conditions of Test 1:

$$[\textrm{Hg}_{(ex)}]_0=150.0$$ $$[\textrm{Hg}_{(in)}]_0=0.0$$ $$[\textrm{M}_{(in)}]_0=[\textrm{M}_{(ex)}]_0=0.0$$ $$[\textrm{MHg}_{(in)}^{(1)}]_0=[\textrm{MHg}_{(ex)}^{(1)}]_0=0.0$$ $$[\textrm{MHg}_{(in)}^{(2)}]_0=[\textrm{MHg}_{(ex)}^{(2)}]_0=0.0$$

For values of $a $ and $ b \in [1,2] $, systems were simulated for an arbitrary period $ \Delta t = 10.0 $, with a time step $ \delta t = 10 ^ {- 4} $, noting the difference between external concentrations $ \Delta C = [\textrm{Hg}_{(ex)}]_{\textrm{Peking}} - [\textrm{Hg}_{(ex)}]_{\textrm{Iara}}$, after the time interval $ \Delta t $, meaning that if the result is positive, our system presents small external mercury concentration. In the end of the simulations, the Iara system has lower external Hg concentration than the system without secretion.

TEST 2: WITH INITIAL MBP CONCENTRATION

In the second test, the parameters vary as test 1. The initial conditions of the Iara system now include an initial concentration of MBP inside and outside the cell.

Initial conditions for test 2:

Peking

$$[\textrm{Hg}_{(ex)}]_0=150.0$$ $$[\textrm{Hg}_{(in)}]_0=0.0$$ $$[\textrm{M}_{(in)}]_0=[\textrm{M}_{(ex)}]_0=0.0$$ $$[\textrm{MHg}_{(in)}^{(1)}]_0=[\textrm{MHg}_{(ex)}^{(1)}]_0=0.0$$ $$[\textrm{MHg}_{(in)}^{(2)}]_0=[\textrm{MHg}_{(ex)}^{(2)}]_0=0.0$$

Iara

$$[\textrm{Hg}_{(ex)}]_0=150.0$$ $$[\textrm{Hg}_{(in)}]_0=0.0$$ $$[\textrm{M}_{(in)}]_0=[\textrm{M}_{(ex)}]_0=25.0$$ $$[\textrm{MHg}_{(in)}^{(1)}]_0=[\textrm{MHg}_{(ex)}^{(1)}]_0=0.0$$ $$[\textrm{MHg}_{(in)}^{(2)}]_0=[\textrm{MHg}_{(ex)}^{(2)}]_0=0.0$$

DEATH TIME

Adopting the initial external concentration of mercury as four times higher than the maximum internal concentration allowed, we simulated the cell's death time for both systems, varying the parameters $a$ and $b$.

These tests were realized in two different initial conditions: in the first, MBP starts to be synthesized at $t=0$ for both systems; in the second, the Iara system starts with an initial concentration of MBP. Here, positive results mean that our bacterium takes longer to die.

TEST 1: WITHOUT INITIAL MBP CONCENTRATION

In the first scenario, Peking's system survives for a longer time than the Iara system. This makes sense, since the defense mechanism of the Iara system is being exported.

TEST 2: WITH INITIAL MBP CONCENTRATION

In the second scenario, however, the Iara system has more survivability.

DISCUSSION

These results indicate that the metal collection is more effective when MBP is exported and produced before the addition of Hg. If we adopt realistic values of $a$ and $b$, the diffusion of Hg through the membrane occurs at a much higher rate than MBP's production.

The simulated results confirmed our presumption that it was necessary to grow the culture in a Hg free medium so that the already secreted proetins could fixate mercury more eficiently. This confirmation helped us to comprehend better our system and to decide the construction steps of our biofilter.

Thus, the previous biofilm growth and maturation (time to secrete the Iaraα chimera) was held as one of the most important steps in regard to the efficiency of our biofilter.