Understanding the sub-cellular dynamics of silicon transportation and synthesis in diatoms using population-level data and computational optimization

Abstract

Controlled synthesis of silicon is a major challenge in nanotechnology and material science. Diatoms, the unicellular algae, are an inspiring example of silica biosynthesis, producing complex and delicate nano-structures. This happens in several cell compartments, including cytoplasm and silica deposition vesicle (SDV). Considering the low concentration of silicic acid in oceans, cells have developed silicon transporter proteins (SIT). Moreover, cells change the level of active SITs during one cell cycle, likely as a response to the level of external nutrients and internal deposition rates. Despite this topic being of fundamental interest, the intracellular dynamics of nutrients and cell regulation strategies remain poorly understood. One reason is the difficulties in measurements and manipulation of these mechanisms at such small scales, and even when possible, data often contain large errors. Therefore, using computational techniques seems inevitable. We have constructed a mathematical model for silicon dynamics in the diatom Thalassiosira pseudonana in four compartments: external environment, cytoplasm, SDV and deposited silica. The model builds on mass conservation and Michaelis-Menten kinetics as mass transport equations. In order to find the free parameters of the model from sparse, noisy experimental data, an optimization technique (global and local search), together with enzyme related penalty terms, has been applied. We have connected population-level data to individual-cell-level quantities including the effect of early division of non-synchronized cells. Our model is robust, proven by sensitivity and perturbation analysis, and predicts dynamics of intracellular nutrients and enzymes in different compartments. The model produces different uptake regimes, previously recognized as surge, externally-controlled and internally-controlled uptakes. Finally, we imposed a flux of SITs to the model and compared it with previous classical kinetics. The model introduced can be generalized in order to analyze different biomineralizing organisms and to test different chemical pathways only by switching the system of mass transport equations.

Figure 1. Cell cycle and cell division of diatoms.

(A) Schematic picture of different stages of the asexual cell cycle of a typical diatom: Silica wall is composed of valves and girdle bands. 1: a full-size diatom cell, 2: Cell during the DNA replication (S-phase), 3; immediately after this silica deposition in SDV starts, 4&5: the valve formation continues until it is ready for the new sibling cells and then the cell divides (G2+M-phase), 6: new cells grow accompanied with silica girdle bands formation (G1) until it achieves the full size. (B)&(C) SEM images of diatom Thalassiosira pseudonana. The image of silica frustules of one diatom including valves and girdle bands (B) and a diatom during cell division (C).

Figure 2. Biological control over biosilicification.

Cell controls mineralization through two types of mechanisms: By formation of membranes in order to make specialized compartments like SDV and by producing biomolecules such as silicon transporter proteins and biomolecules involved in silica deposition. The latter includes molecules with the role of scaffold or catalyzer.

Figure 3. Four compartments considered in the computational modeling and kinetics of their communication.

Compartment 1 is the environment from which diatom uptakes silicic acid by means of silicon transporter proteins (SIT) located on the cell membrane. Then silicon most likely stored in compartment 2, the cytoplasm, and transported to SDV, the compartment 3, again by means of SITs. In SDV the soluble silicon finally deposits and forms the silica frustule, which is compartment 4. Green arrows show the cell regulation over amount of SITs. Once the cell nucleus receives the related signals based on information from inside and outside the cell, with some mechanisms which are mostly unknown, it changes the level of SITs to control the inward flux of silicon to compartments. (This figure is a schematic graph. In reality the location of SITs is not completely clear.)

Figure 4. A flowchart for finding solutions to the inverse (optimization) problem.

After designing an optimization problem, the first step is to check structural identifiability of parameters: to test if the inverse problem is solvable assuming that data are accurate and significant. After that the optimization problem should be solved (globally or locally) and at the last step posteriori analysis should be applied to ensure the results are meaningful and the model is robust. In the end, if the model can be validated, it can predict the mechanisms in the system, which, in turn, can provide a better model and, therefore, a better understanding of the system.

Figure 5. Objective function value.

Best objective function during searching for optimized solution until the algorithm cannot find a better value for a long time. The simulation time was around 12 hours.

Figure 6. Relative error in conservation rules.

(A) Conservation of nutrient (eq. 2), (B) Conservation of enzymes (eqs. 9–11)

Figure 7. Scatter plot of accepted parameters.

(A) Parameter values in the acceptance range of objective function for case 1, when there is no penalty term in the model. Most of parameters are scattered in this case, which means that there are many solutions to the optimization problem. (B) Parameter values in acceptance range of objective function for case 2, when penalty terms added to the objective function. In this case, besides and , most of the parameter values are not scattered much.

Figure 8. Model output of forward problem resulted from different solutions of the inverse problem.

The temporal dynamics of 11 variables of the model in (A) case 1 with no penalty term and (B) case 2 with penalty terms. Even though there are different curves for variables of both cases, using penalty term made the silicon dynamics (1,4,7,10,11) much more unique and identified.

Figure 9. Silicon concentration in the medium – fitting to experimental data.

Data from the experiment and model output of silicon concentration in the culture medium is shown in points and a blue curve. This curve depicts the total silicon consumption by all cells. The red curve is the integration of uptake rate coming from term 1 in equation 5. After 150 minutes, the difference between this value and the total silicon consumption becomes big as a result of non-synchronized cell culture. Adding the black curve, integral of term 2, compensates for this difference.

Figure 10. Sensitivity analysis.

Parameter sensitivity matrix around the solution, for variables that have been used in fitting procedure. (A) Normalized sensitivity of silicon in the environment, , shows it is most sensitive to changes in and . (B) Normalized sensitivity of deposited silica, , shows it is most sensitive to the changes of .

Figure 11. Perturbation analysis.

Changes in, (A) objective function and (B) intracellular dynamics of nutrients, due to perturbation of system by 10% change in parameters , and (solid line: original quantities - dotted lines: perturbed quantities).

Figure 12. Dynamics of silicon concentrations in different compartments.

(A) Silicon transport with the assumption of a constant amount of enzymes in all compartments. (B) Silicon transport considering a flux of proteins for SITs during the cell cycle.

Figure 13. Protein flux used in the model.

The relative changes in SIT protein expression level during the cell cycle of diatoms that has been applied to the model. This curve is inferred using the data from the experiment [44 - Figure 3]. SITs are least expressed during S-phase when silicon deposition is almost stopped.

Figure 14. Silicon uptake rates of diatoms versus silicic acid concentration in medium in different time steps.

(A) Constant amount of total enzyme. (B) Considering changes in total amount of SIT enzymes. The uptake rates have saturated forms in high concentrations of dissolved silicon in water. 2 minutes after adding silicon to the starved cells, the uptake rate is very high (surge uptake). By passing time the uptake rate has a big drop in value. In (B) this drop accrues slower and the rate increases again during valve formation.