Summary of Mathematical Modeling
Given the important role of satellite phages in regulating bacterial communities and their enormous potential for bioengineering and synthetic biology, it is essential to model the population dynamics of satellite phage systems and their host bacteria in natural environments (Ibarra-Chávez, 2021). Surprisingly, there is currently only a single publication that models satellite phages, one limited to the P2/P4 system (Mitarai, 2019). To address the need for mathematical models that will promote satellites to serve as effective systems for synthetic biology, we have written several partial differential equations (PDEs) which model the growth, fluid flow, and the infection dynamics of satellites. Our project entailed the deployment of two types of phage satellites in the model colon and soil microcosms. The first type of satellite system is a P2/P4 system in E. coli with replicative and non-replicative variations of a kanamycin targeting CRISPR system, a negative control non-targeting CRISPR system, and a red fluorescent protein (RFP) satellite phage system. The second satellite phage system type is a Mycobacterium aichiense satellite phage. For each of these systems we wrote a PDE model which models the advection-diffusion fluid dynamics, population dynamics, and Monod growth dynamics of each system. These models were essential in informing our experimental design, and delineating the behaviors of our satellite systems in a variety of natural environments. These models demonstrate that satellite phages and transducing units persist and are infective in natural environments and are therefore potentially useful in synthetic biology and fieldable applications.
Criteria for Modeling Award
What kind of modeling is being done and what information it will provide: We have written multiple one dimensional partial differential equation models which include advection-diffusion, growth, and population dynamics. This provides us with a model of how satellite phages behave in natural environments, where they will ultimately be deployed.
What assumptions were made and why: In order to perform PDE modeling, we must assume that the model gut and soil matrix are spatially uniform. This aligns with the assumptions of Egilmez et al. (2021). Another important assumption is that satellite phages follow similar fluid dynamics and population dynamics as phages due to the lack of characterization of these parameters for satellites. Finally, we assumed that these systems were radially symmetric in order to model spread in one spatial dimension.
What kind of data was used to build/assess the model: We conducted an extensive literature review to find preliminary parameters which we then adjusted accordingly to the results of our microcosms.
How the model results affected the project design and development: We used preliminary results from the model which suggested that the non-replicative system would be significantly less effective than the replicative system to inform our experimental design. We tested both replicative and non-replicative systems. We also used preliminary results to determine minimum inoculation populations. Our modeling was essential for this project since these types of experiments had not been previously conducted.
Review of the Literature and Basis for our Model
There was one existing paper which modeled satellite phages. The paper ‘How pirate phage interfaces with helper phage’ by Mitarai (2019) examined the population dynamics of pirate and helper phage for staphylococcal pathogenicity islands (SaPIs) and the P2-P4 system. One important feature of the P2-P4 system was that P4 can lyse P2 lysogens, unlike SaPIs. This paper used a Lotka-Volterra ODE model for population dynamics including advection-diffusion for soft agar media. This model uses standard Monod growth dynamics to model the growth of host bacteria dependent on the availability of a growth-limiting substrate. This model looked at state variables: uninfected bacteria concentration, pirate phage lysogen concentration, helper phage lysogen concentration, pirate phage concentration, helper phage concentration, intermediate state concentrations, and nutrient concentration. The intermediate state concentrations included in this model allow for latency steps after infections (Mitarai, 2019).
Other previous literature informed the fluid dynamics and population dynamics. The paper ‘Hydrodynamic flow and concentration gradients in the gut enhance neutral bacterial diversity’ by Labavic et al. (2022) was a one-dimensional population dynamic model of bacterial flow in the human gastrointestinal tract (Labavic et al, 2022).
| State variables | Constants | ||
| F | Food | D | Diffusion rate |
| B | Wild-type bacteria | v | Flow velocity |
| M | Mutant bacteria | r | Maximum growth rate |
| k | Monod constant |
This model is a one-dimensional advection-diffusion PDE model following standard Monod growth dynamics based on the availability of nutrients from food (Labavic et al, 2022). This model also includes two populations of bacteria: wild-type and mutant (Labavic et al, 2022).
Finally, ‘A Mathematical Model for Transport and Growth of Microbes in Unsaturated Porous Soil’ by Timsina et al. (2021) was informative to the development of our PDE models for the soil microcosms as it models the fluid flow of bacteria through a soil matrix (Timsina et al. 2021). This model is distinct from other fluid flow models as it includes populations for both bacteria attached to the soil matrix and a free-flowing bacterial population (Timsina et al. 2021).
| State Variables | |||
| θ | Volumetric Moisture | S | Substrate |
| M | Free flowing bacteria | Ψ | Pressure Head |
| Ma | Bacteria attached to the soil | K | Hydraulic conductivity |
| Constants | |||
| D | Diffusion rate of bacteria | ka | Attachment coefficient |
| Ds | Diffusion rate of substrate | Mma | Maximum concentration of attached bacteria |
| vw | Water flux | kf | Freundlich isotherm slope parameter |
| v | Bacterial flux | mum | Growth rate |
| τ | Tortuosity | Y | Growth yield coefficient |
| ky | Detachment coefficient |
Mathematical Model Details
In order to develop a model for satellite phages, we have written several partial differential equations (PDEs) describing the growth, fluid dynamics, and population dynamics of satellite phages. The growth dynamics of each model follow Monod growth dynamics which limit growth to the availability of a growth limiting substrate, in this case carbon. The fluid dynamics follow advection-diffusion equations and for the soil microcosm models, include both populations free flowing and attached to the soil matrix.
For both the soil microcosms and the model colon experiments, we wrote a PDE model of each of the satellite phage systems we have implemented in the wet lab. For a detailed description of the biology, please see our engineering and experiments pages. For both the soil microcosms and the colon model, these include a model for both the replicative and non-replicative kanamycin targeting CRISPR system which cuts in the kanamycin resistance gene resulting in a killing effect for the indicator strain E. coli HL713. We have also written models for the replicative and non-replicative non-targeting CRISPR system which conveys chloramphenicol resistance, and the replicative and non-replicative RFP systems which convey RFP.
Parameters
| Variable Name | Symbol | Units | Initial values for Colon | Initial Values for Soil |
| E. coli HL713/M. aichiense | B | CFU | 6.00E+05 | 1.00E+07 |
| Attached E. coli HL713/M. aichiense | BA | CFU | NA | 1.00E+07 |
| Infected E. coli HL713/M.aichiense | BI | CFU | 1 | 1 |
| Attached Infected E. coli HL713/M. aichiense | BAI | CFU | NA | 1 |
| Transducing units/"Phagelets"/M. aichiense phage | T | PFU | 6.00E+09 | 1.00E+09 |
| P2 Replicative System | P | CFU | 6.00E+05 | 1.00E+07 |
| Attached P2 Replicative System | PA | CFU | NA | 1 |
| Infected P2 Replicative System | PI | CFU | 1 | 1 |
| Infected Attached P2 Replicative System | PAI | CFU | NA | 1 |
| Nutrient concentration | n | % | 0.18 | 0.03 |
| L-rhamnose | L | % | 0.002 | 0.002 |
| Community | C | CFU | 6.00E+06 | 1.00E+09 |
Table. State Parameters
Populations with an initial value of 1 have no initial population, however, to avoid division by zero errors they are initialized with an initial population of 1. This should not significantly affect the results of the computation.
| Variable Name | Symbol | Units | Values |
| Adsorption of transducing units | kT | /s | 9.17E-15 |
| Specific Growth Rate | gmax | /s | 0.00166666666 |
| Death rate of bacteria | d | /s | 0.00000502 |
| Death rate of transducing units | δ | /s | 3.86e-8 |
| Diffusion rate of nutrients | Dn | cm2/s | 1.51e-10 |
| Diffusion rate of bacteria | DB | cm2/s | 1.30E-06 |
| Diffusion rate of Transducing Units | DT | cm2/s | 3.86E-08 |
| Chemotactic velocity | v | cm/s | 4.00E-04 |
| Flow velocity of water | vw | cm/s | 1.75E-03 |
| Volumetric moisture | θ | unitless | -61.5 |
| Outflow rate | O | /s | 3.47E-06 |
| Inflow rate | I | /s | 3.47E-06 |
| Burst size | b | Unitless | 10 |
| Lysis rate | λ | /s | 0.00057 |
Table. Constants
Model Mechanics: Equations
Below are the partial differential equations (PDEs) describing the growth, fluid dynamics, and population dynamics of satellite phages. Each model follows Monod growth dynamics, which is a commonly used model for growth dependent on the availability of a growth limiting substrate. For these models, the growth limiting substrate is considered organic carbon. The fluid dynamics follow advection-diffusion equations and for the soil microcosm models, include both populations free flowing and attached to the soil matrix. The below equations are color coded according to the legend below.
Flow and Diffusion |
|---|
Infection |
Growth and Death |
Inflow and Outflow |
Monod growth rate:
Equations for Colon Models:
Colon Model: Replicative Empty CRISPR/RFP
This equation describes the fluid dynamics, consumption, inflow, and outflow of the growth limiting substrate carbon in the colon.
This equation describes the fluid dynamics, outflow, and consumption of L-rhamnose by the replicative system.
This equation describes the fluid dynamics, infection of the indicator strain, re-infection of the replicative system, deterioration of transducing units, and outflow of the transducing units.
This equation describes the fluid dynamics, infection, growth, and outflow of the indicator strain.
This equation describes the fluid dynamics, infection, growth, and outflow of the infected indicator strain.
This equation describes the fluid dynamics, infection, growth, and outflow of the replicative system.
This equation describes the fluid dynamics, infection, lysis, and outflow of the infected replicative system.
This equation describes the fluid dynamics, growth, and outflow of the bacterial community in the colon.
Colon Model: Non-Replicative Empty CRISPR/RFP
This equation describes the fluid dynamics, consumption, inflow, and outflow of the growth limiting substrate carbon in the colon.
This equation describes the fluid dynamics, infection of the indicator strain, deterioration of transducing units, and outflow of the transducing units.
This equation describes the fluid dynamics, infection, growth, and outflow of the indicator strain.
This equation describes the fluid dynamics, infection, growth, and outflow of the infected indicator strain.
This equation describes the fluid dynamics, growth, and outflow of the bacterial community in the colon.
Colon Model: Replicative CRISPR
This equation describes the fluid dynamics, consumption, inflow, and outflow of the growth limiting substrate carbon in the colon.
This equation describes the fluid dynamics, outflow, and consumption of L-rhamnose by the replicative system.
This equation describes the fluid dynamics, infection of the indicator strain, re-infection of the replicative system, deterioration of transducing units, and outflow of the transducing units.
This equation describes the fluid dynamics, infection, growth, and outflow of the indicator strain.
This equation describes the fluid dynamics, infection, growth, and outflow of the replicative system.
This equation describes the fluid dynamics, infection, lysis, and outflow of the infected replicative system.
This equation describes the fluid dynamics, growth, and outflow of the bacterial community in the colon.
Colon Model: Non-Replicative CRISPR
This equation describes the fluid dynamics, consumption, inflow, and outflow of the growth limiting substrate carbon in the colon.
This equation describes the fluid dynamics, infection of the indicator strain, deterioration of transducing units, and outflow of the transducing units.
This equation describes the fluid dynamics, infection, growth, and outflow of the indicator strain.
This equation describes the fluid dynamics, growth, and outflow of the bacterial community in the colon.
Equations for Soil Models:
Soil Microcosm Model: E. coli Replicative Empty CRISPR/RFP
This equation describes the fluid dynamics and consumption in the soil microcosms.
This equation describes the fluid dynamics and consumption of L-rhamnose by the replicative system.
This equation describes the fluid dynamics, infection of the indicator strain, and deterioration of transducing units of the transducing units.
This equation describes the fluid dynamics, infection, attachment and detachment, and growth of the indicator strain.
This equation describes the attachment and detachment, infection, and growth of the attached indicator strain.
This equation describes the fluid dynamics, attachment and detachment, infection, and growth of the indicator strain.
This equation describes the attachment and detachment, infection, and growth of the attached indicator strain.
This equation describes the fluid dynamics, attachment and detachment, infection, and growth of the replicative system.
This equation describes the attachment and detachment, infection, and growth of the attached replicative system.
This equation describes the fluid dynamics, attachment and detachment, infection, and growth of the infected replicative system.
This equation describes the attachment and detachment, infection, and growth of the attached infected replicative system.
This equation describes the fluid dynamics and growth of the bacterial community in soil.
Soil Microcosm Model: E. coli Non-Replicative Empty CRISPR/RFP
This equation describes the fluid dynamics and consumption in the soil microcosms.
This equation describes the fluid dynamics, infection of the indicator strain, and deterioration of transducing units of the transducing units.
This equation describes the fluid dynamics, infection, attachment and detachment, and growth of the indicator strain.
This equation describes the attachment and detachment, infection, and growth of the attached indicator strain.
This equation describes the fluid dynamics, attachment and detachment, infection, and growth of the infected indicator strain.
This equation describes the attachment and detachment, infection, and growth of the attached infected indicator strain.
This equation describes the fluid dynamics and growth of the bacterial community in soil.
Soil Microcosm Model: M. aichiense and "Phagelets" or Non-replicative E. coli Kanamycin Targeting CRISPR system
This equation describes the fluid dynamics and consumption in the soil microcosms.
This equation describes the fluid dynamics, infection of the indicator strain, and deterioration of the transducing units.
This equation describes the fluid dynamics, infection, attachment and detachment, and growth of the indicator strain.
This equation describes the attachment and detachment, infection, and growth of the attached indicator strain.
This equation describes the fluid dynamics and growth of the bacterial community in soil.
Soil Microcosm Model: Replicative E. coli Kanamycin Targeting CRISPR system
This equation describes the fluid dynamics and consumption in the soil microcosms.
This equation describes the fluid dynamics, infection of the indicator strain, and deterioration of transducing units of the transducing units.
This equation describes the fluid dynamics, infection, attachment and detachment, and growth of the indicator strain.
This equation describes the attachment and detachment, infection, and growth of the attached indicator strain.
This equation describes the fluid dynamics, attachment and detachment, infection, and growth of the replicative system.
This equation describes the attachment and detachment, infection, and growth of the attached replicative system.
This equation describes the fluid dynamics, attachment and detachment, infection, and growth of the infected replicative system.
This equation describes the attachment and detachment, infection, and growth of the attached infected replicative system.
This equation describes the fluid dynamics and growth of the bacterial community in soil.
Colon Model: Replicative Empty CRISPR/RFP
This equation describes the fluid dynamics, consumption, inflow, and outflow of the growth limiting substrate carbon in the colon.
This equation describes the fluid dynamics, outflow, and consumption of L-rhamnose by the replicative system.
This equation describes the fluid dynamics, infection of the indicator strain, re-infection of the replicative system, deterioration of transducing units, and outflow of the transducing units.
This equation describes the fluid dynamics, infection, growth, and outflow of the indicator strain.
This equation describes the fluid dynamics, infection, growth, and outflow of the infected indicator strain.
This equation describes the fluid dynamics, infection, growth, and outflow of the replicative system.
This equation describes the fluid dynamics, infection, lysis, and outflow of the infected replicative system.
This equation describes the fluid dynamics, growth, and outflow of the bacterial community in the colon.
Colon Model: Non-Replicative Empty CRISPR/RFP
This equation describes the fluid dynamics, consumption, inflow, and outflow of the growth limiting substrate carbon in the colon.
This equation describes the fluid dynamics, infection of the indicator strain, deterioration of transducing units, and outflow of the transducing units.
This equation describes the fluid dynamics, infection, growth, and outflow of the indicator strain.
This equation describes the fluid dynamics, infection, growth, and outflow of the infected indicator strain.
This equation describes the fluid dynamics, growth, and outflow of the bacterial community in the colon.
Colon Model: Replicative CRISPR
This equation describes the fluid dynamics, consumption, inflow, and outflow of the growth limiting substrate carbon in the colon.
This equation describes the fluid dynamics, outflow, and consumption of L-rhamnose by the replicative system.
This equation describes the fluid dynamics, infection of the indicator strain, re-infection of the replicative system, deterioration of transducing units, and outflow of the transducing units.
This equation describes the fluid dynamics, infection, growth, and outflow of the indicator strain.
This equation describes the fluid dynamics, infection, growth, and outflow of the replicative system.
This equation describes the fluid dynamics, infection, lysis, and outflow of the infected replicative system.
This equation describes the fluid dynamics, growth, and outflow of the bacterial community in the colon.
Colon Model: Non-Replicative CRISPR
This equation describes the fluid dynamics, consumption, inflow, and outflow of the growth limiting substrate carbon in the colon.
This equation describes the fluid dynamics, infection of the indicator strain, deterioration of transducing units, and outflow of the transducing units.
This equation describes the fluid dynamics, infection, growth, and outflow of the indicator strain.
This equation describes the fluid dynamics, growth, and outflow of the bacterial community in the colon.
Soil Microcosm Model: E. coli Replicative Empty CRISPR/RFP
This equation describes the fluid dynamics and consumption in the soil microcosms.
This equation describes the fluid dynamics and consumption of L-rhamnose by the replicative system.
This equation describes the fluid dynamics, infection of the indicator strain, and deterioration of transducing units of the transducing units.
This equation describes the fluid dynamics, infection, attachment and detachment, and growth of the indicator strain.
This equation describes the attachment and detachment, infection, and growth of the attached indicator strain.
This equation describes the fluid dynamics, attachment and detachment, infection, and growth of the indicator strain.
This equation describes the attachment and detachment, infection, and growth of the attached indicator strain.
This equation describes the fluid dynamics, attachment and detachment, infection, and growth of the replicative system.
This equation describes the attachment and detachment, infection, and growth of the attached replicative system.
This equation describes the fluid dynamics, attachment and detachment, infection, and growth of the infected replicative system.
This equation describes the attachment and detachment, infection, and growth of the attached infected replicative system.
This equation describes the fluid dynamics and growth of the bacterial community in soil.
Soil Microcosm Model: E. coli Non-Replicative Empty CRISPR/RFP
This equation describes the fluid dynamics and consumption in the soil microcosms.
This equation describes the fluid dynamics, infection of the indicator strain, and deterioration of transducing units of the transducing units.
This equation describes the fluid dynamics, infection, attachment and detachment, and growth of the indicator strain.
This equation describes the attachment and detachment, infection, and growth of the attached indicator strain.
This equation describes the fluid dynamics, attachment and detachment, infection, and growth of the infected indicator strain.
This equation describes the attachment and detachment, infection, and growth of the attached infected indicator strain.
This equation describes the fluid dynamics and growth of the bacterial community in soil.
Soil Microcosm Model: M. aichiense and "Phagelets" or Non-replicative E. coli Kanamycin Targeting CRISPR system
This equation describes the fluid dynamics and consumption in the soil microcosms.
This equation describes the fluid dynamics, infection of the indicator strain, and deterioration of the transducing units.
This equation describes the fluid dynamics, infection, attachment and detachment, and growth of the indicator strain.
This equation describes the attachment and detachment, infection, and growth of the attached indicator strain.
This equation describes the fluid dynamics and growth of the bacterial community in soil.
Soil Microcosm Model: Replicative E. coli Kanamycin Targeting CRISPR system
This equation describes the fluid dynamics and consumption in the soil microcosms.
This equation describes the fluid dynamics, infection of the indicator strain, and deterioration of transducing units of the transducing units.
This equation describes the fluid dynamics, infection, attachment and detachment, and growth of the indicator strain.
This equation describes the attachment and detachment, infection, and growth of the attached indicator strain.
This equation describes the fluid dynamics, attachment and detachment, infection, and growth of the replicative system.
This equation describes the attachment and detachment, infection, and growth of the attached replicative system.
This equation describes the fluid dynamics, attachment and detachment, infection, and growth of the infected replicative system.
This equation describes the attachment and detachment, infection, and growth of the attached infected replicative system.
This equation describes the fluid dynamics and growth of the bacterial community in soil.
Assumptions
The most significant assumption in all models, but primarily the models of the soil microcosms, is that the media is uniform. The granularity of the soil matrix must be assumed to be insignificant. This aligns with the assumptions of Egilmez et al. (2021). This allows for modeling these environments using a PDE reaction-diffusion model.
Another important assumption is that satellite phage behave similarly to bacteriophage for several of the parameters in this model. Satellite phage fluid dynamics have not been fully characterized, especially not in soil or other natural environments. Therefore, for initial results, several of the parameters were assumed to be consistent with the parameters for phages. Our experiments in soil microcosms and a model colon allowed us to refine these parameters.
Finally, the majority of our modeling was in one spatial dimension because we assumed them to be radially symmetric.
Computation
These models were computed using the py-pde PDE solver. We chose this solver for two major reasons. First, py-pde is open source and relatively accessible compared to other PDE solvers. This solver has extensive documentation. Second, using the numba package, py-pde is relatively computationally efficient. It was built for research applications and ease of use. It computes with the method of lines with a finite difference approximation, which means that it makes an algebraic approximation of the boundaries and differences. This package also has graphics to allow easy plotting and gifs.
Validation of Model
An important step in PDE model computation which is often overlooked is validation of results. PDE solvers, such as py-pde, use a finite difference approximation which require a suitably small granularity of space (dx) and the granularity of time (dt) must be
If the time resolution is not sufficiently small, the results will be unstable and vary significantly between different time resolutions. Therefore, we validated that our results were consistent over several time resolutions by re-running the same preliminary model at dt = 1, 0.1, 0.01, 0.001, and 0.0001. To exemplify the necessity of time resolution, we tested our model at dt = 1 which is greater than the minimum dt. This model failed at this resolution.
| dt | 1 | 0.1 | 0.01 | 0.001 | 0.0001 |
| Indicator | fails | 0.553 | 0.553 | 0.553 | 0.553 |
| Attached Indicator | fails | -9.51E-07 | -9.51E-07 | -9.51E-07 | -9.51E-07 |
| Satellite | fails | 9.80E+08 | 9.80E+08 | 9.80E+08 | 9.80E+08 |
| Community | fails | 1.00E+09 | 1.00E+09 | 1.00E+09 | 1.00E+09 |
| Nutrient | fails | 3.00E-02 | 3.00E-02 | 3.00E-02 | 3.00E-02 |
Table. Results of the model with a dt between 1 and 1e-4 for t=1000s.
| dt % difference with results at 0.0001 | % difference 0.1 and 0.0001 | % difference 0.01 and 0.0001 | % difference 0.001 and 0.0001 |
| Indicator | -4.82E-07 | 5.93E-08 | -1.03E-06 |
| Attached Indicator | 3.46E-08 | -7.60E-09 | 1.09E-07 |
| Satellite | 3.20E-09 | 3.10E-09 | -2.84E-08 |
| Community | 6.60E-09 | 5.00E-10 | 1.40E-09 |
| Nutrient | 6.10E-09 | 7.30E-09 | -6.70E-08 |
Table. Percent difference between solutions for dt = 0.1, 0.01, 0.001 with 0.0001.
The percent differences in the table above indicate that the temporal resolution is appropriately small for any dt from 0.0001 and 0.1 and therefore the models can be solved with dt=0.1.
Results
Colon Model Key Results:
- Both replicative and non-replicative versions of each system significantly impact the indicator strain population, showing an infected population on the order of 1x105 for the RFP and non-targeting CRISPR system and a similarly effective killing effect for the kanamycin targeting CRISPR system.
- A preliminary result indicated that the non-replicative system would have less of an effect than the replicative system which informed our experimental design to test the replicative system.
- The replicative system has more of an effect than the non-replicative system.
- The effects of each system in the model human colon did not disperse to the end of the colon which aligns with some of our experimental results.
We developed a novel partial differential equation (PDE) model for our satellite phage systems in the human colon. This model uses advection-diffusion fluid dynamics, Monod growth kinematics, and population dynamics to describe the interactions and efficacy of our satellite phage systems. Our results demonstrate that both our replicative and non-replicative systems have a measurable effect and our replicative systems have more of an effect than the non-replicative satellite part systems.
Influence on Experimental Design
An important early result that we found from a preliminary model of the colon was that given the titer and efficiency of the transducing units, the non-replicative systems would not have a significant effect. Therefore, we tested both the non-replicative and replicative systems to confirm this result and show a more significant effect.
Figure. Early result of the PDE model for the non-replicative kanamycin targeting CRISPR system in the model colon.
Colon Model Results
The models of the colon, particularly the RFP systems, clearly demonstrate significant infectivity reaching an infected population on the order of 1x105 CFU in a one week time span. This PDE model also aligns with our experimental findings that the RFP system was effective, however it did not reach as significantly to later segments of the colon. This result is also seen in the soil microcosm models with infected populations reaching 1x107 CFU.
Figure. Difference between infection of replicative and non-replicative systems. Left: E. coli RFP replicative system infected indicator strain in the colon. Right: E. coli RFP non-replicative system infected indicator strain in the colon.
Figure. Model of E. coli Replicative RFP System in the Colon.In order the subplots are: the indicator strain (CFU), infected indicator strain (CFU), satellite (PFU), replicative system (CFU), nutrient (%), and colon microbial community (CFU).
The model of the colon shows significant efficacy for the satellite phage systems. The replicative model for the RFP system in the colon shows higher infectivity than the non-replicative system.
Figure. Model of E. coli Replicative RFP System in the Colon. In order the subplots are: the indicator strain (CFU), infected indicator strain (CFU), satellite (PFU), nutrient (%), and colon microbial community (CFU).
Figure. Model of E. coli Non-Replicative RFP System in the Colon. In order the subplots are: the indicator strain (CFU), satellite (PFU), replicative system (CFU), nutrient (%), and colon microbial community (CFU).
Interestingly, in contrast to the results of the previous models, the E. coli replicative kanamycin targeting CRISPR system has a higher population of indicator strain (CFU) than the non-replicative system.
Figure. Model of E. coli Non-Replicative Kanamycin Targeting CRISPR System in the Colon. In order the subplots are: the indicator strain (CFU), satellite (PFU), nutrient (%), and colon microbial community (CFU).
Soil Microcosm Models Key Results:
- Both replicative and non-replicative of each system significant impact with infected populations ranging from 2x105 to 3.5x105
- The replicative system has more of an effect than the non-replicative system. The replicative and non-replicative RFP models show clear results that the non-replicative system has less effect than the replicative system showing peak populations of approximately 2x105 and 2.5x105 CFU of infected indicator strain respectively.
We developed a novel partial differential equation (PDE) model for our satellite phage systems in the soil. This model uses advection-diffusion fluid dynamics, Monod growth kinematics, and population dynamics to describe the interactions and efficacy of our satellite phage systems. Our results demonstrate that both our replicative and non-replicative systems have a measurable effect and our replicative systems have more of an effect than the non-replicative satellite part systems.
Similar results can be seen in the models of the soil microcosms as in the results for the colon models. The replicative and non-replicative RFP models show clear results that the non-replicative system has less effect than the replicative system showing peak populations of approximately 2x105 and 2.5x105 CFU of infected indicator strain respectively.
Figure. Difference between infection of replicative and non-replicative systems. Left: Indicator population of E. coli Replicative Kanamycin Targeting CRISPR System. Right: Indicator population of E. coli Non-Replicative Kanamycin Targeting CRISPR System in Soil.
The soil microcosm models of the E. coli replicative and non-replicative kanamycin targeting CRISPR system show similar higher killing efficiency for the replicative system than the non-replicative system.
Soil Microcosm Model Results
Figure. Model of E. coli Replicative RFP System in Soil. In order the subplots are: the indicator strain (CFU), attached indicator strain (CFU), infected indicator strain (CFU), infected attached indicator strain (CFU), satellite (PFU), replicative system (CFU), attached replicative system (CFU), soil microbial community (CFU), and nutrient (%).
Figure. Model of E. coli Non-Replicative RFP System in Soil. In order the subplots are: the indicator strain (CFU), attached indicator strain (CFU), infected indicator strain (CFU), infected attached indicator strain (CFU), satellite (PFU), soil microbial community (CFU), and nutrient (%).
The above two figures show the results for the replicative and non-replicative E. coli RFP system in the soil microcosms. The non-replicative system has a peak population of infected indicator strain between 2x105 and 2.5x105. The replicative system has a peak population of infected indicator strain between 2.5x105 and 3x105.
Figure. Model of E. coli Replicative Kanamycin Targeting CRISPR System/ M. aichiense and "Phagelet" System in Soil. In order the subplots are: the indicator strain (CFU), attached indicator strain (CFU), satellite (PFU), replicative system (CFU), attached replicative system (CFU), soil microbial community (CFU), and nutrient (%).
Figure. Model of E. coli Non-Replicative Kanamycin Targeting CRISPR System in Soil. In order the subplots are: the indicator strain (CFU), attached indicator strain (CFU), satellite (PFU), soil microbial community (CFU), and nutrient (%).
The two above figures show the replicative and non-replicative kanamycin targeting CRISPR system. The replicative system has a greater killing effect than the non-replicative system with a minimum population of indicator strain around 3x105 and 3.5x105 respectively.
Refined parameters from comparison with Microcosm Data
After preliminary results from the soil microcosms we were able to adjust the model parameters to better represent the dynamics.
| Variable Name | Symbol | Units | Values |
| Diffusion rate of nutrients | Dn | cm2/s | 5.00E-02 |
| Diffusion rate of bacteria | DB | cm2/s | 2.00E-03 |
| Diffusion rate of Transducing Units | DT | cm2/s | 2.00E-03 |
| Burst size | b | Unitless | 1.00E+01 |
Table. Constants estimated from experimental data.
The diffusion rate of bacteria and transducing units was determined by spread from the ‘center’ to the ‘edge’ measurements using the following equation to estimate the diffusion coefficient
The burst size was re-estimated from the effect size. The diffusion rate of nutrients was re-adjusted accordingly to the new diffusion rate of bacteria and transducing units.
Figure. Model of E. coli Replicative RFP System in Soil. In order the subplots are the indicator strain (CFU), attached indicator strain (CFU), infected indicator strain (CFU), attached infected indicator strain (CFU), satellite (PFU), replicative system (CFU), attached replicative system (CFU), soil microbial community (CFU), and nutrient (%).
Figure. Model of E. coli Non-Replicative RFP System in Soil. In order the subplots are the indicator strain (CFU), attached indicator strain (CFU), infected indicator strain (CFU), attached infected indicator strain (CFU), satellite (PFU), soil microbial community (CFU), and nutrient (%).
Figure. Model of E. coli Replicative Kanamycin Targeting CRISPR System in Soil. In order the subplots are the indicator strain (CFU), attached indicator strain (CFU), satellite (PFU), replicative system (CFU), attached replicative system (CFU), soil microbial community (CFU), and nutrient (%).
Figure. Model of E. coli Non-Replicative Kanamycin Targeting CRISPR System in Soil. Model of E. coli Non-Replicative Kanamycin Targeting CRISPR System in Soil. In order the subplots are the indicator strain (CFU), attached indicator strain (CFU), satellite (PFU), soil microbial community (CFU), and nutrient (%).
These new parameters from the soil microcosm experiments show a further spread which matches our experimental results.
Continuation and Improvements
The most significant place for improvement to these PDE models is to better characterize the fluid dynamics of satellite phages. Our experiments in the model soil microcosms and model colon have yielded estimations for some of these parameters to get more accurate results which were used to adjust the model parameters.
Contributions to Knowledge
We developed a novel partial differential equation (PDE) model for our satellite phage systems in the natural environments of the soil and the human colon. This model uses advection-diffusion fluid dynamics, Monod growth kinematics, and population dynamics to describe the interactions and efficacy of our satellite phage systems. Our results demonstrate that both our replicative and non-replicative systems have a measurable effect and our engineered replicative systems have more of an effect than the non-replicative satellite part systems. These models have the potential to be predictive with improved parameters which would contribute greatly to introducing satellite phages into both synthetic biology and future applications as therapies in natural environments. This model describes the utility of satellite phages and satellite phage systems as vectors in synthetic biology and is foundation to future modeling as satellites are better characterized.