### A Pluto.jl notebook ###
# v0.18.1

using Markdown
using InteractiveUtils

# ╔═╡ 2b2d7259-c670-4765-95d1-d33595729dc2
begin

	# load some packages -
	using PlutoUI
	using Distributions
	using Plots

	# setup paths -
	_PATH_TO_ROOT = pwd()
	_PATH_TO_DATA = joinpath(_PATH_TO_ROOT, "data")
	_PATH_TO_FIGS = joinpath(_PATH_TO_ROOT, "figs")
	_PATH_TO_SRC = joinpath(_PATH_TO_ROOT, "src")

	# show -
	nothing
end

# ╔═╡ 3024cd1a-61dd-4feb-bd21-2f6b38ee36b5
md"""
### Single cell gene expression dynamics

In this lecture we'll continue our discussion of gene expression dynamics. In particular:

* We'll derive a _starter_ model for the kinetic rate of transcription $r_{X,i}$
* We'll introduce the basics of the stochastic simulation algorithms developed by [Gillespie](https://en.wikipedia.org/wiki/Daniel_Gillespie), [Petzold](https://engineering.ucsb.edu/people/linda-petzold) and coworkers. 
* We'll use the [τ leaping algorithm](https://en.wikipedia.org/wiki/Tau-leaping) to simulate the expresssion of a single gene in an _Escherichia coli_ cell.

"""

# ╔═╡ 1342bde2-37d7-4bef-beac-9303742e786b
PlutoUI.LocalResource(joinpath(_PATH_TO_FIGS,"Fig-TXTL-Starter.png"))

# ╔═╡ 922c72ec-aefd-46d6-a465-cbf72a2cc05f
md"""
### A simple model for transcription kinetics

##### Assumptions and derivation
The rate of transcription is the product between the _kinetic limit_ of transcription $r_{X,i}$ and the control variable $u_{i}\left(\dots\right)$, which describes the activity of promoter $i$. One simple model of $r_{X, i}$ can be developed by thinking about RNA polymerase (RNAP) as an enzyme that produces mRNA but does not consume its substrate, gene $i$ ($G_{i}$). Suppose we proposed the elementary steps:

$$\begin{eqnarray}
	RNAP+G_{i}&\rightleftharpoons&{RNAP:G_{i}}\\
	{RNAP:G_{i}}&\longrightarrow&RNAP+G_{i}+m_{i}
\end{eqnarray}$$

The kinetics of each elementary step can be written using mass-action kinetics, i.e.,

$$\begin{eqnarray}
	r_{1,i} & = & k_{1,i}\left[RNAP\right]\left[G_{i}\right]\\
	r_{2,i} & = & k_{2,i}\left[RNAP:G_{i}\right]\\
	r_{3,i} & = & k_{3,i}\left[RNAP:G_{i}\right]
\end{eqnarray}$$

where $\left[\cdot\right]$ denotes a species concentration, and $k_{\star,i}$ denotes the rate constant governing the $\star$ elementary reaction:

* The rate $r_{1,i}$ describes the _association_ rate between RNAP and the ith gene,
* The rate $r_{2,i}$ represents the rate of _dissociation_ of the RNAP-gene complex, and
* The $r_{3,i}$ denotes the rate of _transcription_ of gene $i$ forming mRNA $m_{i}$

RNAP must obey the relationship:

$$\left[RNAP\right]_{T} = \left[RNAP\right] + \left[RNAP:G_{i}\right]$$

where $\left[RNAP\right]_{T}$ denotes the total RNAP concentration in the system, 
$\left[RNAP\right]$ denotes the free RNAP concentration (not bound to gene) while
$\left[RNAP:G_{i}\right]$ denotes the enzyme-substrate complex. 

To estimate the _overall_ rate of transcription $r_{X, i}$, we stipulate a single rate-limiting step out of the set of elementary reactions. Let's assume that the rate of elongation ($r_{3,i}$) is the slowest step. Thus, the overall transcription rate is then given by:

$$v = k_{3,i}\left[RNAP:G_{i}\right]$$

Following the same procedure as the derivation of Michaelis-Menten enzyme kinetics gives the expression:

$$r_{X,i} = V_{max,i}\frac{\left[G_{i}\right]}{K_{X,i}+\left[G_{i}\right]}$$

where $V_{max,i}\equiv{k_{3,i}}\left[RNAP\right]_{T}$, and $K_{X,i}$ denotes the saturation constant governing the expression of gene $i$:

##### A model for the elongation rate constant
We can estimate a value for $V_{max,i}$, and in particular the rate of elongation $k_{3,i}$ (units: 1/s), from first principles. The elongation rate constant can modeled as an average elongation rate of the polymerase $\langle e_{X} \rangle$ (units: nt/s) times the inverse of the length of gene $i$ $L_{i}$:

$$k_{3,i} \sim \langle e_{X} \rangle L_{i}^{-1}$$

Experimental measurements of the elongation rate $\langle e_{X} \rangle$ for multiple organisms are availble on [bionumbers](https://bionumbers.hms.harvard.edu/search.aspx?trm=RNAP+elongation+rate).

##### What is the saturation constant of transcription?

The saturation constant of transcription $K_{X,i}$ (units: conc) is defined as the ratio of elementary rate constants:

$$K_{X,i}\equiv\frac{k_{2,i}+k_{3,i}}{k_{1,i}}$$

In the limit of small $k_{3,i}$, the saturation constant $K_{X,i}$ is the inverse of the affinity (and directly proportional to the dissociation constant) of the polymerase for the gene. In other words:

* High affinity: tight binding $k_{1,i}\gg k_{2,i}$ implies $K_{X,i}$ is small, conversely 
* Low affinity: loose binding $k_{1,i}\ll k_{2,i}$ which implies $K_{X,i}$ is large.

We know that the RNAP dissociation constant $K_{D,i}$:

$$K_{D,i} = \frac{k_{2,i}}{k_{1,i}}$$

for the lac promoter in _E. coli_ is [$K_{D}\sim 550$ nM (units: nM or molecules/cell)](https://bionumbers.hms.harvard.edu/search.aspx?trm=KD+for+RNAP+to+lac+E.+coli). 
"""

# ╔═╡ 2d47c9da-c182-4e7f-9f4d-bfff4b97e3a4
md"""
### A crash course in Stochastic Simulation Algorithms (SSAs)

__Key references__
* [Gillespie DT. Stochastic simulation of chemical kinetics. Annu Rev Phys Chem. 2007;58:35-55. doi: 10.1146/annurev.physchem.58.032806.104637. PMID: 17037977.](https://pubmed.ncbi.nlm.nih.gov/17037977/)
* [Gillespie, D. T. (2001). Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem Phys. 115: 1716–1733. Bibcode:2001JChPh.115.1716G. doi:10.1063/1.1378322](https://aip.scitation.org/doi/10.1063/1.1378322)
"""

# ╔═╡ f9c5fea7-f713-403e-ad82-75c68ecfe81d
md"""
##### Example: Expression of a gene in a single _Escherichia coli_ cell
"""

# ╔═╡ c4cdac00-b344-4246-a445-23f710047a8c
begin

	# setup the parameters -
	parameters = Dict{String, Any}()

	# let's compute a literature value -
	k₁ = 100.0 	# units: 1/conc-t
	eₓ = 35.0 	# units: nt/s
	L = 1000.0 	# units: nt
	KD = 1000.0  # units: nM
	Kₓ = (eₓ*(1/L) + KD*k₁)/(k₁)

	# get parameters from bionumbers -
	parameters["RNAP_copy_number"] = 4600.0 					# units: copies/cell BNID 108601
	parameters["RNAP_elongation_rate"] = eₓ 					# units: nt/s BNID 111871
	parameters["gene_coding_length"] = L 						# units: nt BNID 111610
	parameters["mRNA_half_life"] = 0.5*(60)						# units: s BIND 104324
	parameters["gene_copy_number"] = 20.0 						# units: copies/cell problem
	parameters["saturation_constant_transcription"] = Kₓ 		# units: copies/cell problem
	
	# show -
	nothing
end

# ╔═╡ 782c6ddb-f5fa-46db-8a1e-5ae01285e3ef
Kₓ

# ╔═╡ 3c48cfd0-5403-4faf-a1f9-6460a18df49c
md"""
### Summary and conclusions
In this lecture we:

* Derived a _starter_ model for the kinetic rate of transcription $r_{X,i}$
* Introduced the basics of the stochastic simulation algorithms developed by [Gillespie](https://en.wikipedia.org/wiki/Daniel_Gillespie), [Petzold](https://engineering.ucsb.edu/people/linda-petzold) and coworkers. 
* Used the [τ leaping algorithm](https://en.wikipedia.org/wiki/Tau-leaping) to simulate the expresssion of a single gene in an _Escherichia coli_ cell.
"""

# ╔═╡ 8eae03bc-f2f4-4546-81eb-112be811f5dd
md"""
### Next time

* We'll move back to the continuous world to simulate gene expression for cell-free synthetic circuits and 
* We'll couple the rate of gene expression with the availability of metabolic resources. 

__Reference__:

[Allen TE, Palsson BØ. Sequence-based analysis of metabolic demands for protein synthesis in prokaryotes. J Theor Biol. 2003 Jan 7;220(1):1-18. doi: 10.1006/jtbi.2003.3087. PMID: 12453446.](https://pubmed.ncbi.nlm.nih.gov/12453446/)
"""

# ╔═╡ c7949300-f103-424d-996e-907cc0f04d92
md"""
### Function library
"""

# ╔═╡ cc2e34c3-e86c-40ca-be7a-0ba1d6f92201
function generate_step_size(S,X,R)::Float64

	# setup -
	ϵ = 0.03
	τ_array = Vector{Float64}()
	
	# compute the auxilary variables -
	μ = S*R
	σ = (S.^2)*R

	# compute the highest order event, and then τ -
	(ℳ,ℛ) = size(S)
	for i ∈ 1:ℳ

		# find the rates in which this species is a reactant -
		idx_negative = findall(x->x<0.0, S[i,:])

		# get the values for these rates -
		gᵢ = maximum(R[idx_negative])

		# terms -
		term_1 = max(ϵ*X[i]/gᵢ, 1)/abs(μ[i])
		term_2 = (max(ϵ*X[i]/gᵢ, 1)^2)/σ[i]
		τᵢ = min(term_1, term_2)
		push!(τ_array, τᵢ)
	end
	
	return minimum(τ_array)
end

# ╔═╡ de95690b-a564-471d-8597-0d9e6dcf2f1b
function propensity(X::Vector{Float64}, parameters::Dict{String, Any})

	# get values from the parameters dictionary -
	RNAP = parameters["RNAP_copy_number"]
	eₓ = parameters["RNAP_elongation_rate"]
	Kₓ = parameters["saturation_constant_transcription"]
	G = parameters["gene_copy_number"]
	t_half_life = parameters["mRNA_half_life"]
	L = parameters["gene_coding_length"]

	# compute Vmax and the degrdation constant -
	Vmax = RNAP*(eₓ)*(1/L)
	θₓ = -(log(0.5)/t_half_life)

	# compute the propensity -
	a = Array{Float64,1}(undef,2)
	a[1] = Vmax*(G/(Kₓ + G))
	a[2] = θₓ*X[1]

	# return -
	return a
end

# ╔═╡ ed06b789-ece0-4d4e-a9bc-4431806f3b43
function tau_leaping_solve(S::Matrix{Float64}, Xₒ::Array{Float64,1}, Tₒ::Float64, T₁::Float64,
	parameters::Dict{String,Any}; τ::Union{Nothing,Float64} = nothing, should_update_τ::Bool = false)
	
	# initialize some stuff -
	loop_counter = 1
	should_we_continue = true
	state_array = Array{Vector{Float64},1}()
	time_array = Vector{Float64}()
	T = Tₒ
	X = Xₒ
	K_array = Vector{Float64}()

	push!(time_array,Tₒ)
	push!(state_array,Xₒ)
	
	# generate an initial propensity function -
	R = propensity(Xₒ, parameters)

	# generate a step size -
	if (should_update_τ == true)
		τ = generate_step_size(S, X, R)
	elseif (isnothing(τ) == true)
		τ = 0.01	
	end
	
	# main loop -
	while (should_we_continue)

		# K-array -
		for r ∈ R
			# for each channel / compute a random number of "firings" -
			λ = r*τ
			K = rand(Poisson(λ))
			push!(K_array, K)
		end

		# update the state -
		Δ = S*K_array
		X += Δ

		# correct for negative numbers if there are any -
		#X = max.(0.0, X)

		# recompute the propensity with the new state -
		R = propensity(X, parameters)
	
		# package into the state array -
		push!(state_array, X)
			
		# update time -
		T += τ
		push!(time_array, T)
		
		# clean up -
		empty!(K_array)

		# should we go around again?
		if (T>=T₁)
			should_we_continue = false
		else

			if (should_update_τ == true)
				τ = generate_step_size(S, X, R)
			elseif (isnothing(τ) == true)
				τ = 0.01	
			end
		end
	end

	SA = vcat(state_array...)
	TA = time_array
	
	# return -
	return (TA, SA)
end

# ╔═╡ b8bed16f-e4b3-4ee8-8c73-a04d1df43205
begin

	# setup problem -
	Xₒ = ones(1) # suppose we start with a single mRNA molecule -
	number_of_cells = 3
	cell_dictionary = Dict()

	# we have 1 species (mRNA) and two reaction "channels" -
	S = Array{Float64,2}(undef, 1,2)
	S[1,1] = 1.0
	S[1,2] = -1.0
	
	# setup time -
	Tₒ = 0.0
	T₁ = 120.0 # simulate number of s 

	# run solver -
	for i ∈ 1:number_of_cells
		cell_dictionary[i] = tau_leaping_solve(S, Xₒ, Tₒ, T₁, parameters; τ = 0.01)
	end
	
	# show -
	nothing
end

# ╔═╡ 5e3fb403-4f13-451b-a996-2466504971e7
begin

	for i ∈ 1:number_of_cells
		(T,X) = cell_dictionary[i]
		if (i == 1)
			plot(T,X, legend=false)
		else
			plot!(T,X)
		end
	end
	xlabel!("Time (s)", fontsize=18)
	ylabel!("Copies (mRNA/cell)", fontsize=18)
	current()
end

# ╔═╡ 8cfc75c7-5893-4505-8825-b2b351088aea
TableOfContents(title="📚 Lecture Outline", indent=true, depth=5, aside=true)

# ╔═╡ cdf5e092-499c-4010-a2a3-c050a9395fb4
html"""
<style>
main {
    max-width: 860px;
    width: 70%;
    margin: auto;
    font-family: "Roboto, monospace";
}

a {
    color: blue;
    text-decoration: none;
}
</style>"""

# ╔═╡ ea0020f6-a051-11ec-2fb6-ef39a663134b
html"""
<script>
	// initialize -
	var section = 0;
	var subsection = 0;
	var subsubsection = 0;
	var headers = document.querySelectorAll('h3, h5, h6');
	
	// main loop -
	for (var i=0; i < headers.length; i++) {
	    
		var header = headers[i];
	    var text = header.innerText;
	    var original = header.getAttribute("text-original");
	    if (original === null) {
	        
			// Save original header text
	        header.setAttribute("text-original", text);
	    } else {
	        
			// Replace with original text before adding section number
	        text = header.getAttribute("text-original");
	    }
	
	    var numbering = "";
	    switch (header.tagName) {
	        case 'H3':
	            section += 1;
	            numbering = section + ".";
	            subsection = 0;
	            break;
	        case 'H5':
	            subsection += 1;
	            numbering = section + "." + subsection;
	            break;
			case 'H6':
	            subsubsection += 1;
	            numbering = section + "." + subsection + "." + subsubsection;
	            break;
	    }
		// update the header text 
		header.innerText = numbering + " " + text;
	};
</script>"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
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git-tree-sha1 = "bf0a1121af131d9974241ba53f601211e9303a9e"
uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
version = "0.7.37"

[[deps.Preferences]]
deps = ["TOML"]
git-tree-sha1 = "de893592a221142f3db370f48290e3a2ef39998f"
uuid = "21216c6a-2e73-6563-6e65-726566657250"
version = "1.2.4"

[[deps.Printf]]
deps = ["Unicode"]
uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"

[[deps.Qt5Base_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Fontconfig_jll", "Glib_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "OpenSSL_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libxcb_jll", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_keysyms_jll", "Xorg_xcb_util_renderutil_jll", "Xorg_xcb_util_wm_jll", "Zlib_jll", "xkbcommon_jll"]
git-tree-sha1 = "ad368663a5e20dbb8d6dc2fddeefe4dae0781ae8"
uuid = "ea2cea3b-5b76-57ae-a6ef-0a8af62496e1"
version = "5.15.3+0"

[[deps.QuadGK]]
deps = ["DataStructures", "LinearAlgebra"]
git-tree-sha1 = "78aadffb3efd2155af139781b8a8df1ef279ea39"
uuid = "1fd47b50-473d-5c70-9696-f719f8f3bcdc"
version = "2.4.2"

[[deps.REPL]]
deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"

[[deps.Random]]
deps = ["SHA", "Serialization"]
uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[deps.RecipesBase]]
git-tree-sha1 = "6bf3f380ff52ce0832ddd3a2a7b9538ed1bcca7d"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
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[[deps.RecipesPipeline]]
deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase"]
git-tree-sha1 = "995a812c6f7edea7527bb570f0ac39d0fb15663c"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.5.1"

[[deps.Reexport]]
git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "1.2.2"

[[deps.RelocatableFolders]]
deps = ["SHA", "Scratch"]
git-tree-sha1 = "cdbd3b1338c72ce29d9584fdbe9e9b70eeb5adca"
uuid = "05181044-ff0b-4ac5-8273-598c1e38db00"
version = "0.1.3"

[[deps.Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "838a3a4188e2ded87a4f9f184b4b0d78a1e91cb7"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.3.0"

[[deps.Rmath]]
deps = ["Random", "Rmath_jll"]
git-tree-sha1 = "bf3188feca147ce108c76ad82c2792c57abe7b1f"
uuid = "79098fc4-a85e-5d69-aa6a-4863f24498fa"
version = "0.7.0"

[[deps.Rmath_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "68db32dff12bb6127bac73c209881191bf0efbb7"
uuid = "f50d1b31-88e8-58de-be2c-1cc44531875f"
version = "0.3.0+0"

[[deps.SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"

[[deps.Scratch]]
deps = ["Dates"]
git-tree-sha1 = "0b4b7f1393cff97c33891da2a0bf69c6ed241fda"
uuid = "6c6a2e73-6563-6170-7368-637461726353"
version = "1.1.0"

[[deps.Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[deps.SharedArrays]]
deps = ["Distributed", "Mmap", "Random", "Serialization"]
uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383"

[[deps.Showoff]]
deps = ["Dates", "Grisu"]
git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de"
uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f"
version = "1.0.3"

[[deps.Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[deps.SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "b3363d7460f7d098ca0912c69b082f75625d7508"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.0.1"

[[deps.SparseArrays]]
deps = ["LinearAlgebra", "Random"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[deps.SpecialFunctions]]
deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "5ba658aeecaaf96923dce0da9e703bd1fe7666f9"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "2.1.4"

[[deps.StaticArrays]]
deps = ["LinearAlgebra", "Random", "Statistics"]
git-tree-sha1 = "74fb527333e72ada2dd9ef77d98e4991fb185f04"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.4.1"

[[deps.Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[deps.StatsAPI]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "c3d8ba7f3fa0625b062b82853a7d5229cb728b6b"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.2.1"

[[deps.StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "8977b17906b0a1cc74ab2e3a05faa16cf08a8291"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.16"

[[deps.StatsFuns]]
deps = ["ChainRulesCore", "HypergeometricFunctions", "InverseFunctions", "IrrationalConstants", "LogExpFunctions", "Reexport", "Rmath", "SpecialFunctions"]
git-tree-sha1 = "25405d7016a47cf2bd6cd91e66f4de437fd54a07"
uuid = "4c63d2b9-4356-54db-8cca-17b64c39e42c"
version = "0.9.16"

[[deps.StructArrays]]
deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"]
git-tree-sha1 = "57617b34fa34f91d536eb265df67c2d4519b8b98"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.6.5"

[[deps.SuiteSparse]]
deps = ["Libdl", "LinearAlgebra", "Serialization", "SparseArrays"]
uuid = "4607b0f0-06f3-5cda-b6b1-a6196a1729e9"

[[deps.TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"

[[deps.TableTraits]]
deps = ["IteratorInterfaceExtensions"]
git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39"
uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c"
version = "1.0.1"

[[deps.Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "OrderedCollections", "TableTraits", "Test"]
git-tree-sha1 = "5ce79ce186cc678bbb5c5681ca3379d1ddae11a1"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.7.0"

[[deps.Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"

[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[deps.URIs]]
git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.3.0"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unzip]]
git-tree-sha1 = "34db80951901073501137bdbc3d5a8e7bbd06670"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.1.2"

[[deps.Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.12+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
# ╟─3024cd1a-61dd-4feb-bd21-2f6b38ee36b5
# ╠═1342bde2-37d7-4bef-beac-9303742e786b
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