diff --git a/dev/404.html b/dev/404.html index c89aeac..7ace7bf 100644 --- a/dev/404.html +++ b/dev/404.html @@ -6,16 +6,16 @@ 404 | BioMarkovChains.jl - + - - + + -
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+ \ No newline at end of file diff --git a/dev/api.html b/dev/api.html index cb72cef..46d4735 100644 --- a/dev/api.html +++ b/dev/api.html @@ -6,18 +6,18 @@ BioMarkovChains.jl - + - - - - - + + + + + -
BioMarkovChains.jl

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# BioMarkovChains.BMC — Type.
julia
BMC

Alias for the type BioMarkovChain.

source


# BioMarkovChains.BioMarkovChain — Type.
julia
struct BioMarkovChain{S<:DataType, M<:AbstractMatrix, I<:AbstractVector, N<:Integer} <: AbstractBioMarkovChain

A BioMarkovChain represents a Markov chain used in biological sequence analysis. It contains a transition probability matrix (tpm) and an initial distribution of probabilities (inits) and also the order of the Markov chain.

Fields

  • alphabet::A: Is the state space of the sequence whether DNA, RNA AminoAcid DataTypes.

  • tpm::M: The transition probability matrix.

  • inits::I: The initial distribution of probabilities.

  • n::N: The order of the Markov chain.

Constructors

  • BioMarkovChain(tpm::M, inits::I, n::N=1) where {M<:AbstractMatrix, I<:AbstractVector, N<:Integer}: Constructs a BioMarkovChain object with the provided transition probability matrix, initial distribution, and order.

  • BioMarkovChain(sequence::LongNucOrView{4}, n::Int64=1): Constructs a BioMarkovChain object based on the DNA sequence and transition order.

Example

sequence = LongDNA{4}("ACTACATCTA")
+    
BioMarkovChains.jl
Appearance
# BioMarkovChains.BMC — Type. 
julia
BMC
Alias for the type BioMarkovChain.
source

# BioMarkovChains.BioMarkovChain — Type. 
julia
struct BioMarkovChain{S<:DataType, M<:AbstractMatrix, I<:AbstractVector, N<:Integer} <: AbstractBioMarkovChain
A BioMarkovChain represents a Markov chain used in biological sequence analysis. It contains a transition probability matrix (tpm) and an initial distribution of probabilities (inits) and also the order of the Markov chain.
Fields
  • alphabet::A: Is the state space of the sequence whether DNA, RNA AminoAcid DataTypes.
  • tpm::M: The transition probability matrix.
  • inits::I: The initial distribution of probabilities.
  • n::N: The order of the Markov chain.
Constructors
  • BioMarkovChain(tpm::M, inits::I, n::N=1) where {M<:AbstractMatrix, I<:AbstractVector, N<:Integer}: Constructs a BioMarkovChain object with the provided transition probability matrix, initial distribution, and order.
  • BioMarkovChain(sequence::LongNucOrView{4}, n::Int64=1): Constructs a BioMarkovChain object based on the DNA sequence and transition order.
Example
sequence = LongDNA{4}("ACTACATCTA")
 
 model = BioMarkovChain(sequence, 2)
 BioMarkovChain:
@@ -77,8 +77,8 @@
  0.0  0.0  1.0  0.0
  0.0  0.0  0.0  1.0
  0.0  1.0  0.0  0.0
- 1.0  0.0  0.0  0.0
source


-    
+ 1.0  0.0  0.0  0.0

source


+ \ No newline at end of file diff --git a/dev/article01.html b/dev/article01.html new file mode 100644 index 0000000..bd5f911 --- /dev/null +++ b/dev/article01.html @@ -0,0 +1,42 @@ + + + + + + Using other functions | BioMarkovChains.jl + + + + + + + + + + + + + +
BioMarkovChains.jl

Appearance

Using other functions ​

We can now calculate a transition probability matrix from a LongDNA sequence using the transition_probability_matrix and initials methods for a given LongDNA sequence:

julia
using BioSequences, GeneFinder
+
+sequence = dna"CCTCCCGGACCCTGGGCTCGGGAC"
+
+tpm = transition_probability_matrix(sequence)
+initials = initials(sequence)
+
+println(tpm)
+println(initials)
4×4 Matrix{Float64}:
+ 0.0   1.0       0.0       0.0
+ 0.0   0.5       0.2       0.3
+ 0.25  0.125     0.625     0.0
+ 0.0   0.666667  0.333333  0.0
+
+4-element Vector{Float64}:
+ 0.08695652173913043
+ 0.43478260869565216
+ 0.34782608695652173
+ 0.13043478260869565

More conveniently, we can now use the transition_model method and obtain the transition probabilities and the initial distribution and build a transition model (BioMarkovChain):

+ + + + \ No newline at end of file diff --git a/dev/assets/api.md.BpffH_a8.js b/dev/assets/api.md.CI7p70XE.js similarity index 99% rename from dev/assets/api.md.BpffH_a8.js rename to dev/assets/api.md.CI7p70XE.js index b649fe3..e60df66 100644 --- a/dev/assets/api.md.BpffH_a8.js +++ b/dev/assets/api.md.CI7p70XE.js @@ -1,4 +1,4 @@ -import{_ as T,c as e,m as a,a as t,a7 as Q,o as s}from"./chunks/framework.DAOdO8p8.js";const c6=JSON.parse('{"title":"","description":"","frontmatter":{},"headers":[],"relativePath":"api.md","filePath":"api.md","lastUpdated":null}'),l={name:"api.md"},n=Q(`
# BioMarkovChains.BMC — Type.
julia
BMC

Alias for the type BioMarkovChain.

source


# BioMarkovChains.BioMarkovChain — Type.
julia
struct BioMarkovChain{S<:DataType, M<:AbstractMatrix, I<:AbstractVector, N<:Integer} <: AbstractBioMarkovChain

A BioMarkovChain represents a Markov chain used in biological sequence analysis. It contains a transition probability matrix (tpm) and an initial distribution of probabilities (inits) and also the order of the Markov chain.

Fields

Constructors

Example

sequence = LongDNA{4}("ACTACATCTA")
+import{_ as T,c as e,m as a,a as t,a7 as Q,o as s}from"./chunks/framework.CVYz1rUz.js";const c6=JSON.parse('{"title":"","description":"","frontmatter":{},"headers":[],"relativePath":"api.md","filePath":"api.md","lastUpdated":null}'),l={name:"api.md"},n=Q(`
# BioMarkovChains.BMC — Type. 
julia
BMC
Alias for the type BioMarkovChain.
source

# BioMarkovChains.BioMarkovChain — Type. 
julia
struct BioMarkovChain{S<:DataType, M<:AbstractMatrix, I<:AbstractVector, N<:Integer} <: AbstractBioMarkovChain
A BioMarkovChain represents a Markov chain used in biological sequence analysis. It contains a transition probability matrix (tpm) and an initial distribution of probabilities (inits) and also the order of the Markov chain.
Fields
  • alphabet::A: Is the state space of the sequence whether DNA, RNA AminoAcid DataTypes.
  • tpm::M: The transition probability matrix.
  • inits::I: The initial distribution of probabilities.
  • n::N: The order of the Markov chain.
Constructors
  • BioMarkovChain(tpm::M, inits::I, n::N=1) where {M<:AbstractMatrix, I<:AbstractVector, N<:Integer}: Constructs a BioMarkovChain object with the provided transition probability matrix, initial distribution, and order.
  • BioMarkovChain(sequence::LongNucOrView{4}, n::Int64=1): Constructs a BioMarkovChain object based on the DNA sequence and transition order.
Example
sequence = LongDNA{4}("ACTACATCTA")
 
 model = BioMarkovChain(sequence, 2)
 BioMarkovChain:
diff --git a/dev/assets/api.md.BpffH_a8.lean.js b/dev/assets/api.md.CI7p70XE.lean.js
similarity index 99%
rename from dev/assets/api.md.BpffH_a8.lean.js
rename to dev/assets/api.md.CI7p70XE.lean.js
index bef5249..e97867c 100644
--- a/dev/assets/api.md.BpffH_a8.lean.js
+++ b/dev/assets/api.md.CI7p70XE.lean.js
@@ -1 +1 @@
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diff --git a/dev/assets/app.CO7flB_1.js b/dev/assets/app.CzYKMekC.js
similarity index 81%
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rename to dev/assets/app.CzYKMekC.js
index 691c187..e854e05 100644
--- a/dev/assets/app.CO7flB_1.js
+++ b/dev/assets/app.CzYKMekC.js
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-import{j as o,a8 as p,a9 as u,aa as l,ab as c,ac as f,ad as d,ae as m,af as h,ag as g,ah as A,Y as P,d as _,u as v,l as R,z as w,ai as y,aj as C,ak as E,a6 as b}from"./chunks/framework.DAOdO8p8.js";import{R as T}from"./chunks/theme.Bo19b1i6.js";function i(e){if(e.extends){const a=i(e.extends);return{...a,...e,async enhanceApp(t){a.enhanceApp&&await a.enhanceApp(t),e.enhanceApp&&await e.enhanceApp(t)}}}return e}const s=i(T),S=_({name:"VitePressApp",setup(){const{site:e,lang:a,dir:t}=v();return R(()=>{w(()=>{document.documentElement.lang=a.value,document.documentElement.dir=t.value})}),e.value.router.prefetchLinks&&y(),C(),E(),s.setup&&s.setup(),()=>b(s.Layout)}});async function j(){globalThis.__VITEPRESS__=!0;const e=L(),a=D();a.provide(u,e);const t=l(e.route);return a.provide(c,t),a.component("Content",f),a.component("ClientOnly",d),Object.defineProperties(a.config.globalProperties,{$frontmatter:{get(){return t.frontmatter.value}},$params:{get(){return t.page.value.params}}}),s.enhanceApp&&await s.enhanceApp({app:a,router:e,siteData:m}),{app:a,router:e,data:t}}function D(){return h(S)}function L(){let e=o,a;return g(t=>{let n=A(t),r=null;return n&&(e&&(a=n),(e||a===n)&&(n=n.replace(/\.js$/,".lean.js")),r=P(()=>import(n),[])),o&&(e=!1),r},s.NotFound)}o&&j().then(({app:e,router:a,data:t})=>{a.go().then(()=>{p(a.route,t.site),e.mount("#app")})});export{j as createApp};
+import{j as o,a9 as p,aa as u,ab as l,ac as c,ad as f,ae as d,af as m,ag as h,ah as g,ai as A,Y as P,d as _,u as v,l as R,z as w,aj as y,ak as C,al as E,a6 as b}from"./chunks/framework.CVYz1rUz.js";import{R as T}from"./chunks/theme.CUlsXZi2.js";function i(e){if(e.extends){const a=i(e.extends);return{...a,...e,async enhanceApp(t){a.enhanceApp&&await a.enhanceApp(t),e.enhanceApp&&await e.enhanceApp(t)}}}return e}const s=i(T),S=_({name:"VitePressApp",setup(){const{site:e,lang:a,dir:t}=v();return R(()=>{w(()=>{document.documentElement.lang=a.value,document.documentElement.dir=t.value})}),e.value.router.prefetchLinks&&y(),C(),E(),s.setup&&s.setup(),()=>b(s.Layout)}});async function j(){globalThis.__VITEPRESS__=!0;const e=L(),a=D();a.provide(u,e);const t=l(e.route);return a.provide(c,t),a.component("Content",f),a.component("ClientOnly",d),Object.defineProperties(a.config.globalProperties,{$frontmatter:{get(){return t.frontmatter.value}},$params:{get(){return t.page.value.params}}}),s.enhanceApp&&await s.enhanceApp({app:a,router:e,siteData:m}),{app:a,router:e,data:t}}function D(){return h(S)}function L(){let e=o,a;return g(t=>{let n=A(t),r=null;return n&&(e&&(a=n),(e||a===n)&&(n=n.replace(/\.js$/,".lean.js")),r=P(()=>import(n),[])),o&&(e=!1),r},s.NotFound)}o&&j().then(({app:e,router:a,data:t})=>{a.go().then(()=>{p(a.route,t.site),e.mount("#app")})});export{j as createApp};
diff --git a/dev/assets/article01.md.-w_lHw60.js b/dev/assets/article01.md.-w_lHw60.js
new file mode 100644
index 0000000..a1643b0
--- /dev/null
+++ b/dev/assets/article01.md.-w_lHw60.js
@@ -0,0 +1,19 @@
+import{_ as s,c as a,o as n,a7 as i}from"./chunks/framework.CVYz1rUz.js";const u=JSON.parse('{"title":"Using other functions","description":"","frontmatter":{},"headers":[],"relativePath":"article01.md","filePath":"article01.md","lastUpdated":null}'),e={name:"article01.md"},t=i(`
Using other functions ​
We can now calculate a transition probability matrix from a LongDNA sequence using the transition_probability_matrix and initials methods for a given LongDNA sequence:
julia
using BioSequences, GeneFinder
+
+sequence = dna"CCTCCCGGACCCTGGGCTCGGGAC"
+
+tpm = transition_probability_matrix(sequence)
+initials = initials(sequence)
+
+println(tpm)
+println(initials)
4×4 Matrix{Float64}:
+ 0.0   1.0       0.0       0.0
+ 0.0   0.5       0.2       0.3
+ 0.25  0.125     0.625     0.0
+ 0.0   0.666667  0.333333  0.0
+
+4-element Vector{Float64}:
+ 0.08695652173913043
+ 0.43478260869565216
+ 0.34782608695652173
+ 0.13043478260869565
More conveniently, we can now use the transition_model method and obtain the transition probabilities and the initial distribution and build a transition model (BioMarkovChain):
`,5),p=[t];function l(o,h,c,r,d,k){return n(),a("div",null,p)}const E=s(e,[["render",l]]);export{u as __pageData,E as default};
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new file mode 100644
index 0000000..6cdf871
--- /dev/null
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@@ -0,0 +1 @@
+import{_ as s,c as a,o as n,a7 as i}from"./chunks/framework.CVYz1rUz.js";const u=JSON.parse('{"title":"Using other functions","description":"","frontmatter":{},"headers":[],"relativePath":"article01.md","filePath":"article01.md","lastUpdated":null}'),e={name:"article01.md"},t=i("",5),p=[t];function l(o,h,c,r,d,k){return n(),a("div",null,p)}const E=s(e,[["render",l]]);export{u as __pageData,E as default};
diff --git a/dev/assets/biomarkovchains.md.C4HXJoA5.js b/dev/assets/biomarkovchains.md.C4HXJoA5.js
new file mode 100644
index 0000000..b0d2571
--- /dev/null
+++ b/dev/assets/biomarkovchains.md.C4HXJoA5.js
@@ -0,0 +1 @@
+import{_ as s,c as a,m as t,a as Q,a7 as e,o as T}from"./chunks/framework.CVYz1rUz.js";const h6=JSON.parse('{"title":"DNA as a Markov chain","description":"","frontmatter":{},"headers":[],"relativePath":"biomarkovchains.md","filePath":"biomarkovchains.md","lastUpdated":null}'),l={name:"biomarkovchains.md"},o=e('
DNA as a Markov chain ​
Several packages (e.g. MarkovChainsHammer.jl, DiscreteMarkovChains.jl, etc.) in the Julia ecosystem have been implemented to work with Markov chains with a state space of integers, those could be efficient in many ways, but they are clumsy to work with a specialized biological types as in the BioJulia ecosystem. Therefore, in the GeneFinder package we dedicated some implementations to work with BioSequence types so that we can expand the functionality in an efficient way (see complete API).
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"),t("em",null,"Comparative Gene Finding"),Q(". Vol. 20. Computational Biology. London: Springer London. "),t("a",{href:"http://link.springer.com/10.1007/978-1-4471-6693-1",target:"_blank",rel:"noreferrer"},"http://link.springer.com/10.1007/978-1-4471-6693-1"),Q(".")],-1);function s6(l6,o6,n6,d6,m6,r6){return T(),a("div",null,[o,t("p",null,[Q("One important step towards many gene finding algorithms is to represent a DNA sequence as a Markov chain. In this representation a DNA sequence of a reduced alphabet "),t("mjx-container",n,[(T(),a("svg",d,r)),i]),Q(" is draw as a four-vertex graph, where each letter of "),t("mjx-container",h,[(T(),a("svg",c,g)),x]),Q(" is a "),w,Q(" and the edges of the graph represent "),u,Q(" from one nucleotide to another in a sequence (e.g. "),t("mjx-container",H,[(T(),a("svg",_,f)),M]),Q(" represent a single nucleotide to nucleotide transition). This is also considered more specifically as a Discrete Markov chain (Axelson-Fisk 2015). The complete set of transitions and states of a DNA sequence of alphabet "),t("mjx-container",k,[(T(),a("svg",y,b)),V]),Q(".")]),t("p",null,[Q("More formally a Markov chain is a random process where each state is a random variable "),t("mjx-container",v,[(T(),a("svg",D,A)),C]),Q(" where "),t("mjx-container",R,[(T(),a("svg",S,P)),X]),Q(" is a discrete time in a finite sequence "),t("mjx-container",E,[(T(),a("svg",I,G)),N]),Q(" and the probability to jump from one state into another is "),z,Q(". Therefore a definition of this "),J,Q(" is given by:")]),q,F,t("p",null,[Q("where "),t("mjx-container",$,[(T(),a("svg",U,K)),Y]),Q(" . This property led us to generalize a way to calculate the probability of a sequence "),t("mjx-container",t1,[(T(),a("svg",Q1,T1)),e1]),Q(" from a process "),t("mjx-container",s1,[(T(),a("svg",l1,n1)),d1]),Q(" where each random variable is a nucleotide from "),t("mjx-container",m1,[(T(),a("svg",r1,h1)),c1]),Q(" so that:")]),t("mjx-container",p1,[(T(),a("svg",g1,w1)),u1]),t("p",null,[Q("Note that previous equations has two terms, a initial probability "),t("mjx-container",H1,[(T(),a("svg",_1,f1)),M1]),Q(" and the the product of all transitions beginning at "),t("mjx-container",k1,[(T(),a("svg",y1,b1)),V1]),Q(". So, to calculate the initial probability distribution of each of the nucleotides of a string "),t("mjx-container",v1,[(T(),a("svg",D1,A1)),C1]),Q(" with the alphabet 𝒜 we can first calculate the transition probability matrix "),t("mjx-container",R1,[(T(),a("svg",S1,P1)),X1]),Q(" out of the frequency count of the transitions. In an alphabet 𝒜 we got "),t("mjx-container",E1,[(T(),a("svg",I1,G1)),N1]),Q(" transitions of one order, that is the "),t("mjx-container",z1,[(T(),a("svg",J1,F1)),$1]),Q(" which coincides with the frequency of the dinucleotides in the sequence. So we can later in fact build a "),t("mjx-container",U1,[(T(),a("svg",W1,Y1)),t2]),Q(" matrix representing all the transitions. For instance in a DNA sequence "),t("mjx-container",Q2,[(T(),a("svg",a2,e2)),s2]),Q(" of 24 nucleotides:")]),t("mjx-container",l2,[(T(),a("svg",o2,d2)),m2]),t("p",null,[Q("We can calculate each frequency nucleotide to any other nucleotide "),t("mjx-container",r2,[(T(),a("svg",i2,c2)),p2]),Q(" where "),t("mjx-container",g2,[(T(),a("svg",x2,u2)),H2]),Q(" is the actual count of the dinucleotide, and therefore "),t("mjx-container",_2,[(T(),a("svg",L2,M2)),k2]),Q(" is the counts of the nucleotide "),t("mjx-container",y2,[(T(),a("svg",Z2,V2)),v2]),Q(" to any other nucleotide and build the transition probability matrix:")]),t("mjx-container",D2,[(T(),a("svg",j2,C2)),R2]),t("p",null,[Q("It is noteworthy that initial probabilities can also be obtained from the counts of each nucleotide transitions "),t("mjx-container",S2,[(T(),a("svg",O2,X2)),E2]),Q(" over the total sum of the dinucleotide counts "),t("mjx-container",I2,[(T(),a("svg",B2,N2)),z2]),Q(":")]),t("mjx-container",J2,[(T(),a("svg",q2,$2)),U2]),t("p",null,[Q("That way for the previous example example we can can calculate the initial probabilities "),t("mjx-container",W2,[(T(),a("svg",K2,t6)),Q6]),Q(". Both set of probabilities composed a "),a6,Q(" that can be used to predict the probability of any DNA sequence using equation (2).")]),T6,e6])}const c6=s(l,[["render",s6]]);export{h6 as __pageData,c6 as default};
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"),t("em",null,"Comparative Gene Finding"),Q(". Vol. 20. Computational Biology. London: Springer London. "),t("a",{href:"http://link.springer.com/10.1007/978-1-4471-6693-1",target:"_blank",rel:"noreferrer"},"http://link.springer.com/10.1007/978-1-4471-6693-1"),Q(".")],-1);function s6(l6,o6,n6,d6,m6,r6){return T(),a("div",null,[o,t("p",null,[Q("One important step towards many gene finding algorithms is to represent a DNA sequence as a Markov chain. In this representation a DNA sequence of a reduced alphabet "),t("mjx-container",n,[(T(),a("svg",d,r)),i]),Q(" is draw as a four-vertex graph, where each letter of "),t("mjx-container",h,[(T(),a("svg",c,g)),x]),Q(" is a "),w,Q(" and the edges of the graph represent "),u,Q(" from one nucleotide to another in a sequence (e.g. "),t("mjx-container",H,[(T(),a("svg",_,f)),M]),Q(" represent a single nucleotide to nucleotide transition). This is also considered more specifically as a Discrete Markov chain (Axelson-Fisk 2015). The complete set of transitions and states of a DNA sequence of alphabet "),t("mjx-container",k,[(T(),a("svg",y,b)),V]),Q(".")]),t("p",null,[Q("More formally a Markov chain is a random process where each state is a random variable "),t("mjx-container",v,[(T(),a("svg",D,A)),C]),Q(" where "),t("mjx-container",R,[(T(),a("svg",S,P)),X]),Q(" is a discrete time in a finite sequence "),t("mjx-container",E,[(T(),a("svg",I,G)),N]),Q(" and the probability to jump from one state into another is "),z,Q(". Therefore a definition of this "),J,Q(" is given by:")]),q,F,t("p",null,[Q("where "),t("mjx-container",$,[(T(),a("svg",U,K)),Y]),Q(" . This property led us to generalize a way to calculate the probability of a sequence "),t("mjx-container",t1,[(T(),a("svg",Q1,T1)),e1]),Q(" from a process "),t("mjx-container",s1,[(T(),a("svg",l1,n1)),d1]),Q(" where each random variable is a nucleotide from "),t("mjx-container",m1,[(T(),a("svg",r1,h1)),c1]),Q(" so that:")]),t("mjx-container",p1,[(T(),a("svg",g1,w1)),u1]),t("p",null,[Q("Note that previous equations has two terms, a initial probability "),t("mjx-container",H1,[(T(),a("svg",_1,f1)),M1]),Q(" and the the product of all transitions beginning at "),t("mjx-container",k1,[(T(),a("svg",y1,b1)),V1]),Q(". So, to calculate the initial probability distribution of each of the nucleotides of a string "),t("mjx-container",v1,[(T(),a("svg",D1,A1)),C1]),Q(" with the alphabet 𝒜 we can first calculate the transition probability matrix "),t("mjx-container",R1,[(T(),a("svg",S1,P1)),X1]),Q(" out of the frequency count of the transitions. In an alphabet 𝒜 we got "),t("mjx-container",E1,[(T(),a("svg",I1,G1)),N1]),Q(" transitions of one order, that is the "),t("mjx-container",z1,[(T(),a("svg",J1,F1)),$1]),Q(" which coincides with the frequency of the dinucleotides in the sequence. So we can later in fact build a "),t("mjx-container",U1,[(T(),a("svg",W1,Y1)),t2]),Q(" matrix representing all the transitions. For instance in a DNA sequence "),t("mjx-container",Q2,[(T(),a("svg",a2,e2)),s2]),Q(" of 24 nucleotides:")]),t("mjx-container",l2,[(T(),a("svg",o2,d2)),m2]),t("p",null,[Q("We can calculate each frequency nucleotide to any other nucleotide "),t("mjx-container",r2,[(T(),a("svg",i2,c2)),p2]),Q(" where "),t("mjx-container",g2,[(T(),a("svg",x2,u2)),H2]),Q(" is the actual count of the dinucleotide, and therefore "),t("mjx-container",_2,[(T(),a("svg",L2,M2)),k2]),Q(" is the counts of the nucleotide "),t("mjx-container",y2,[(T(),a("svg",Z2,V2)),v2]),Q(" to any other nucleotide and build the transition probability matrix:")]),t("mjx-container",D2,[(T(),a("svg",j2,C2)),R2]),t("p",null,[Q("It is noteworthy that initial probabilities can also be obtained from the counts of each nucleotide transitions "),t("mjx-container",S2,[(T(),a("svg",O2,X2)),E2]),Q(" over the total sum of the dinucleotide counts "),t("mjx-container",I2,[(T(),a("svg",B2,N2)),z2]),Q(":")]),t("mjx-container",J2,[(T(),a("svg",q2,$2)),U2]),t("p",null,[Q("That way for the previous example example we can can calculate the initial probabilities "),t("mjx-container",W2,[(T(),a("svg",K2,t6)),Q6]),Q(". Both set of probabilities composed a "),a6,Q(" that can be used to predict the probability of any DNA sequence using equation (2).")]),T6,e6])}const c6=s(l,[["render",s6]]);export{h6 as __pageData,c6 as default};
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DNA as a Markov chain ​
Several packages (e.g. MarkovChainsHammer.jl, DiscreteMarkovChains.jl, etc.) in the Julia ecosystem have been implemented to work with Markov chains with a state space of integers, those could be efficient in many ways, but they are clumsy to work with a specialized biological types as in the BioJulia ecosystem. Therefore, in the GeneFinder package we dedicated some implementations to work with BioSequence types so that we can expand the functionality in an efficient way (see complete API).
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Markov Chains using BioSequences ​
We can now calculate a transition probability matrix from a LongDNA sequence using the transition_probability_matrix and initials methods for a given LongDNA sequence:
julia
using BioSequences, GeneFinder
-
-sequence = dna"CCTCCCGGACCCTGGGCTCGGGAC"
-
-tpm = transition_probability_matrix(sequence)
-initials = initials(sequence)
-
-println(tpm)
-println(initials)
4×4 Matrix{Float64}:
- 0.0   1.0       0.0       0.0
- 0.0   0.5       0.2       0.3
- 0.25  0.125     0.625     0.0
- 0.0   0.666667  0.333333  0.0
-
-4-element Vector{Float64}:
- 0.08695652173913043
- 0.43478260869565216
- 0.34782608695652173
- 0.13043478260869565
More conveniently, we can now use the transition_model method and obtain the transition probabilities and the initial distribution and build a transition model (BioMarkovChain):
julia
BioMarkovChain(sequence)
BioMarkovChain of DNAAlphabet{4}():
-  - Transition Probability Matrix --> Matrix{Float64}(4 × 4):
-   0.0     1.0     0.0     0.0
-   0.0     0.5     0.2     0.3
-   0.25    0.125   0.625   0.0
-   0.0     0.6667  0.3333  0.0
-  - Initial Probabilities -> Vector{Float64}(4 × 1):
-   0.087
-   0.4348
-   0.3478
-   0.1304
-  - Markov Chain Order -> Int64:
-   1
Note that, sometimes the dinucleotides transition do not harbor important biological meaning, whereas trinucleotides or codons are, in fact, the building block of proteins. Therefore, sometimes the transition model we want to build is usually a second-order Markov chain, that represents the possible transitions of a trinucleotide.
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julia
BioMarkovChain(sequence, 2)
BioMarkovChain of DNAAlphabet{4}():
-  - Transition Probability Matrix --> Matrix{Float64}(4 × 4):
-   0.0     0.5     0.2     0.3
-   0.05    0.475   0.325   0.15
-   0.1562  0.3906  0.4156  0.0375
-   0.0833  0.375   0.3417  0.2
-  - Initial Probabilities -> Vector{Float64}(4 × 1):
-   0.087
-   0.4348
-   0.3478
-   0.1304
-  - Markov Chain Order -> Int64:
-   2
References ​
Axelson-Fisk, Marina. 2015. Comparative Gene Finding. Vol. 20. Computational Biology. London: Springer London. http://link.springer.com/10.1007/978-1-4471-6693-1.
`,4);function y6(M6,b6,Z6,v6,V6,D6){return e(),Q("div",null,[o,t("p",null,[a("One important step towards many gene finding algorithms is to represent a DNA sequence as a Markov chain. In this representation a DNA sequence of a reduced alphabet "),t("mjx-container",n,[(e(),Q("svg",d,m)),i]),a(" is draw as a four-vertex graph, where each letter of "),t("mjx-container",h,[(e(),Q("svg",p,g)),x]),a(" is a "),w,a(" and the edges of the graph represent "),u,a(" from one nucleotide to another in a sequence (e.g. "),t("mjx-container",H,[(e(),Q("svg",_,L)),f]),a(" represent a single nucleotide to nucleotide transition). This is also considered more specifically as a Discrete Markov chain (Axelson-Fisk 2015). The complete set of transitions and states of a DNA sequence of alphabet "),t("mjx-container",y,[(e(),Q("svg",M,Z)),v]),a(".")]),t("p",null,[a("More formally a Markov chain is a random process where each state is a random variable "),t("mjx-container",V,[(e(),Q("svg",D,C)),A]),a(" where "),t("mjx-container",R,[(e(),Q("svg",S,E)),P]),a(" is a discrete time in a finite sequence "),t("mjx-container",B,[(e(),Q("svg",I,G)),N]),a(" and the probability to jump from one state into another is "),z,a(". Therefore a definition of this "),F,a(" is given by:")]),t("mjx-container",J,[(e(),Q("svg",q,W)),U]),t("p",null,[a("where "),t("mjx-container",K,[(e(),Q("svg",Y,a1)),Q1]),a(" . This property led us to generalize a way to calculate the probability of a sequence "),t("mjx-container",e1,[(e(),Q("svg",T1,l1)),o1]),a(" from a process "),t("mjx-container",n1,[(e(),Q("svg",d1,m1)),i1]),a(" where each random variable is a nucleotide from "),t("mjx-container",h1,[(e(),Q("svg",p1,g1)),x1]),a(" so that:")]),t("mjx-container",w1,[(e(),Q("svg",u1,_1)),k1]),t("p",null,[a("Note that previous equations has two terms, a initial probability "),t("mjx-container",L1,[(e(),Q("svg",f1,M1)),b1]),a(" and the the product of all transitions beginning at "),t("mjx-container",Z1,[(e(),Q("svg",v1,D1)),j1]),a(". So, to calculate the initial probability distribution of each of the nucleotides of a string "),t("mjx-container",C1,[(e(),Q("svg",A1,S1)),O1]),a(" with the alphabet 𝒜 we can first calculate the transition probability matrix "),t("mjx-container",E1,[(e(),Q("svg",P1,I1)),X1]),a(" out of the frequency count of the transitions. In an alphabet 𝒜 we got "),t("mjx-container",G1,[(e(),Q("svg",N1,F1)),J1]),a(" transitions of one order, that is the "),t("mjx-container",q1,[(e(),Q("svg",$1,U1)),K1]),a(" which coincides with the frequency of the dinucleotides in the sequence. So we can later in fact build a "),t("mjx-container",Y1,[(e(),Q("svg",t2,Q2)),e2]),a(" matrix representing all the transitions. For instance in a DNA sequence "),t("mjx-container",T2,[(e(),Q("svg",s2,o2)),n2]),a(" of 24 nucleotides:")]),t("mjx-container",d2,[(e(),Q("svg",r2,i2)),h2]),t("p",null,[a("We can calculate each frequency nucleotide to any other nucleotide "),t("mjx-container",p2,[(e(),Q("svg",c2,x2)),w2]),a(" where "),t("mjx-container",u2,[(e(),Q("svg",H2,k2)),L2]),a(" is the actual count of the dinucleotide, and therefore "),t("mjx-container",f2,[(e(),Q("svg",y2,b2)),Z2]),a(" is the counts of the nucleotide "),t("mjx-container",v2,[(e(),Q("svg",V2,j2)),C2]),a(" to any other nucleotide and build the transition probability matrix:")]),t("mjx-container",A2,[(e(),Q("svg",R2,O2)),E2]),t("p",null,[a("It is noteworthy that initial probabilities can also be obtained from the counts of each nucleotide transitions "),t("mjx-container",P2,[(e(),Q("svg",B2,X2)),G2]),a(" over the total sum of the dinucleotide counts "),t("mjx-container",N2,[(e(),Q("svg",z2,J2)),q2]),a(":")]),t("mjx-container",$2,[(e(),Q("svg",W2,K2)),Y2]),t("p",null,[a("That way for the previous example example we can can calculate the initial probabilities "),t("mjx-container",t6,[(e(),Q("svg",a6,e6)),T6]),a(". Both set of probabilities composed a "),s6,a(" that can be used to predict the probability of any DNA sequence using equation (2).")]),l6,t("p",null,[a("A very nice nice property of the transition probability matrix is that the "),o6,a(),t("mjx-container",n6,[(e(),Q("svg",d6,m6)),i6]),a(", that is the _n_th power of "),t("mjx-container",h6,[(e(),Q("svg",p6,g6)),x6]),a(" represents "),t("mjx-container",w6,[(e(),Q("svg",u6,_6)),k6]),a(" transitions in "),L6,a(" steps. We can also have higher order transition models as:")]),f6])}const A6=s(l,[["render",y6]]);export{C6 as __pageData,A6 as default};
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-*/function st(o,e){var t=Object.keys(o);if(Object.getOwnPropertySymbols){var r=Object.getOwnPropertySymbols(o);e&&(r=r.filter(function(n){return Object.getOwnPropertyDescriptor(o,n).enumerable})),t.push.apply(t,r)}return t}function ut(o){for(var e=1;e0){var r=e[e.length-1];r!==t&&r.pause()}var n=e.indexOf(t);n===-1||e.splice(n,1),e.push(t)},deactivateTrap:function(e,t){var r=e.indexOf(t);r!==-1&&e.splice(r,1),e.length>0&&e[e.length-1].unpause()}},Ar=function(e){return e.tagName&&e.tagName.toLowerCase()==="input"&&typeof e.select=="function"},Tr=function(e){return(e==null?void 0:e.key)==="Escape"||(e==null?void 0:e.key)==="Esc"||(e==null?void 0:e.keyCode)===27},ge=function(e){return(e==null?void 0:e.key)==="Tab"||(e==null?void 0:e.keyCode)===9},Nr=function(e){return ge(e)&&!e.shiftKey},Cr=function(e){return ge(e)&&e.shiftKey},ct=function(e){return setTimeout(e,0)},ft=function(e,t){var r=-1;return e.every(function(n,a){return t(n)?(r=a,!1):!0}),r},ye=function(e){for(var t=arguments.length,r=new Array(t>1?t-1:0),n=1;n1?p-1:0),I=1;I=0)c=r.activeElement;else{var f=i.tabbableGroups[0],p=f&&f.firstTabbableNode;c=p||h("fallbackFocus")}if(!c)throw new Error("Your focus-trap needs to have at least one focusable element");return c},v=function(){if(i.containerGroups=i.containers.map(function(c){var f=br(c,a.tabbableOptions),p=wr(c,a.tabbableOptions),C=f.length>0?f[0]:void 0,I=f.length>0?f[f.length-1]:void 0,M=p.find(function(m){return le(m)}),z=p.slice().reverse().find(function(m){return le(m)}),P=!!f.find(function(m){return se(m)>0});return{container:c,tabbableNodes:f,focusableNodes:p,posTabIndexesFound:P,firstTabbableNode:C,lastTabbableNode:I,firstDomTabbableNode:M,lastDomTabbableNode:z,nextTabbableNode:function(x){var $=arguments.length>1&&arguments[1]!==void 0?arguments[1]:!0,K=f.indexOf(x);return K<0?$?p.slice(p.indexOf(x)+1).find(function(q){return le(q)}):p.slice(0,p.indexOf(x)).reverse().find(function(q){return le(q)}):f[K+($?1:-1)]}}}),i.tabbableGroups=i.containerGroups.filter(function(c){return c.tabbableNodes.length>0}),i.tabbableGroups.length<=0&&!h("fallbackFocus"))throw new Error("Your focus-trap must have at least one container with at least one tabbable node in it at all times");if(i.containerGroups.find(function(c){return c.posTabIndexesFound})&&i.containerGroups.length>1)throw new Error("At least one node with a positive tabindex was found in one of your focus-trap's multiple containers. 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x=m===0?i.tabbableGroups.length-1:m-1,$=i.tabbableGroups[x];M=se(f)>=0?$.lastTabbableNode:$.lastDomTabbableNode}else ge(p)||(M=P.nextTabbableNode(f,!1))}else{var K=ft(i.tabbableGroups,function(V){var U=V.lastTabbableNode;return f===U});if(K<0&&(P.container===f||Re(f,a.tabbableOptions)&&!le(f,a.tabbableOptions)&&!P.nextTabbableNode(f))&&(K=z),K>=0){var q=K===i.tabbableGroups.length-1?0:K+1,H=i.tabbableGroups[q];M=se(f)>=0?H.firstTabbableNode:H.firstDomTabbableNode}else ge(p)||(M=P.nextTabbableNode(f))}}else M=h("fallbackFocus");return M},S=function(c){var f=Ae(c);if(!(l(f,c)>=0)){if(ye(a.clickOutsideDeactivates,c)){s.deactivate({returnFocus:a.returnFocusOnDeactivate});return}ye(a.allowOutsideClick,c)||c.preventDefault()}},T=function(c){var f=Ae(c),p=l(f,c)>=0;if(p||f instanceof Document)p&&(i.mostRecentlyFocusedNode=f);else{c.stopImmediatePropagation();var C,I=!0;if(i.mostRecentlyFocusedNode)if(se(i.mostRecentlyFocusedNode)>0){var M=l(i.mostRecentlyFocusedNode),z=i.containerGroups[M].tabbableNodes;if(z.length>0){var P=z.findIndex(function(m){return m===i.mostRecentlyFocusedNode});P>=0&&(a.isKeyForward(i.recentNavEvent)?P+1=0&&(C=z[P-1],I=!1))}}else i.containerGroups.some(function(m){return m.tabbableNodes.some(function(x){return se(x)>0})})||(I=!1);else I=!1;I&&(C=g({target:i.mostRecentlyFocusedNode,isBackward:a.isKeyBackward(i.recentNavEvent)})),b(C||i.mostRecentlyFocusedNode||d())}i.recentNavEvent=void 0},F=function(c){var f=arguments.length>1&&arguments[1]!==void 0?arguments[1]:!1;i.recentNavEvent=c;var p=g({event:c,isBackward:f});p&&(ge(c)&&c.preventDefault(),b(p))},L=function(c){if(Tr(c)&&ye(a.escapeDeactivates,c)!==!1){c.preventDefault(),s.deactivate();return}(a.isKeyForward(c)||a.isKeyBackward(c))&&F(c,a.isKeyBackward(c))},R=function(c){var f=Ae(c);l(f,c)>=0||ye(a.clickOutsideDeactivates,c)||ye(a.allowOutsideClick,c)||(c.preventDefault(),c.stopImmediatePropagation())},B=function(){if(i.active)return lt.activateTrap(n,s),i.delayInitialFocusTimer=a.delayInitialFocus?ct(function(){b(d())}):b(d()),r.addEventListener("focusin",T,!0),r.addEventListener("mousedown",S,{capture:!0,passive:!1}),r.addEventListener("touchstart",S,{capture:!0,passive:!1}),r.addEventListener("click",R,{capture:!0,passive:!1}),r.addEventListener("keydown",L,{capture:!0,passive:!1}),s},N=function(){if(i.active)return r.removeEventListener("focusin",T,!0),r.removeEventListener("mousedown",S,!0),r.removeEventListener("touchstart",S,!0),r.removeEventListener("click",R,!0),r.removeEventListener("keydown",L,!0),s},_=function(c){var f=c.some(function(p){var C=Array.from(p.removedNodes);return C.some(function(I){return I===i.mostRecentlyFocusedNode})});f&&b(d())},A=typeof window<"u"&&"MutationObserver"in window?new MutationObserver(_):void 0,O=function(){A&&(A.disconnect(),i.active&&!i.paused&&i.containers.map(function(c){A.observe(c,{subtree:!0,childList:!0})}))};return s={get active(){return i.active},get paused(){return i.paused},activate:function(c){if(i.active)return this;var f=u(c,"onActivate"),p=u(c,"onPostActivate"),C=u(c,"checkCanFocusTrap");C||v(),i.active=!0,i.paused=!1,i.nodeFocusedBeforeActivation=r.activeElement,f==null||f();var I=function(){C&&v(),B(),O(),p==null||p()};return C?(C(i.containers.concat()).then(I,I),this):(I(),this)},deactivate:function(c){if(!i.active)return this;var f=ut({onDeactivate:a.onDeactivate,onPostDeactivate:a.onPostDeactivate,checkCanReturnFocus:a.checkCanReturnFocus},c);clearTimeout(i.delayInitialFocusTimer),i.delayInitialFocusTimer=void 0,N(),i.active=!1,i.paused=!1,O(),lt.deactivateTrap(n,s);var p=u(f,"onDeactivate"),C=u(f,"onPostDeactivate"),I=u(f,"checkCanReturnFocus"),M=u(f,"returnFocus","returnFocusOnDeactivate");p==null||p();var z=function(){ct(function(){M&&b(E(i.nodeFocusedBeforeActivation)),C==null||C()})};return M&&I?(I(E(i.nodeFocusedBeforeActivation)).then(z,z),this):(z(),this)},pause:function(c){if(i.paused||!i.active)return this;var f=u(c,"onPause"),p=u(c,"onPostPause");return i.paused=!0,f==null||f(),N(),O(),p==null||p(),this},unpause:function(c){if(!i.paused||!i.active)return this;var f=u(c,"onUnpause"),p=u(c,"onPostUnpause");return i.paused=!1,f==null||f(),v(),B(),O(),p==null||p(),this},updateContainerElements:function(c){var f=[].concat(c).filter(Boolean);return i.containers=f.map(function(p){return typeof p=="string"?r.querySelector(p):p}),i.active&&v(),O(),this}},s.updateContainerElements(e),s};function kr(o,e={}){let t;const{immediate:r,...n}=e,a=oe(!1),i=oe(!1),s=d=>t&&t.activate(d),u=d=>t&&t.deactivate(d),l=()=>{t&&(t.pause(),i.value=!0)},h=()=>{t&&(t.unpause(),i.value=!1)};return $e(()=>kt(o),d=>{d&&(t=Dr(d,{...n,onActivate(){a.value=!0,e.onActivate&&e.onActivate()},onDeactivate(){a.value=!1,e.onDeactivate&&e.onDeactivate()}}),r&&s())},{flush:"post"}),Ot(()=>u()),{hasFocus:a,isPaused:i,activate:s,deactivate:u,pause:l,unpause:h}}class 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document with ID ".concat(this._documentIds.get(e),' has changed before removal: term "').concat(r,'" was not present in field "').concat(u,'". 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qr(16);zt(()=>[h.value,v.value,y.value],async([m,x,$],K,q)=>{var be,qe,He,Qe;(K==null?void 0:K[0])!==m&&F.clear();let H=!1;if(q(()=>{H=!0}),!m)return;g.value=m.search(x).slice(0,16),S.value=!0;const B=$?await Promise.all(g.value.map(Q=>L(Q.id))):[];if(H)return;for(const{id:Q,mod:ae}of B){const ne=Q.slice(0,Q.indexOf("#"));let te=F.get(ne);if(te)continue;te=new Map,F.set(ne,te);const X=ae.default??ae;if(X!=null&&X.render||X!=null&&X.setup){const ie=Yt(X);ie.config.warnHandler=()=>{},ie.provide(Zt,i),Object.defineProperties(ie.config.globalProperties,{$frontmatter:{get(){return i.frontmatter.value}},$params:{get(){return i.page.value.params}}});const Ye=document.createElement("div");ie.mount(Ye),Ye.querySelectorAll("h1, h2, h3, h4, h5, h6").forEach(he=>{var et;const we=(et=he.querySelector("a"))==null?void 0:et.getAttribute("href"),Ze=(we==null?void 0:we.startsWith("#"))&&we.slice(1);if(!Ze)return;let Xe="";for(;(he=he.nextElementSibling)&&!/^h[1-6]$/i.test(he.tagName);)Xe+=he.outerHTML;te.set(Ze,Xe)}),ie.unmount()}if(H)return}const U=new Set;if(g.value=g.value.map(Q=>{const[ae,ne]=Q.id.split("#"),te=F.get(ae),X=(te==null?void 0:te.get(ne))??"";for(const ie in Q.match)U.add(ie);return{...Q,text:X}}),await de(),H)return;await new Promise(Q=>{var ae;(ae=T.value)==null||ae.unmark({done:()=>{var ne;(ne=T.value)==null||ne.markRegExp(M(U),{done:Q})}})});const Z=((be=r.value)==null?void 0:be.querySelectorAll(".result .excerpt"))??[];for(const Q of Z)(qe=Q.querySelector('mark[data-markjs="true"]'))==null||qe.scrollIntoView({block:"center"});(Qe=(He=n.value)==null?void 0:He.firstElementChild)==null||Qe.scrollIntoView({block:"start"})},{debounce:200,immediate:!0});async function L(m){const x=Xt(m.slice(0,m.indexOf("#")));try{if(!x)throw new Error(`Cannot find file for id: ${m}`);return{id:m,mod:await yt(()=>import(x),[])}}catch($){return console.error($),{id:m,mod:{}}}}const R=oe(),V=Fe(()=>{var m;return((m=v.value)==null?void 0:m.length)<=0});function N(m=!0){var x,$;(x=R.value)==null||x.focus(),m&&(($=R.value)==null||$.select())}_e(()=>{N()});function _(m){m.pointerType==="mouse"&&N()}const A=oe(-1),O=oe(!1);$e(g,m=>{A.value=m.length?0:-1,w()});function w(){de(()=>{const m=document.querySelector(".result.selected");m&&m.scrollIntoView({block:"nearest"})})}Ee("ArrowUp",m=>{m.preventDefault(),A.value--,A.value<0&&(A.value=g.value.length-1),O.value=!0,w()}),Ee("ArrowDown",m=>{m.preventDefault(),A.value++,A.value>=g.value.length&&(A.value=0),O.value=!0,w()});const c=Pt();Ee("Enter",m=>{if(m.isComposing||m.target instanceof HTMLButtonElement&&m.target.type!=="submit")return;const x=g.value[A.value];if(m.target instanceof HTMLInputElement&&!x){m.preventDefault();return}x&&(c.go(x.id),t("close"))}),Ee("Escape",()=>{t("close")});const p=ar({modal:{displayDetails:"Display detailed list",resetButtonTitle:"Reset search",backButtonTitle:"Close search",noResultsText:"No results for",footer:{selectText:"to select",selectKeyAriaLabel:"enter",navigateText:"to navigate",navigateUpKeyAriaLabel:"up arrow",navigateDownKeyAriaLabel:"down arrow",closeText:"to close",closeKeyAriaLabel:"escape"}}});_e(()=>{window.history.pushState(null,"",null)}),Bt("popstate",m=>{m.preventDefault(),t("close")});const C=Vt($t?document.body:null);_e(()=>{de(()=>{C.value=!0,de().then(()=>s())})}),Wt(()=>{C.value=!1});function I(){v.value="",de().then(()=>N(!1))}function M(m){return new RegExp([...m].sort((x,$)=>$.length-x.length).map(x=>`(${er(x)})`).join("|"),"gi")}return(m,x)=>{var $,K,q,H;return ee(),Kt(qt,{to:"body"},[k("div",{ref_key:"el",ref:r,role:"button","aria-owns":($=g.value)!=null&&$.length?"localsearch-list":void 0,"aria-expanded":"true","aria-haspopup":"listbox","aria-labelledby":"localsearch-label",class:"VPLocalSearchBox"},[k("div",{class:"backdrop",onClick:x[0]||(x[0]=B=>m.$emit("close"))}),k("div",Qr,[k("form",{class:"search-bar",onPointerup:x[4]||(x[4]=B=>_(B)),onSubmit:x[5]||(x[5]=Jt(()=>{},["prevent"]))},[k("label",{title:E.value,id:"localsearch-label",for:"localsearch-input"},Xr,8,Yr),k("div",ea,[k("button",{class:"back-button",title:j(p)("modal.backButtonTitle"),onClick:x[1]||(x[1]=B=>m.$emit("close"))},aa,8,ta)]),Ut(k("input",{ref_key:"searchInput",ref:R,"onUpdate:modelValue":x[2]||(x[2]=B=>Gt(v)?v.value=B:null),placeholder:E.value,id:"localsearch-input","aria-labelledby":"localsearch-label",class:"search-input"},null,8,na),[[jt,j(v)]]),k("div",ia,[b.value?Se("",!0):(ee(),re("button",{key:0,class:rt(["toggle-layout-button",{"detailed-list":j(y)}]),type:"button",title:j(p)("modal.displayDetails"),onClick:x[3]||(x[3]=B=>A.value>-1&&(y.value=!j(y)))},ua,10,oa)),k("button",{class:"clear-button",type:"reset",disabled:V.value,title:j(p)("modal.resetButtonTitle"),onClick:I},fa,8,la)])],32),k("ul",{ref_key:"resultsEl",ref:n,id:(K=g.value)!=null&&K.length?"localsearch-list":void 0,role:(q=g.value)!=null&&q.length?"listbox":void 0,"aria-labelledby":(H=g.value)!=null&&H.length?"localsearch-label":void 0,class:"results",onMousemove:x[7]||(x[7]=B=>O.value=!1)},[(ee(!0),re(nt,null,at(g.value,(B,U)=>(ee(),re("li",{key:B.id,role:"option","aria-selected":A.value===U?"true":"false"},[k("a",{href:B.id,class:rt(["result",{selected:A.value===U}]),"aria-label":[...B.titles,B.title].join(" > "),onMouseenter:Z=>!O.value&&(A.value=U),onFocusin:Z=>A.value=U,onClick:x[6]||(x[6]=Z=>m.$emit("close"))},[k("div",null,[k("div",pa,[ya,(ee(!0),re(nt,null,at(B.titles,(Z,be)=>(ee(),re("span",{key:be,class:"title"},[k("span",{class:"text",innerHTML:Z},null,8,ma),ga]))),128)),k("span",ba,[k("span",{class:"text",innerHTML:B.title},null,8,wa)])]),j(y)?(ee(),re("div",xa,[B.text?(ee(),re("div",Fa,[k("div",{class:"vp-doc",innerHTML:B.text},null,8,Ea)])):Se("",!0),Sa,Aa])):Se("",!0)])],42,va)],8,da))),128)),j(v)&&!g.value.length&&S.value?(ee(),re("li",Ta,[ve(pe(j(p)("modal.noResultsText"))+' "',1),k("strong",null,pe(j(v)),1),ve('" ')])):Se("",!0)],40,ha),k("div",Na,[k("span",null,[k("kbd",{"aria-label":j(p)("modal.footer.navigateUpKeyAriaLabel")},Da,8,Ca),k("kbd",{"aria-label":j(p)("modal.footer.navigateDownKeyAriaLabel")},_a,8,ka),ve(" "+pe(j(p)("modal.footer.navigateText")),1)]),k("span",null,[k("kbd",{"aria-label":j(p)("modal.footer.selectKeyAriaLabel")},La,8,Ra),ve(" "+pe(j(p)("modal.footer.selectText")),1)]),k("span",null,[k("kbd",{"aria-label":j(p)("modal.footer.closeKeyAriaLabel")},"esc",8,za),ve(" "+pe(j(p)("modal.footer.closeText")),1)])])])],8,Hr)])}}}),Ja=tr(Pa,[["__scopeId","data-v-f5c68218"]]);export{Ja as default};
diff --git a/dev/assets/chunks/framework.DAOdO8p8.js b/dev/assets/chunks/framework.CVYz1rUz.js
similarity index 65%
rename from dev/assets/chunks/framework.DAOdO8p8.js
rename to dev/assets/chunks/framework.CVYz1rUz.js
index f0c6e7f..5628588 100644
--- a/dev/assets/chunks/framework.DAOdO8p8.js
+++ b/dev/assets/chunks/framework.CVYz1rUz.js
@@ -2,16 +2,16 @@
 * @vue/shared v3.4.21
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diff --git a/dev/assets/chunks/theme.Bo19b1i6.js b/dev/assets/chunks/theme.CUlsXZi2.js
similarity index 99%
rename from dev/assets/chunks/theme.Bo19b1i6.js
rename to dev/assets/chunks/theme.CUlsXZi2.js
index 0daf78f..0bd1b2a 100644
--- a/dev/assets/chunks/theme.Bo19b1i6.js
+++ b/dev/assets/chunks/theme.CUlsXZi2.js
@@ -1,7 +1,7 @@
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diff --git a/dev/assets/index.md.By85Oifm.js b/dev/assets/index.md.By85Oifm.js
deleted file mode 100644
index 7771382..0000000
--- a/dev/assets/index.md.By85Oifm.js
+++ /dev/null
@@ -1 +0,0 @@
-import{_ as e,c as a,o as t,a7 as o}from"./chunks/framework.DAOdO8p8.js";const _=JSON.parse('{"title":"BioMarkovChains","description":"","frontmatter":{},"headers":[],"relativePath":"index.md","filePath":"index.md","lastUpdated":null}'),i={name:"index.md"},n=o('
BioMarkovChains ​
Representing biological sequences as Markov chains
Overview ​
This package aim to represent BioSequences types as Markov chains to perform different operations and predictions
Installation ​
You can install BioMarkovChains from the julia REPL. Press ] to enter pkg mode, and enter the following:
add BioMarkovChains
If you are interested in the cutting edge of the development, please check out the master branch to try new features before release.
',9),s=[n];function r(l,c,d,h,p,v){return t(),a("div",null,s)}const k=e(i,[["render",r]]);export{_ as __pageData,k as default};
diff --git a/dev/assets/index.md.By85Oifm.lean.js b/dev/assets/index.md.By85Oifm.lean.js
deleted file mode 100644
index b81c281..0000000
--- a/dev/assets/index.md.By85Oifm.lean.js
+++ /dev/null
@@ -1 +0,0 @@
-import{_ as e,c as a,o as t,a7 as o}from"./chunks/framework.DAOdO8p8.js";const _=JSON.parse('{"title":"BioMarkovChains","description":"","frontmatter":{},"headers":[],"relativePath":"index.md","filePath":"index.md","lastUpdated":null}'),i={name:"index.md"},n=o("",9),s=[n];function r(l,c,d,h,p,v){return t(),a("div",null,s)}const k=e(i,[["render",r]]);export{_ as __pageData,k as default};
diff --git a/dev/assets/index.md.DAS101VW.js b/dev/assets/index.md.DAS101VW.js
new file mode 100644
index 0000000..33d46e4
--- /dev/null
+++ b/dev/assets/index.md.DAS101VW.js
@@ -0,0 +1,25 @@
+import{_ as i,c as e,m as a,a as t,a7 as s,o as n,a8 as o}from"./chunks/framework.CVYz1rUz.js";const P=JSON.parse('{"title":"BioMarkovChains","description":"","frontmatter":{},"headers":[],"relativePath":"index.md","filePath":"index.md","lastUpdated":null}'),l={name:"index.md"},r=s('
BioMarkovChains ​

Representing biological sequences as Markov chains
Overview ​
This package aim to represent BioSequences types as Markov chains to perform different operations and predictions
Installation ​
You can install BioMarkovChains from the julia REPL. Press ] to enter pkg mode, and enter the following:
add BioMarkovChains
If you are interested in the cutting edge of the development, please check out the master branch to try new features before release.
Create a BioMarkovChain of DNA ​
julia
using BioSequences, BioMarkovChains
+
+sequence = dna"CCTCCCGGACCCTGGGCTCGGGAC"
+
+BioMarkovChain(sequence)
BioMarkovChain of DNAAlphabet{4}() and order 1:
+  - Transition Probability Matrix --> Matrix{Float64}(4 × 4):
+   0.0     1.0     0.0     0.0
+   0.0     0.5     0.2     0.3
+   0.25    0.125   0.625   0.0
+   0.0     0.6667  0.3333  0.0
+  - Initial Probabilities -> Vector{Float64}(4 × 1):
+   0.087
+   0.4348
+   0.3478
+   0.1304
Note that, sometimes the dinucleotides transition do not harbor important biological meaning, whereas trinucleotides or codons are, in fact, the building block of proteins. Therefore, sometimes the transition model we want to build is usually a second-order Markov chain, that represents the possible transitions of a trinucleotide.
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julia
BioMarkovChain(sequence, 2)
BioMarkovChain of DNAAlphabet{4}() and order 2:
+  - Transition Probability Matrix --> Matrix{Float64}(4 × 4):
+   0.0     0.5     0.2     0.3
+   0.05    0.475   0.325   0.15
+   0.1562  0.3906  0.4156  0.0375
+   0.0833  0.375   0.3417  0.2
+  - Initial Probabilities -> Vector{Float64}(4 × 1):
+   0.087
+   0.4348
+   0.3478
+   0.1304
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new file mode 100644
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--- /dev/null
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diff --git a/dev/assets/style.CaReSTrk.css b/dev/assets/style.CaReSTrk.css
deleted file mode 100644
index 53e5ceb..0000000
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BioMarkovChains.jl
Appearance
DNA as a Markov chain ​
Several packages (e.g. MarkovChainsHammer.jl, DiscreteMarkovChains.jl, etc.) in the Julia ecosystem have been implemented to work with Markov chains with a state space of integers, those could be efficient in many ways, but they are clumsy to work with a specialized biological types as in the BioJulia ecosystem. Therefore, in the GeneFinder package we dedicated some implementations to work with BioSequence types so that we can expand the functionality in an efficient way (see complete API).
One important step towards many gene finding algorithms is to represent a DNA sequence as a Markov chain. In this representation a DNA sequence of a reduced alphabet A = {A, C, G, T} is draw as a four-vertex graph, where each letter of A is a state (vertex) and the edges of the graph represent transitions from one nucleotide to another in a sequence (e.g. A → T represent a single nucleotide to nucleotide transition). This is also considered more specifically as a Discrete Markov chain (Axelson-Fisk 2015). The complete set of transitions and states of a DNA sequence of alphabet A.
More formally a Markov chain is a random process where each state is a random variable Xt where t∈T is a discrete time in a finite sequence T and the probability to jump from one state into another is only dependent of the current state. Therefore a definition of this Markov property is given by:
P(Xt=j|Xt−1=i)
where i, j ∈ A . This property led us to generalize a way to calculate the probability of a sequence T from a process (X1...XT) where each random variable is a nucleotide from A so that:
P(X1=i1,...,XT=iT)=P(X1=i1)∏t=2TP(Xt=it|Xt−1=it−1)
Note that previous equations has two terms, a initial probability P(X1=i1) and the the product of all transitions beginning at t=2. So, to calculate the initial probability distribution of each of the nucleotides of a string T with the alphabet 𝒜 we can first calculate the transition probability matrix M^ out of the frequency count of the transitions. In an alphabet 𝒜 we got 42 transitions of one order, that is the AA, AC, AG, ... which coincides with the frequency of the dinucleotides in the sequence. So we can later in fact build a 4x4 matrix representing all the transitions. For instance in a DNA sequence T of 24 nucleotides:
CCTCCCGGACCCTGGGCTCGGGAC
We can calculate each frequency nucleotide to any other nucleotide m^ij=cijci where cij is the actual count of the dinucleotide, and therefore ci is the counts of the nucleotide i to any other nucleotide and build the transition probability matrix:
[ACGTA0.001.000.000.00C0.000.560.220.30G0.250.120.620.00T0.000.670.330.00]
It is noteworthy that initial probabilities can also be obtained from the counts of each nucleotide transitions cij over the total sum of the dinucleotide counts ck:
π^i=ci∑kck
That way for the previous example example we can can calculate the initial probabilities π^ = (0.08,0.43,0.34,0.13). Both set of probabilities composed a transition model that can be used to predict the probability of any DNA sequence using equation (2).
Markov Chains using BioSequences ​
We can now calculate a transition probability matrix from a LongDNA sequence using the transition_probability_matrix and initials methods for a given LongDNA sequence:
julia
using BioSequences, GeneFinder
-
-sequence = dna"CCTCCCGGACCCTGGGCTCGGGAC"
-
-tpm = transition_probability_matrix(sequence)
-initials = initials(sequence)
-
-println(tpm)
-println(initials)
4×4 Matrix{Float64}:
- 0.0   1.0       0.0       0.0
- 0.0   0.5       0.2       0.3
- 0.25  0.125     0.625     0.0
- 0.0   0.666667  0.333333  0.0
-
-4-element Vector{Float64}:
- 0.08695652173913043
- 0.43478260869565216
- 0.34782608695652173
- 0.13043478260869565
More conveniently, we can now use the transition_model method and obtain the transition probabilities and the initial distribution and build a transition model (BioMarkovChain):
julia
BioMarkovChain(sequence)
BioMarkovChain of DNAAlphabet{4}():
-  - Transition Probability Matrix --> Matrix{Float64}(4 × 4):
-   0.0     1.0     0.0     0.0
-   0.0     0.5     0.2     0.3
-   0.25    0.125   0.625   0.0
-   0.0     0.6667  0.3333  0.0
-  - Initial Probabilities -> Vector{Float64}(4 × 1):
-   0.087
-   0.4348
-   0.3478
-   0.1304
-  - Markov Chain Order -> Int64:
-   1
Note that, sometimes the dinucleotides transition do not harbor important biological meaning, whereas trinucleotides or codons are, in fact, the building block of proteins. Therefore, sometimes the transition model we want to build is usually a second-order Markov chain, that represents the possible transitions of a trinucleotide.
A very nice nice property of the transition probability matrix is that the n-step transition probability matrix Mn=(mij(n)), that is the _n_th power of M represents i →j transitions in n steps. We can also have higher order transition models as:
julia
BioMarkovChain(sequence, 2)
BioMarkovChain of DNAAlphabet{4}():
-  - Transition Probability Matrix --> Matrix{Float64}(4 × 4):
-   0.0     0.5     0.2     0.3
-   0.05    0.475   0.325   0.15
-   0.1562  0.3906  0.4156  0.0375
-   0.0833  0.375   0.3417  0.2
-  - Initial Probabilities -> Vector{Float64}(4 × 1):
-   0.087
-   0.4348
-   0.3478
-   0.1304
-  - Markov Chain Order -> Int64:
-   2
References ​
Axelson-Fisk, Marina. 2015. Comparative Gene Finding. Vol. 20. Computational Biology. London: Springer London. http://link.springer.com/10.1007/978-1-4471-6693-1.

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BioMarkovChains.jl
Appearance
DNA as a Markov chain ​
Several packages (e.g. MarkovChainsHammer.jl, DiscreteMarkovChains.jl, etc.) in the Julia ecosystem have been implemented to work with Markov chains with a state space of integers, those could be efficient in many ways, but they are clumsy to work with a specialized biological types as in the BioJulia ecosystem. Therefore, in the GeneFinder package we dedicated some implementations to work with BioSequence types so that we can expand the functionality in an efficient way (see complete API).
One important step towards many gene finding algorithms is to represent a DNA sequence as a Markov chain. In this representation a DNA sequence of a reduced alphabet A = {A, C, G, T} is draw as a four-vertex graph, where each letter of A is a state (vertex) and the edges of the graph represent transitions from one nucleotide to another in a sequence (e.g. A → T represent a single nucleotide to nucleotide transition). This is also considered more specifically as a Discrete Markov chain (Axelson-Fisk 2015). The complete set of transitions and states of a DNA sequence of alphabet A.
More formally a Markov chain is a random process where each state is a random variable Xt where t∈T is a discrete time in a finite sequence T and the probability to jump from one state into another is only dependent of the current state. Therefore a definition of this Markov property is given by:
$
\begin{align} P(X_{t} = j |X_{t−1} = i) \end{align} $
where i, j ∈ A . This property led us to generalize a way to calculate the probability of a sequence T from a process (X1...XT) where each random variable is a nucleotide from A so that:
P(X1=i1,...,XT=iT)=P(X1=i1)∏t=2TP(Xt=it|Xt−1=it−1)
Note that previous equations has two terms, a initial probability P(X1=i1) and the the product of all transitions beginning at t=2. So, to calculate the initial probability distribution of each of the nucleotides of a string T with the alphabet 𝒜 we can first calculate the transition probability matrix M^ out of the frequency count of the transitions. In an alphabet 𝒜 we got 42 transitions of one order, that is the AA, AC, AG, ... which coincides with the frequency of the dinucleotides in the sequence. So we can later in fact build a 4x4 matrix representing all the transitions. For instance in a DNA sequence T of 24 nucleotides:
CCTCCCGGACCCTGGGCTCGGGAC
We can calculate each frequency nucleotide to any other nucleotide m^ij=cijci where cij is the actual count of the dinucleotide, and therefore ci is the counts of the nucleotide i to any other nucleotide and build the transition probability matrix:
[ACGTA0.001.000.000.00C0.000.560.220.30G0.250.120.620.00T0.000.670.330.00]
It is noteworthy that initial probabilities can also be obtained from the counts of each nucleotide transitions cij over the total sum of the dinucleotide counts ck:
π^i=ci∑kck
That way for the previous example example we can can calculate the initial probabilities π^ = (0.08,0.43,0.34,0.13). Both set of probabilities composed a transition model that can be used to predict the probability of any DNA sequence using equation (2).
References ​
Axelson-Fisk, Marina. 2015. Comparative Gene Finding. Vol. 20. Computational Biology. London: Springer London. http://link.springer.com/10.1007/978-1-4471-6693-1.

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     BioMarkovChains | BioMarkovChains.jl
     
     
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BioMarkovChains.jl
Appearance
BioMarkovChains ​
Representing biological sequences as Markov chains
Overview ​
This package aim to represent BioSequences types as Markov chains to perform different operations and predictions
Installation ​
You can install BioMarkovChains from the julia REPL. Press ] to enter pkg mode, and enter the following:
add BioMarkovChains
If you are interested in the cutting edge of the development, please check out the master branch to try new features before release.

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BioMarkovChains.jl
Appearance
BioMarkovChains ​

Representing biological sequences as Markov chains
Overview ​
This package aim to represent BioSequences types as Markov chains to perform different operations and predictions
Installation ​
You can install BioMarkovChains from the julia REPL. Press ] to enter pkg mode, and enter the following:
add BioMarkovChains
If you are interested in the cutting edge of the development, please check out the master branch to try new features before release.
Create a BioMarkovChain of DNA ​
julia
using BioSequences, BioMarkovChains
+
+sequence = dna"CCTCCCGGACCCTGGGCTCGGGAC"
+
+BioMarkovChain(sequence)
BioMarkovChain of DNAAlphabet{4}() and order 1:
+  - Transition Probability Matrix --> Matrix{Float64}(4 × 4):
+   0.0     1.0     0.0     0.0
+   0.0     0.5     0.2     0.3
+   0.25    0.125   0.625   0.0
+   0.0     0.6667  0.3333  0.0
+  - Initial Probabilities -> Vector{Float64}(4 × 1):
+   0.087
+   0.4348
+   0.3478
+   0.1304
Note that, sometimes the dinucleotides transition do not harbor important biological meaning, whereas trinucleotides or codons are, in fact, the building block of proteins. Therefore, sometimes the transition model we want to build is usually a second-order Markov chain, that represents the possible transitions of a trinucleotide.
A very nice nice property of the transition probability matrix is that the n-step transition probability matrix Mn=(mij(n)), that is the _n_th power of M represents i →j transitions in n steps. We can also have higher order transition models as:
julia
BioMarkovChain(sequence, 2)
BioMarkovChain of DNAAlphabet{4}() and order 2:
+  - Transition Probability Matrix --> Matrix{Float64}(4 × 4):
+   0.0     0.5     0.2     0.3
+   0.05    0.475   0.325   0.15
+   0.1562  0.3906  0.4156  0.0375
+   0.0833  0.375   0.3417  0.2
+  - Initial Probabilities -> Vector{Float64}(4 × 1):
+   0.087
+   0.4348
+   0.3478
+   0.1304

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