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Estimation of kinetic parameters for cellular secretion from in vitro measurements

Alexander Stepanov
July 3, 2024

Introduction

Secretion in eukaryotic cells is a fundamental process involving regulated release of a substance out of the cell. Cell secretion investigated in vitro in response to stimuli can help us to uncover complex intracellular microenvironment in biosystems.

A typical in vitro secretion assay begins with seeding a number of cell with a culture medium according to a protocol. After a while the medium containing secreted products is collected to quantify the exact amount of the products.

Neglecting cell proliferation or death (C=const) during in vitro experiment, cell product secretion is determined by the following equation:

(1)dPdt=kC

In the cell secretion process, we also assume that the initial cell product concentration in media after stimulation is zero. Therefore, the initial condition is P=0 at t=0.

Integrating ODE (Equation (1)) we can obtain an equation for k:

(2)k=1Ct0Pt2Pt1t2t1

So, having experimental timeseries in vitro and initial cell concentration we can estimate secretion rate constant.
This rate constant is a black box hiding the effects of different invironmental factors including media substances. Under controlled conditions we can pull out of the black box some influencers.

Consider non-obligatory effector when in in vitro experiment secretion is occur without investigated effector but it just accelerate or inhibit baseline secretion.
Based on InSysBio's approach to describe multiple effects of cytokines on cell dynamics processes we can obtain two equations (3) and (4) for k for obligatory effector (kobl) or non-obligatory effector A (knob). In both equations effector A is affected by a modulator M.

(3)knob=kbase1+EmaxA1+EMAM/EC50M1+M/EC50MAEC50A/1+ECMAM/EC50M1+M/EC50M1+AEC50A/1+ECMAM/EC50M1+M/EC50M (4)kobl=kmaxA1+EMAMEC50M1+MEC50MAEC50A/1+ECMAM/EC50M1+M/EC50M1+AEC50A/1+ECMAM/EC50M1+M/EC50M

Above equations contain unknown parameters depending on experimental conditions. How can we estimate these parameters? Let’s analyse them.

Consider several experimental cases to estimate parameters.

1. Control (blank) in vitro experiment without an effector

An effector is absent, that is A=0 a modifier Math input errorM at the same time may be present in the system. Then,

(5)knobA0=kbase(6)koblA0=0

As expected, without an effector the product doesn’t secreted when an effector is obligatory but one is secreted with baseline rate constant when an effector is optional. Both cases are modulator-independent.

Remember, we assumed in this case constant cell concentration and absence of any effector. In this conditions we can estimate baseline secretion rate constant (for non-obligatory effectors):

(7)kbase=1Ct0Pt2A0Pt1A0t2t1

Recommendation: due to linear assumption we need two measurements in linear region of the experimental curve.

2. In vitro experiment with one effector and without any modulator

An effector is added in saturable concentration without addition of a modifier. In experiment saturable concentration means effect doesn’t change at effector concentration above saturation or even the effect can decrease. If effector is an activator, dose-response curve have often bell-like shape due to tixicity of huge concentration of the effector. But we consider only first half of bell-shaped curve (for activators) in phisiological range of concentrations. From mathematical point of view saturable concentration is great enough to apply the concept of “limit at infinity”. So, experimental conditions (3) and (4) in math mode we can denote: A and M=0. Then, firstly, rewrite the equations at condition M=0:

(8)knobM0=kbase1+EmaxAAEC50A1+AEC50A (9)koblM0=kmaxAAEC50A1+AEC50A

Second,

(10)limAknobM0=kbaseEmaxA (11)limAkoblM0=kmaxA

Finally,

(12)knobA,M0=kbaseEmaxA

(13)koblA,M0=kmaxA

Using Equation (2) and assumptions applied in this case, we can estimate kmaxA and EmaxA as follow (note, kbase is estimated in Equation (7)):

(14)EmaxA=1kbase1Ct0Pt2APt1At2t1 (15)kmaxA=1Ct0Pt2APt1At2t1

If required experimental data are absent we still able to roughly estimate parameter assume some simplification.

Simplification 1

If initial cell concentration is not provided, then kbase or kmaxA cannot be calculated but EmaxA does. Insert Equation (4) into Equation (14)

(16)EmaxA=Pt2APt1APt2A0Pt1A0t2A0t1A0t2At1A

Here we just need to know four experimental measurements without effector and at saturable effector cancentration.

Simplification 2

When the first time point measurement coincide with the beginning of the experiment t1=t0=0 when the product is absent and product measurements occur at the same time after the beginning of the experiment at different concentration of the effector t2A=t2A0. Then,

(17)EmaxA=PAPA0

3. In vitro experiment with one effector and one modulator

in this case both an effector and a modifier are added in saturable concentration. At first step let’s find constant at concentration of the effector in saturation:

(18)knobA=limAknob=kbaseEmaxA1+EMAMEC50M1+MEC50M (19)koblA=limAkobl=kmaxA1+EMAMEC50M1+MEC50M

Then,

(20)knobA,M=limMknobA=kbaseEmaxAEMA (21)koblA,M=limMkoblA=kmaxAEMA

Using Equation (2) and assumptions applied in this case, we can estimate EMA. For an obligatory effector:

(22)EMA=koblA,MkoblA,M0=Pt2A,MPt1A,MPt2A,M0Pt1A,M0

For a non-obligatory effector:

(23)EMA=knobA,MkbaseEmaxA=Pt2A,MPt1A,MPt2A,M0Pt1A,M0

EMA for both types of effectors are equal.

Simplification 1

When the first time point measurement coincide with the beginning of the experiment t1=t0=0 when the product is absent and product measurements occur at the same time after the beginning of the experiment at different concentration of the effector t2A=t2A0. Then,

(24)EMA=Pt2A,MPt2A,M0

4. In vitro experiment at half the saturation concentration of an effector without modulator

In fourth case an effector is added at concentration caused a half of effect (A=A50) without addition of a modifier (M=0). Definition: A50 is an effector concentration corresponding to its half-maximal effect on reaction corrected to baseline. In other words, at A50 secretion rate is a half of that at saturable effector concentration. Or, in mathematical notation, VA50=1/2(VAVA0) at any time point. In these experimental conditions

(25)knobM0(A50)=kbase1+EmaxAA50EC50A1+A50EC50A (26)koblM0(A50)=kmaxAA50EC50A1+A50EC50A

On the other hand,

(27)V(A50)=12(V(A)+V(A0))

(28)2k(A50)C=k(A)C+k(A0)C

(29)2k(A50)=k(A)+k(A0)

For an obligatory effector Equation (29) takes the form:

(30)2kmaxAA50EC50A1+A50EC50A=kmaxA+0

(31)2A50EC50A=1+A50EC50A

(32)EC50A=A50

For a non-obligatory effector Equation (29) takes the form:

(33)2kbase1+EmaxAA50EC50A1+A50EC50A=kbaseEmaxA+kbase

(34)2(1+EmaxAA50EC50A)=(EmaxA+1)(1+A50EC50A)

(35)A50EC50A(EmaxA1)=EmaxA1

(36)EC50A=A50

So, for both cases EC50A=A50.

5. In vitro experiment at half the saturation concentration of a modulator and saturable concentration of an effector

In fifth case a modifier is added at concentration caused a half of effect (M=M50) and an effector is added at saturable concentration (A). This case and derivation are similar to previous fourth case. First, consider saturable effector concentration Equations (18) and (19):

(37)knobA=limAknob=kbaseEmaxA1+EMAMEC50M1+MEC50M (38)koblA=limAkobl=kmaxA1+EMAMEC50M1+MEC50M

Now, rewrite these equations for M50:

(39)knobA(M50)=kbaseEmaxA1+EMAM50EC50M1+M50EC50M (40)koblA(M50)=kmaxA1+EMAM50EC50M1+M50EC50M

On the other hand, by definition

(41)VA(M50)=12(VA(M)+VA(M0))

(42)2kA(M50)C=kA(M)C+kA(M0)C

(43)2kA(M50)=kA(M)+kA(M0)

For an obligatory effector Equation (43) takes the form (using Equations (12),(20),(39)):

(44)2kmaxA1+EMAM50EC50M1+M50EC50M=kmaxAEMA+kmaxA

(45)2(1+EMAM50EC50M)=(EMA+1)(1+M50EC50M)

(46)M50EC50M(EMA1)=EMA1

(47)EC50M=M50

For a non-obligatory effector Equation (43) takes the form (using Equations (13),(21),(40)):

(48)2kbaseEmaxA1+EMAM50EC50M1+M50EC50M=kbaseEmaxAEMA+kbaseEmaxA

(49)2+2EMAM50EC50M=(EMA+1)(1+M50EC50M)

(50)M50EC50M(EMA1)=EMA1

(51)EC50M=M50

What if the effector is given at not saturable concentration? Consider a case number 6.

6. In vitro experiment at half the saturation concentration of a modulator and arbitrary concentration of an effector

Rewrite the system of Equation (3) and (4) for M50:

(52)knob(M50)=kbase1+EmaxA1+EMAM50/EC50M1+M50/EC50MAEC50A/1+ECMAM50/EC50M1+M50/EC50M1+AEC50A/1+ECMAM50/EC50M1+M50/EC50M

(53)kobl(M50)=kmaxA1+EMAM50EC50M1+M50EC50MAEC50A/1+ECMAM50/EC50M1+M50/EC50M1+AEC50A/1+ECMAM50/EC50M1+M50/EC50M

On the other hand, by definition

(54)V(M50)=12(V(M)+V(M0))

(55)2k(M50)C=k(M)C+k(M0)C

(56)2k(M50)=k(M)+k(M0)

(57)knobA,M=limMknobA=kbaseEmaxAEMAAEC50A+EC50MAEC50A+EC50M

(58)koblA,M=limMkoblA=kmaxAEMAAECMAEC50A+A

Define new variables: A=A/EC50A and M=M/EC50M.

For an obligatory effector

kbase1+EmaxA1+EMAM50/EC50M1+M50/EC50MAEC50A/1+ECMAM50/EC50M1+M50/EC50M1+AEC50A/1+ECMAM50/EC50M1+M50/EC50M=kbaseEmaxAEMAAEC50A+EC50MAEC50A+EC50M+kbase1+EmaxAAEC50A1+AEC50A

1+EmaxA1+EMAM50/EC50M1+M50/EC50MAEC50A/1+ECMAM50/EC50M1+M50/EC50M1+AEC50A/1+ECMAM50/EC50M1+M50/EC50M=EmaxAEMAAEC50A+EC50MAEC50A+EC50M+1+EmaxAAEC50A1+AEC50A

21+EmaxA1+EMAM1+MA/1+ECMAM1+M1+A/1+ECMAM1+M=EmaxAEMAA+EC50MA+EC50M+1+EmaxAA1+A

2(1+EmaxAA1+EMAM1+ECMAM)(1+ECMAM)1+A+M(A+ECMA)=(1+A)(EmaxAEMAA+EC50M)+(1+EmaxAA)(A+EC50M)(A+EC50M)(1+A)

21+EMAA+M(ECMA+AEmaxAEMA)1+A+M(A+ECMA)=(1+A)(EmaxAEMAA+EC50M)+(1+EmaxAA)(A+EC50M)(A+EC50M)(1+A)

Finally,

(59)EC50M=M50ECMA+AEC50A1+AEC50A

How one can check the Equation (59)?

limAEC50M=M50

which is the same as obtained in case 5.

Conclusion

So, using time- and concentration-dependences obtained in in vitro experiments we can estimate the following parameters of secretion rate constant:

(60)kbase=1Ct0Pt2A0Pt1A0t2t1

(61)kmaxA=1Ct0Pt2APt1At2t1

(62)EmaxA=1kbase1Ct0Pt2APt1At2t1

(63)EMA=Pt2A,MPt1A,MPt2A,M0Pt1A,M0

(64)EC50A=A50

(65)EC50M=M50

(66)EC50M=M50ECMA+AEC50A1+AEC50A

These equations with predefined the first experimental point ( Pt1=0 at t1=0 ) are implemented in fIVE DB.

Follow us to know how estimate parameters for other cell processes from in vitro experiments…

 

 

 

 

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