# Does the magnitude of gene-expression changes decrease the more downstream a gene is from the origin of change?

If I have a decrease or increase in expression in one gene, will the decrease/increase in expression in the downstream genes always be of a magnitude lower than the previous ones, or can they be higher?

I’m thinking in terms of signal propagation in a network. If I, for example, knockdown gene $$X$$ to 40% of its basal level, and it influences genes $$Y$$ and $$Z$$ in the following manner:

$$X → Y → Z,$$

Will I expect genes $$Y$$ and $$Z$$ to suffer lower decreases/increases than the previous gene, for example:

$$X: 40\% ~⇒~ Y: 60\% ~⇒~ Z: 80\%$$

Or could I have a scenario where a small change in one gene’s expression can lead to higher magnitude changes in downstream ones, for example:

$$X: 40\% ~⇒~ Y: 20\% ~⇒~ Z: 10\%$$

I suppose that in some networks the signal will always decrease in magnitude the farther it is from the source, but I’m wondering if, in a biological network, that will also happen and consistently in each step. I read papers with knockdowns all the time, but I never stopped to analyse if knockdowns always create lower magnitude changes in downstream genes or not. So I was wondering if this intuitive notion holds empirically or not.

• It's going to vary depending on the gene. Some genes exhibit dose dependence, others don't. Nov 10 '20 at 20:09
• @MattDMo I know some genes are dose dependent and some not, but I'm not sure that's the problem. Dose dependence would change downstream genes in acordance with the dose of the source of change, but that doesn't tell me if the magnitude will always be lower, or not. So if 40%X -> 60%Y and 20%X -> 40%Y, that's dose dependent, but will the magnitude always be lower like in this example? Nov 10 '20 at 20:24
• @David well, I work with molecular biology and bioinformatics. And I don't see how that is making a simplistic mathematical assumption about biology. Biology is not simple, and cannot be reduced to simple assumptions sometimes. But other times it demonstrates very well conserved patterns. I'm making no assumption. I'm merely trying to discuss to see if there is a pattern here. It might not have. But if one cannot make very objective questions about whether some pattern exists or not in biological systems, then we should close all systems biology departments, because they would be pointless. Nov 10 '20 at 20:30

Gene expression can be very well described by Hill functions, i.e.:

$$c_Y(c_X) = \frac{c_X^n}{1+c_X^n}$$

when $$X$$ activates $$Y$$ and

$$c_Y(c_X) = \frac{1}{1+c_X^n}$$

when $$X$$ represses $$Y$$ (omitting units and all sorts of constants for simplicity). For the common case that $$n>1$$, these functions look like this: As you can see, they are far from linear, but sigmoids. For $$n→∞$$, these become perfect step functions.

As a result, gene regulation mostly works like a switch: As long as you are not close to the critical concentration ($$1$$ in the above example), changes in the regulator have a less than proportional effect. For instance in the above example, it doesn’t matter whether the concentration of your activator is $$2$$ or $$10$$, the protein production is pretty much $$1$$.

Therefore, “signals” in gene-regulation networks get stabilised instead of diluted. There are also some other self-regulation effects contributing to this such as genes that self-regulate, the availability of amino acids, and the limits of the translation apparatus.

Note that without these sigmoidal relationships, you would also have big influences of the inevitable fluctuations of the regulator. As a result, it would be impossible to keep the metabolism in some stable state – the only stable state in a linear system is death. Simply put, you would have to keep a very precise and regular diet, lest your body stops working.

• Great answer, and exactly the kind of answer I was looking for. I remember reading about the Hill equations in Uri Alon's book many years ago in the begining of my masters in biochemistry. But I had other things in mind and ended up not finishing it, and forgot most, except the network patterns. Guess I should go back to it now. But just some questions: do we know the range and frequency of Hill coefficients in nature? Because it is going to be linear somewhere around n=1.5, right? But I guess the question is whether n=1.5 is a value that exists in practice. Nov 11 '20 at 12:58
• Also, this switch behavior does make sense in terms of stabilizing the signal, but we can clearly observe some dose dependency when inhibiting genes in the lab. So, at the very least, it's not a binary switch, but one with many states (or steps), unless all the dose dependency we are observing is in the linear phase, in which case the functions are far from perfect steps. Is there some literature on that? And third, just stretching a bit, is there some general pattern of how much the expression changes after the "switch" is on? Nov 11 '20 at 12:58
• @LizardMan: 1) I would have to search for a source on the distribution of Hill coefficients myself, but IIRC they are often considerably larger than 1. Of course there also are some cases, where you (or evolution) want the Hill coefficient to be small. 2) Sure, it’s not a binary switch, but for many practical purposes, it’s sufficiently close. Remember that in any cascade, it suffices if you are not in the intermediate region for one expression step. 3) I am not sure, what you wish to know here. The general pattern is that you have no expression if the switch is on and otherwise, it depends. Nov 11 '20 at 13:24
• 1) Ok, I'll try to look for this later. Thanks. 2) I meant binary in the sense of having only one step (two states), but I get it anyway. Just out of curiosity, are you aware of any gene for which all expression steps have been characterized? Something like, gene X goes through one step when it's TF is at concentration A, through another step at concentration B, and so on. Nov 11 '20 at 14:21
• 3) I meant something like this: gene X has a basal level (will not be zero for many); it's TF rises sufficiently to make it go through one step; after the step, X's expression is N fold in relation to it's basal level. Is N some kind of constant (either for each gene per step, or in general for all genes)? Nov 11 '20 at 14:21

Just to expand on @Wrzlprmft's great answer with a concrete example:

Sticking to your simple gene circuit with $$X$$, $$Y$$, and $$Z$$, now consider the possibility that $$Y$$ also activates its own expression via a positive feedback loop:

$$X \to \underset{\circlearrowright}Y \to Z$$

Here activation of $$Y$$ will be only minimally dependent on $$X$$. Once $$Y$$ is activated, the following self-activation makes $$Y$$ practically independent of further $$X$$ activity, causing $$Y$$ to drive its own activation until saturation.

If by decreasing $$X$$ to 50%, its levels are still above the minimum level required for activating $$Y$$, this would have practically no effect on the final (steady state) levels of $$Y$$. Conversely, if a 50% decrease brings $$X$$ below the $$Y$$ activation level, it would lead to a complete shutdown of $$Y$$.

• Note that you need an additional ingredient for a positive feedback loop to act as a robust bistable switch which isn’t switched on by the slightest noise: Either $Y$’s self activation doesn’t suffice for a self-amplifying effect at low concentrations of $Y$ or you have something like “$Y$ suppresses $W$ suppresses $Y$”. Nov 11 '20 at 14:54
• Indeed, good clarification! I was thinking of giving some examples of positive feedback in biology (cell cycle and lac operon, for example) where such "protective mechanisms" indeed exist, but didn't want to go too off-topic. Nov 11 '20 at 14:59